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0 depending on X and D such that for every X0 8D and 0 < < R0 one has
iv)
NT ax
ax
a
da. aa: r E
iv)
LP balanced-degeneracy
As we have previously observed surface measure becomes singular near a char acteristic point. On the other hand, the angle function W vanishes, thus balancing the singularities of ADPx-domains we obtain the following two re For sults which respectively establish the mutual absolute continuity of £-harmonic and surface measure and the solvability of the Dirichlet problem with data in D
LP(8 da). ,
a. a da,
LUCA CAPOGNA, NICOLA GAROFALO, AND DUY-MINH NHIEU THEOREM 1.5. Let D C !R.n be a 0' - ADPx domain. Fix x 1 E D . For every p0' ADPx 1 there exist positive constants C, R 1 , depending on p, M, Ra, x 1 , and on the - parameters, such that for every y E oD and 0 < r < R1 one has 56
>
Moreover, the measures dwx and dO' are mutually absolutely continuous. We mention explicitly that a basic consequence of Theorem 1.5 is that the standard surface measure on the boundary of a 0' - ADPx domain is doubling. D JR.n be a 0' - ADPx domain. For every p 1 there existsTHEOREM a constant1 .6C. Let0 depending on D, X and p such that if f E LP (oD, dO'), then Hf (x) { f (y) P(x , y) dO' (y ) , lew and II Na (Hf ) II LP(8D,da) :S C l l f i i LP (8D,da) · Furthermore, Hf converges nontangentially 0'-a.e. to f on oD. C
>
>
=
Concerning Theorems 1 .3, 1 .4, 1.5 and 1 .6 we mention that large classes of domains to which they apply were found in [CGN2], but one should also see [LUl] for domains satisfying assumption in Definition 1 . 1 . The discussion of these examples is taken up in section 9 . In closing we briefly describe the organization o f the paper. In section we collect some known results on Carnot-Caratheodory metrics which are needed in the paper. In section 3 we discuss some known results on the subelliptic Dirichlet problem which constitute the potential theoretic backbone of the paper. In section we discuss Jerison's mentioned example. Section 5 is devoted to proving some new interior a priori estimates of Cauchy Schauder type. Such estimates are obtained by means of a family of subelliptic mollifiers which were introduced by Danielli and two of us in [CDGl], see also [CDG2] . The main results are Theorems 5 . 1 , 5.5, and Corollary 5.3. We feel that, besides being instrumental to the present paper, these results will prove quite useful in future research on the subject. In section 6 we use the interior estimates in Theorem 5 . 1 to prove that if a domain satisfies a uniform outer tangent X-ball condition, then the horizontal gra dient of the Green function is bounded up to the boundary, hence, in particular, near I;, see Theorem 6.6. The proof of such result rests in an essential way on the linear growth estimate provided by Theorem 6.3. Another crucial ingredient is Lemma 6.1 which allows a delicate control of some ad-hoc subelliptic barriers. In the final part of the section we show that, by requesting the uniform outer X-ball condition only in a neighborhood of the characteristic set I;, we are still able to obtain the boundedness of the horizontal gradient of up to the characteristic set, although we now loose the uniformity in the estimates, see Theorem 6.9, 6.10 and Corollary 6 . 1 1 . In section we establish a Poisson type representation formula for domains which satisfy the uniform outer X-ball condition in a neighborhood of the charac teristic set. This result generalizes a similar Poisson type formula in the Heisenberg group JH[n obtained by Lanconelli and Uguzzoni in [LUl] , and extended in [CGN2]
iii)
2
4
G
G
7
MUTUAL ABSOLUTE CONTINUITY to Carnot groups of Heisenberg type. If generically the Green function of a smooth domain had bounded horizontal gradient up to the characteristic set, then such 57
Poisson formula would follow in an elementary way from integration by parts. As we previously stressed, however, things are not so simple and the boundedness of XC fails in general near the characteristic set. However, when D C Rn satisfies the uniform outer X-ball condition in a neighborhood of the characteristic set, then combining Theorem 6.6 with the estimate K(x, y) :::; IXG (x, y) l
'
X E D, y E aD ,
see (7.7) , we prove the boundedness of the Poisson kernel y -+ K (x, y) on aD. The main result in section 7 is Theorem 7.10. This representation formula with the estimates of the Green function in sections and 6 lead to a priori estimates in for the solution to (1 .3) when the datum ¢ E C(aD). Solvability of ( 1 . 3 ) with data in Lebesgue classes requires, however, a much deeper analysis. The first observation is that the outer ball condition alone does not guarantee the development of a rich potential theory. For instance, it may not be possible to find: a) Good nontangential regions of approach to the boundary from within the domain; b) Appropriate interior Harnack chains of nontangential balls. This is where the basic results on NTAx domains from [CGl] enter the picture. In the opening of section 8 we recall the definition of NTAx-domain along with those results from [CGl] which constitute the foundations of the present study. Using these results we establish Theorem 8.9. The remaining part of the section is devoted to proving Theorems 1.3, 1 .4, and 1 .6. Finally, section 9 is devoted to the discussion of examples of ADPx and a ADPx domains and of some open problems.
5
LP
1.5
2. Preliminaries
n�
In Rn , with 3, we consider a system X = {X1 , . . . , Xm } of coo vector fields satisfying Hi::irmander's finite rank condition ( 1 . 1 ) . A piecewise C 1 curve [0 , T] -+ Rn is called [FP] if whenever exists one has for every � E ]Rn
'Y :
sub-unitary
'Y'(t)
m
< 'Y' (t) , � >2 :::; j=l L < Xj ('Y (t) ) , � > 2 . We note explicitly that the above inequality forces "(1 ( t) to belong to the span of {X1 ( . . . , Xm ( ( ) ) } . The sub-unit length of is by definition 8 ( ) = T. l 'Y 'Y(t)), 'Y t 'Y Given x, y E !Rn , denote by Sn (x, y) the collection of all sub-unitary 'Y : [O, T] -+ n which join x to y . The accessibility theorem of Chow and Rashevsky, [RaJ , [Ch] , states that, given a connected open set r2 c !Rn , for every x, y E n there exists 'Y E Sn (x, y) . As a consequence, if we pose dn ( x, y)
=
inf { l s ( 'Y ) I 'Y E Sn ( x, y) } ,
Carnot-Carathiodory distance dJRn (x,on y).n, associated It is clear
we obtain a distance on n, called the with the system X . When n = Rn , we write d(x, y) instead of
LUCA CAPOGNA, NICOLA GAROFALO, AND DUY-MINH NHIEU that d(x, y) ::::; dn(x, y), x, y E n, for every connected open set n Rn. In [NSWJ it was proved that for every connected n Rn there exist C, € > 0 such that (2.1) C lx - y l :::=; dn(x, y) ::::; C- 1 Ix - yl • , x,y E O. This gives d(x,y) ::::; c - 1 l x - yl • , x,y E 0, and therefore is continuous. 58
c
CC
It is easy to see that also the continuity of the opposite inclusion holds [GNl] , hence the metric and the Euclidean topology are compatible. For and > we let < } . The basic properties of these balls were established by Nagel, Stein and Wainger in their seminal paper [NSW] . Denote by Y1 , . . , Yi the collection of the Xj 's and of those commutators which are needed to generate A formal "degree" is assigned to each namely the corresponding order of the commutator. If is a n-tuple of integers, following [NSWJ we let ) and = 2:7= 1 = det ( Yi, , . . . , Yi J . The is defined by
x E Rn r 0,
Bd(x, r) {y E Rn I d(x, y) r . Rn. ::::; }i, I = (i1, ... , i ), 1 n ::::; l d( deg(}i1 , ij J) Nagel-Stein-Wainger polynomial a1(x) A(x,r) L l ai (x) l rd(I), (2.2) r > 0. For a given bounded open set U Rn, we let (2.3) Q sup {d(I) I la i (x) l =J 0, x E U} , Q(x) inf {d(J) l lai (x) l =J 0}, and notice that ::::; Q(x) ::::; Q. The numbers Q and Q(x) are respectively called the local homogeneous dimension of U and the homogeneous dimension at x with =
=
I
c
=
=
n
respect to the system X .
C
THEOREM 2.1 ( [NSW] ) . For every bounded set U Rn, there exist constants C, Ro > 0 such that, for any x E U, and 0 < r ::::; R0, (2.4) c A(x, r) ::::; I Bd(x, r) l ::::; c - l A(x, r). As a consequence, one has with C1 2Q (2.5) for every x E U and 0 < r :::=; R0• The numbers C1 , Ro in (2. 5 ) will be referred to as the characteristic local pa rameters of U. Because of (2.2), if we let E(x,r) = A(xr� r) , (2.6) then the function r ----. E(x, r) is strictly increasing. We denote by F(x, ) the inverse function of E(x, ) , so that F(x, E(x, r)) r. Let r(x, y) = r(y, x) be the positive fundamental solution of the sub-Laplacian =
·
·
=
.c
and consider its level sets
=
m
l: x;xj , j=l
O(x, r) The following definition plays a key role in this paper.
MUTUAL ABSOLUTE CONTINUITY DEFINITION 2.2. For every x E JRn, and r > 0, the set B (x , r) { y E JRn I r (x, y) > E(�, rJ will be called the X -ball, centered at x with radius r.
59
=
We note explicitly that
B(x,r) D.(x,E(x,r)),
and that
=
D.(x, r) = B (x, F(x, r)) .
One of the main geometric properties of the X-balls, is that they are equiv alent to the Carnot-Caratheodory balls. To see this, we recall the following im portant result, established independently in [NSW] , [SC] . Hereafter, the no tation Xu = (X1 u , ... , Xmu) indicates the sub-gradient of a function u , whereas IXu! = (2::;: 1 (Xj u ) 2 ) ! will denote its length.
THEOREM 2.3. Given a bounded set U JRn, there exists R0, depending on U and X, such that for x E U, 0 < d(x,y) s; R0, one has for s E N {0}, and for someonconstant C C(U, X , s) > 0 d(x, y) 2- s IX11 Xh ··· X1sr (x , y) l (2.7) c - 1 I Bd (x, d(x, y) ) ' l 2 d(x,y) r(x , y) � C I Bd(x, d(x , y ) ) l Inallowed the first to actinequality on eitherinx(2.or7),y. one has ]i E {1, ... , m} for i 1 , .. . , s, and X}i is C
U
=
::;_
=
(2. 5 ), a(2.>7),1 , itdepending is now easy to recognize that, given a bounded set such that on U and
I n view of UC there exists
JRn ,
X,
(2.8) for x E U, 0 < r s; R0 • We observe that, as a consequence of (2. 4 ), and of (2.7), one has 1 ) (2. 9) C d(x, y) s; F (x , s; C - 1 d(x , y), x r( , y) for all x E U, 0 < d(x, y) s; R0• We observe that for a Carnot group G of step k, if Vk is a V1 stratification of the Lie algebra of G, then one has A(x,r) = canst r Q , for every x E G and every r > 0, with Q 2::�=1 j dimVj , the homogeneous dimension of the group G. In this case Q ( x ) Q. In the sequel the following properties of a Carnot-Caratheodory space will be g =
EB . . . EB
=
=
useful.
PROPOSITION 2.4. (JR", d) is locally compact. Furthermore, for any bounded set U C lRn there exists Ro R0(U) > 0 such that the closed balls B(x0, R) , with X0 E U and 0 < R < R0, are compact. =
LUCA CAPOGNA, NICOLA GAROFALO, AND DUY-lv!INH NHIEU REMARK 2.5. Compactness of balls of large radii may fail in general, see [GNl] . However, there are important cases in which Proposition 2.4 holds glob in the sense that one can take U to coincide with the whole ambient space and = oo . One example is that of Carnot groups. Another interesting case is that Raally, when the vector fields Xj have coefficients which are globally Lipschitz, see [GNl] , [GN2] . Henceforth, for any given bounded set U C JRn we will always assume that the local parameter Ra has been chosen so to accommodate Proposition 2.4. 60
3. The Dirichlet problem
X = {X11 , ..p, X::;moo} ofwec=denote vector fields in !Rn (1.1),{f LP(D) X fD CLP(D),j by £1· P (D) the 1, ... , m} endowed with its natural m = + l lfl l c1·P(D) l lf i i LP (D) L 1 1 Xj fi i £P (D ) j =l The local space Cz��(D) has the usual meaning, whereas for 1 p < oo the space £6•P (D) is defined as the closure of C0(D) in the norm of £ 1 ·P (D). A function u Cz��(D) is called harmonic in D if for any ¢; E C0(D) one has jD f xjuXj ¢; dx = 0 , j =l i.e., a harmonic function is a weak solution to the equation Cu = 2:;':, 1 XJ Xj u = 0. By Hi:irmander's hypoellipticity theorem [H] , if u is harmonic in D, then u lRn , and a function ¢; c=(D). Given a bounded open set D £1 •2 (D), the Dirichlet problem consists in finding u cf� �(D) such that { uCu =¢; 0E £6'2 (D)in .D (3. 1 ) By adapting classical arguments, see for instance [GT] , one can show that there exists a unique solution u £ 1 • 2 (D) to (3.1). If we assume, in addition, that ¢; C(D), in general we cannot say that the function u takes up the boundary value ¢; with continuity. A Wiener type criterion for sub-Laplacians was proved in [NS]. Subsequently, using the Wiener series in [NS] , Citti obtained in [Ci] an estimate of the modulus of continuity at the boundary of the solution of (3.1). In [D] an integral Wiener type estimate at the boundary was established for a general class In what follows, given a system satisfying and an open set JRn , for Banach space E I j E norm
.
::;
=
·
::;
E
E
c
E
E
-
E
E
of quasilinear equations having p-growth in the sub-gradient. Since such estimate is particularly convenient for the applications, we next state it for the special case of linear equations.
p=2 THEOREM 3.1. Let ¢; E £1• 2 (D) n C(D). Consider the solution u to (3.1). There = C(X) > 0, and Ra = Ra(D, X) > 0, such that given X0 8D, and 0 < r <existR 0 such that for each ball B ( x, ar) C D one has sup u C inf u. (3.4) and Using such Harnack principle one sees that for any x, y E D, the measures are mutually absolutely continuous. For the basic properties of the harmonic :S:
dw x
B(x,r)
:::;
B ( x ,r)
dw x
dwY
measure we refer the reader to the paper [CGl] . Here, it is important to recall that, thanks to the results in [B] , [CGl] , the following result of Brelot type holds.
1 (aD, dwx ), for THEOREM 3. 3 . A function ¢ is resol u tive i f and only i f ¢ E L one (and therefore for all) x E D. The following definition is particularly important for its potential-theoretic im plications. In the sequel, given a condenser (K, 0), we denote by cap(K, 0) the sub-elliptic capacity of K with respect to 0, see [D] . DEFINITION 3. 4 . An open set D c !Rn is called thin at Xo E aD, if capx (Dc n Bd(x0,r), Bd(Xa,2r)) > 0. . . (3.5) lIm lnf capx (Bd(X0, r), Bd(Xo, 2r)) 3. 5 . If a problem. bounded open set D c !Rn is thin at Xo E aD, then Xo is regularTHEOREM for the Dirichlet r--+0
62
LUCA CAPOGNA , NICOLA GAROFALO, AND DUY-MINH NHIEU Proof. If D is thin at X0 &D, then 1 [ capx (Dc n Bd(x0, t), Bd(Xa, 2t)) ] dt capx (Bd(xa, t),Bd(x0, 2t)) t E
-
R
=
o
oo
.
to Theorem 3 . 1 , the divergence of the above integral implies for 0 < r < Thanks R/3 osc {u, D n Bd(x0,r)} :::; osc {¢,&D n Bd(X0,2R)} . Letting R ----> 0 we infer the regularity of X0•
0
A useful, and frequently used, sufficient condition for regularity is provided by the following definition.
An open set n �n is said to have positive density at Xo J O n Bd(xa,r)l > 0. . . lIm mf j Bd(Xo, r) l PROPOSITION 3. 7. If De has positive density at X0, then D is thin at X0•
DEFINITION 3.6.
&n,
c
if one has
E
r-+0
Proof. We recall the Poincare inequality
In I
c I DC n Bd(Xa, r)l (3.6) capx (Bd(Xa, r), Bd(xa, 2r)) r 2 capx (Bd(X0, r), Bd(Xa, 2r)) :::;
C
E
c
=
n
-
-
---'=:-----'--�'--
Now the capacitary estimates in [D] , [CDG3] give
capx (Bd(X0,r),Bd(X0,2r)) c- l rQ - 2 , C(O, X) > 0. Using these estimates in (3.6) we find capx (Dc n Bd(xa, r ), Bd(Xa, 2r)) -> C* IDe Bd(xa, r)l , I Bd(xa, r) I capx (Bd(xa, r), Bd(xa, 2r)) where C* C* (0, X ) > 0. The latter inequality proves that if De has positive density at X0, then D is thin at the same point.
C rQ -2 for some constant C
:::;
:::;
=
n
=
0
A basic example of a class of regular domains for the Dirichlet problem is provided by the (Euclidean) C1 • 1 domains in a Carnot group of step r = It was proved in [CGl] that such domains possess a scale invariant region of non-tangential approach at every boundary point, hence they satisfy the positive density condition in Proposition 3.7. Thus, in particular, every such domain is regular for the Dirichlet problem for any fixed sub-Laplacian on the group. Another important example is provided by the non-tangentially accessible domains (NTA domains, henceforth) studied in [CGl] . Such domains constitute a generalization of those introduced by Jerison and Kenig in the Euclidean setting [JK] , see Section 8.
2.
MUTUAL ABSOLUTE CONTINUITY 63 DEFINITION 3.8. Let D !Rn be a bounded open set. For 0 < a 1, the class r�·0 (D) is defined as the collection of all f E C(D) £=(D), such that l f(x) - f(y) l < oo. sup x,yED , xfy d ( X, Y ) 0 We endow r�·"' (D) with the norm l f(x) f(y) l sup (x,y) 0 x,yED,xfy The meaning of the symbol r?��( D ) is the obvious one, that is, f E r ?��( D ) if, for every w D, one has f E r�·0(w). !Rn denotes a bounded closed set, by f r�·0 (F) we mean that f coincides on the set F with a function g E r�· 0 (D), where D is a bounded open set containing F. The Lipschitz class r�· 1 (D) has a special interest, due to its connection with the Sobolev space .C 1 • 00 (D). In fact, we C
::;
n
d
E
CC
-
If F C
have the following theorem of Rademacher-Stepanov type, established in [GNl] , which will be needed in the proof of Lemma 6.1.
THEOREM 3.9. (i) Given a bounded open set U C !Rn,1 there exist Ro R0(U, X) > 0, and C = C(U, X) > 0, such that if f E .C · = (Bd(x0, 3R)), with and 0R),< soR as< R0, then fforcaneverybe modified on a R)set of dx-measure zero in x0Bd = UBd(xo, to satisfy x, y E Bd(x0, l f (x) - f(y) l :S C d(x, Y) llfll.cr . oo(Bd(x0,3R)) · If,inequality furthermore, f replace E c=(Bd(x0, 3R)), then in the right-hand side of the previous one can the beterman l open lf l l .c' . oo (Bd(x0 , 3R)) with I I X fi i Loo (Bd (xa , 3R)) · (ii) Vice-versa, let D !Rn set such that SUPx ,yE D d(x, y) < oo . If 1 1 f r�· (D), then f E .C •00 (D). note explicitly that part ( i) of Theorem 3. 9 asserts that every function f .C 1 ·=We(Bd(x0,3R)) has a representative which is Lipschitz continuous in Bd(xo,R) with respect to the metric d, i.e., continuing to denote with f such representative, one has f E r0• 1 (Bd(xo,R)). Part (ii) was also obtained independently in [FSS]. The following result was established in [D] . 3.10. Let D !Rn be a bounded open set which is thin at every x0 THEOREM fJD. If ¢ E r0·i3 (D), for some j3 E (0, 1), then there exists a E (0, 1), with a = a ( D , X , /3) such that IHf (x ) - Hf (y) l sup d(x, y) 0 < x,yED,xfy Given a bounded open set D C !Rn, consider the positive Green function = G(y,x) for C and D, constructed in [B] . For every fixed x E D, one Gcan(x,y)represent G(x, · ) as follows (3.7) G(x, ) r(x, ) hx , where hx Hf.(x, .) . Since, by Hormander's hypo-ellipticity theorem, r(x, · ) E c=(JRn \ {x}), we conclude that, if D is thin at every X0 8D, then there exists a E (0, 1) such that, for every E > 0, one has (3.8) G(x, · ) E r�·0( D \ B(x0,E)) . =
E
C
E
E
C
E
,
00 ·
·
=
·
=
-
E
LUCA CAPOGNA, NICOLA GAROFALO, AND DUY-MINH NHIEU
64
We close this section with recalling an important consequence of the results of Kohn and Nirenberg [KNl] ( see Theorem 4), and of Derridj [Del] , [De2] , about smoothness in the Dirichlet problem at non-characteristic points. We recall the following definition.
1 domain D !Rn, a point x0 aD is called DEFINITIONfor3.11. GivenXa =C{X characteristic the system 1 , ... , Xm } iffor j = 1, . . . one has < Xj(X0), N (xo) > = 0 , where N(x0) indicates a normal vector to aD at x0• We indicate with = :Ev, x the collection of all characteristic points. The set is a closed subset of aD. THEOREM 3.12. Let D !Rn be a COCJ domain which is regular for (1.3). Consider the point harmonic withan¢ openCOCJ(8D). If X0 V ofaDX0 issucha non characteristic for£,function then thereH!{,exists neighborhood that H!{ C00(D n V ) . REMARK 3.13.3. 12Wefailsstressin that, as weat characteristic indicated in thepoints. introduction, theit fails concluso sion of Theorem general In fact, D andproblem the boundary are realthatanalytic, incompletel generaly thethatsoleven utionif ofthethedomain Dirichlet H!{ maydatum be not¢better Holder continuous up to the boundary, see Theorem 3. 10. An example of such negative phenomenon indedicated the Heisenberg groupa related lHin was constructed by Jerison in [J 1] . The next section is to it. For example concerning the heat equation see [KN2] . C
E
,m
:E
:E
C
E
E
E
4. The example of D. Jerison Consider the Heisenberg group ( discussed in the introduction) with its left invariant generators .4 of its Lie algebra. Recall that lH!n is equipped with the non-isotropic dilations
(1 )
6\(z, t) = (>.z, >.2t) , whose infinitesimal generator is given by the vector field n (xi-a + Yi a . ) + 2 -a . =L ·i=l ax, 8y, at We say that a function u lH!n IR is homogeneous of degree a lR if for every (z, t) lH!n and every >. > 0 one has u(o>.(z, t)) = u(z, t) . One easily checks that if u C 1 (lH!n) then u is homogeneous of degree a if and only if Zu = a u . We also consider the vector field n ( a a) e = L xi - - Yi- , (4.1) i= l ayi axi which is the infinitesimal generator of the one-parameter group of transformations Ro lH!n lHin, () IR, given by Ro(z, t) = (e z, t), z = X + iy en . z
.
:
E
--+
E
E
-->
>."'
E
:
-
i()
E
MUTUAL ABSOLUTE CONTINUITY n
65
Notice that when 1, then in the z-plane Ro is simply a counterclockwise rotation of angle B, and in such case in the standard polar coordinates (r, B) in O· As a consequence of (4.2), if f o N for some function f : [0, ) IR, then one has the beautiful formula (4.3) Since f(t) t 2- Q satisfies the ode in the right-hand side of (4. 3 ) one can show that a fundamental solution of - 0 with pole at the group identity e (0, 0) IH!n (4.2)
=
=
=
u =
=
C
is given by
(4.4)
oo
f(z, t)
0
=
=
CQ , (z, t) =f. N(z, t) Q -2
___,
E
e,
where CQ > needs to be appropriately chosen. The following example due to D. Jerison [Jl] shows that, even when the domain and the boundary data are real analytic, in general the solution to the subellip tic Dirichlet problem (1.3) may not be any better than f0 ·" near a characteristic boundary point. Consider the domain
M E IR . Since nM is scale invariant with respect to { J>. h>o we might think of nM as the analogue of a (M 2:: or a (M < Introduce the variable
convex cone T
=
concave cone
0),
T(z, t)
=
4t N2
(z, t) =f e .
0).
66
LUCA CAPOGNA, NICOLA GAROFALO, AND DUY-MINH NHIEU It is clear that T is homogeneous of degree zero and therefore ZT 0 . Moreover, with 8 as in (4.1) , one easily checks that =
87
=
0.
T = 'Y} are constituted by the t-axis { 1, t - 4 J1 - "(2 iz l 2 ' if !'YI < 1. Furthermore, the function T takes the constant value T v'l +4M16M2 ' on anM. We now consider a function of the form (4.5) v v(z,t) Na u(T) , where the number a > 0 will be appropriately chosen later on. One has the following result whose verification we leave to the reader. PROPOSITION 4.1. For any a > 0 one has £0v 4¢Na- 2 { (1 - T2 )u11 (T) - � Tu1(T) + a(o:+4Q - 2) u(T) } 41/!Na-2 { (1 - T2 )u"(T) - (n + 1)Tu1(T) + a(a ; 2n) u(T) }· Proposition 4.1 we can now construct a positive harmonic function in nM Using which vanishes on the boundary (this function is a Green function with pole It is important to observe the level sets when 'Y = and by the paraboloids 'Y
=
=
=
=
at an interior point).
E
PROPOSITION 4.2. For any a (0, 1) there exists a number M M(a) < 0 such that the nonconvex cone nM admits a positive sol u tion of £0v 0 of the form (4.5) which vanishes on f)0,M · Proof. From Proposition 4.1 we see that if v of the form (4. 5 ) has to solve the equation £0v = 0, then the function u must be a solution of the Jacobi equation (4.6) (1 - T2 )u11(T) - (n + 1)Tu1(T) + a(a 4+ 2n) U(T) 0 . 1} is degenerate and corresponds to the As we have observed the level { T t-axis {z 0}. One solution of (4. 6 ) which is smooth as T -> 1 (remember, the t-axis is inside nM and thus we want our function v to be smooth around the t-axis since by hypoellipticity has to be in C00(0,M)) is the hypergeometric function n +-1 ; 1 - T ) 9a (T) F (- a2 , n + 2o: ; 2 -2When 0 < o: < 2 one can varify that 9a (1) 1 , and that 9a(T) -> -oo as T -> - 1+ . =
=
=
=
=
V
=
=
MUTUAL ABSOLUTE CONTINUITY Therefore, 9a has a zero Ta . One can check (see Erdelyi, Magnus, Oberhettinger and Tricomi, vol. l , p . l lO (14)), that as o + , then Ta - 1 + . We infer that for > 0 sufficiently close to 0 there exists - 1 < Ta < 0 such that 9a (Ta ) = 0 · If we choose Ta < 0 , M = M(o:) = � 1 - r,; then it is clear that On 8D,M we have T Ta , and therefore the function of the form (6.10), with u(r) = 9 of being harmonic and nonnegative in n,proof.and furthermore ona (r),anMhaswethehaveproperty that v = NOtga (Ta ) 0. This completes the 67
0: �
�
o:
V
=
=
0
a (z, t)ga (r) v = N , a ee Ero8D,M(fiM), D,M.
1),
Since o: belongs the interval (0, then it is clear that belongs at most to the Folland-Stein Holder class but is not any better than metrically Holder in any neighborhood of = (0, 0). What produces this negative phenomenon is the fact that the point is characteristic for
5. Subelliptic interior Schauder estimates
In this section we establish some basic interior Schauder type estimates that, besides from playing an important role in the sequel, also have an obvious indepen dent interest. Such estimates are tailored on the intrinsic geometry of the system . . . , Xm} , and are obtained by means of a family of sub-elliptic mollifiers which were introduced in [CDGl] , see also [CDG2] . For convenience, we state the relevant results in terms of the X-balls introduced in Definition 2.2, but we stress that, thanks to ( 2.8 ) , we could have as well employed the metric balls Since in this paper our focus is on .C-harmonic functions, we do not explicitly treat the non-homogeneous equation .C = with a non-zero right-hand side. Estimates for solutions of the latter equation can, however, be obtained by relatively simple modifications of the arguments in the homogeneous case. The following is the main result in this section.
X = {X1 ,
B(x, r) u f
Bd(x, r).
5 . 1 . Let D C !Rn be a bounded open set and suppose that u is har monicTHEOREM in D. There exists Ro > 0, depending on D and X, such that for every x E D and 0 < r ::=; Ro for which B(x,r) C D, one has for any s E N I Xh Xh· · ·Xjs u(x) l --;rc �ax l u i , for some constant C = C(D, X, s) > 0. In the above estimate, for every i = 1, . . . , s, the index ji runs in the set {1, . . . , m} . 5.2. We emphasize that Theorem 5. 1 cannot be established similarly tooremitsREMARK classical ancestor for harmonic functions, one usesof atheharmonic mean-value the coupled with the trivial observation that anywhere derivative function istionsharmonic. In the present are no longer harmonic!non-commutative setting, derivatives of harmonic func ::=;
B ( x ,r)
A useful consequence of Theorem 5 . 1 is the following.
LUCA CAPOGNA, NICOLA GAROFALO, AND DUY-MINH NHIEU C OROLLARY 5.3. Let D !Rn be a bounded, open set and suppose that u is a non-negative harmonic function in D. There exists Ro > 0, depending on D and X,anysuch that for given s N any x D and 0 < r � Ro for which B(x, 2r) D, one has for for some C = C(D X, s) > 0 . Proof. Since u 2: 0 , we immediately obtain the result from Theorem 5.1 and from the Harnack inequality (3.4) . 68
C
E
E
C
,
0
To prove Theorem 5 . 1 , we use the family of sub-elliptic mollifiers introduced in [CDGI] , see also [CDG2] . Choose a nonnegative function E Cg"(IR) , with We C [1, 2] , and such that = and let = define the kernel
f 1 f(R-1 s). fR(s) R1-) 1Xyr(x,y)l2 KR (X, y) fR (f(x, y) f(x, y)2 u Lfoc (!Rn), following [CDGl] we define the subelliptic mollGiven ifier ofaufunction by JR u(x) = fa,. u(y) KR(x, y) dy, R > 0. ( 5.1 ) We note that for any fixed x !Rn, (5.2 ) supp KR(x, ) f!(x, 2R) \ f!(x, R). One of the important features of JR u is expressed by the following theorem. THEOREM 5.4. Let D C !Rn be open and suppose that u is harmonic in D. There exists Ro > 0 , depending on D and X, such that for any x E D, and every 0 < R Ro for which D(x, 2 R) C D, one has u(x) JR u(x). Proof. Let u and f! (x, R) be in the statement of the theorem. We obtain for 'ljJ C00 (D) and 0 < t :S R, see [CGL] , ( 5.3) 2 1 '1/J (x) = 18fl(x,t) '1/J (y) 1 XIDyf(x,y)i f(X, Y)I dHn-l (Y) + lfl(x,t) £'1/J(y) [r(x,y) - t] dy. Taking 'ljJ = u in ( 5.3) , we find i2 ( 5.4 ) u(x) Jan( { x ,t) u(y) 1 XI Dy f(x,y) f(x, y) i dHn_I (y). We are now going to use ( 5.4) to complete the proof. The idea is to start from the definition of JR u( x), and then use Federer co-area formula [Fe] . One finds y) i 2 dHn l(Y)l dt. JR u(x) = loo fR (t) [l u(y) IXI Dyr(x, r (x, y)I The previous equality, ( 5.4 ) , and the fact that fiR !R (s)ds 1, imply the con clusion. JIR f(s)ds 1,
supp f
E
=
E ·
C
:S
=
as
E
=
0
8fl(x,t)
=
0
MUTUAL ABSOLUTE CONTINUITY
69
The essence of our main a priori estimate is contained in the following theorem. JR.n .
THEOREM 5.5. Fix a bounded set U C There exists a constant Ro > 0, depending only on U and on the system X , such that for any u E Lfoc (JR.n ), x E U, 0 < R :::; R0, and s E N one has for some C C(U, X , s) > 0, 1 1 l u(y) l dy. IXj, Xh · · ·Xi, JR u(x) l :::; R F( R)2 + s !1(x,R) c
=
X,
Proof. We first consider the case s = 1 . From (2.7) , and from the support property (5.2) of KR (x ) we can differentiate under the integral sign in (5. 1 ) , to obtain ,
·
,
u(x) l :::; jrB(x,2R) i u(y) i iXx KR (x , y)i dy. By the definition of KR(x, y) it is easy to recognize that the components of its sub-gradient XxKR ( x , y) are estimated as follows R - 2 I Xr ( x, y) l 3 r ( x , y)- 4 IXj KR ( x y)i < + C R - 1 r(x, y) - 2 L IXi Xkr ( x, y) i i Xkr ( x, y) i k=1 3 y) + R-1 I X r(x, l r (x, y)- 3 = Ih{x , y) + Ik ( x, y) + Ik {x, y) . IX JR
c
,
m
c
To control the three terms in the right-hand side of the above inequality, we use the size estimates (2.7) , along with the observation that, due to the fact that on the support of KR ( , · ) one has 1 1 < r ( x, :::; R ' 2R then Theorem 2.3, and (2.9), give for all x U, 0 < R :::; Ra, and D ( x, 2R) \ D (x, R) c < d( x, y) < c - 1 . (5.5) - F (x, R) -
x
y) E
Using (2.7) , (5.5) , one obtains that for i = 1 , 2, 3 .
yE
c
sup l lk (x, Y) l :::; RF(x, R) yE!1(x,2R)\!1(x,R)
3
for any x E U, provided that 0 < R :::; R0• This completes the proof in the case s = 1. The case 2 is handled recursively by similar considerations based on Theorem 2.3, and we omit details. It may be helpful for the interested reader to note that Theorem 2.3 implies
s�
1Xi, Xi2 ·· ·Xj, r ( x , y ) l < C d( x, y ) - s r ( x , y ) ,
so that by (5.5) one obtains (5.6)
sup IXh X12 . . .Xj, r (x, y)i yE!l(x,2R)\!1(x,R)
d(P' x0) - d(z , P) -> ( 1 -34 {}) r - 94 () r = (1 - -43 {} - -94 B) r = -r2 This proves (6. 2 ). The above considerations allow to apply Theorem 3.9, which, keeping in mind that r(x0, ) E C00(Bd(P, �p) ) , presently gives (6.3) j r (xo,x) - r(x0,y) j S C p sup jX r(x0,�) j . Using (2.7) we obtain for � E Bd(P, �p) 1 j Xr(xo, �)I S C d(x0,�) E(xo,d(xo,�)) '
provided that we take () =
-
-
·
eEBd (P, £ p)
LUCA CAPOGNA, NICOLA GAROFALO, AND DUY-MINH NHIEU where --> E(x0, t) is the function introduced in (2.6). Since by (6.2) we have d(x0,�) ;::: r/2, the latter estimate, combined with the increasingness of E(x0,·), leads t o the conclusion 1 r) sup I X f(xo, � ) I ::::; C ) r Xo, E( eE Bd ( P,�p Inserting this inequality in (6.3), and observing that rE(!o , r) ::::; C I Bd(�o,r)l , we find lf (xo, x) - r(xo , y) l ::::; c I Bd(Xo,r r) l d(x, y). 72
t
This completes the proof of the lemma.
0 The following definition plays a crucial role in the subsequent development.
DEFINITION 6.2. A domain D C !Rn is said to possess an outer X-ball tangent at Xo E aD if for some r > 0 there exists a X -ball B(x 1 , r) such that: (6.4) X0 aB(x1,r), B(x1 ,r) n D We say that D possesses the uniform outer X-ball if one can find Ro > 0 such that for every Xo aD, and any 0 < r < Ro, there exists a X -ball B(x1 , r) for which (6.4) holds. Some comments are in order. First, it should be clear from (2.8) that the existence of an outer X-ball tangent at Xo E aD implies that D is thin at Xo ( the reverse implication is not necessarily true) . Therefore, thanks to Theorem 3.5, X0 is regular for the Dirichlet problem. Secondly, when X = { a�, , . . . , a�n } , then the distance d(x, y) is just the ordinary Euclidean distance l x - Yl · In such case, Definition 6.2 coincides with the notion introduced by Poincare in his classical paper [P] . In this setting a X-ball is just a standard Euclidean ball, then every cu domain and every convex domain possess the uniform outer X-ball condition. When we abandon the Euclidean setting, the construction of examples is technically =
E
0.
E
much more involved and we discuss them in the last section of this paper. We are now ready to state the first key boundary estimate.
THEOREM 6.3. Let D !Rn be a connected open set, and suppose that for some rdepending > 0, D has an outer X-ball B(x 1 ,r) tangent at Xo aD . There exists c > 0, D and on X , such that if ¢ E C(aD) , ¢ 0 in B(x 1 , 2r)naD, then we haveonlyforonevery xED I H�(x)l ::::; C d(x,r Xo) max aD 1 ¢ 1 . Proof. Without loss of generality we assume max l ¢ 1 1. Following the idea in [P] we introduce the function ,x) 1 X E D, 1 ,r)-1 1- -E(xr(x,12r)f(x) - E(xE(x1 , r)(6.5) 1 ' where X f(x1 , x) denotes the positive fundamental solution of £, with singularity at x 1 , and t --> E(x 1 , t) is defined as in (2 .6 ) . Clearly, f is £-harmonic in !Rn \ { x } . Since r(x 1 , · ) ::::; E(x 1 ,r) - 1 outside B(x 1 , r), we see that f ;::: 0 in !R n \ B(x 1 ,r), c
E
=
aD
=
_
-t
l
MUTUAL ABSOLUTE CONTINUITY hence in particular in D. Moreover, f 1 on 8B(x1 , 2r) D, whereas f 2: 1 in (JR.n \ B(x 1 , 2r)) D . By Theorem 3.2 we infer I H.f (x) l � f(x) for every x D. The proof will be completed if we show that (6.6) J(x) � C d(x,x0) r , for every x E D. Consider the function h(t) E(x 1 , t) - 1 . We have for 0 < s < t < R0, h(s) - h(t) (t - s) E(E'(xX1,T1 , T))2 , for some s < T < t . Using the increasingness of the function r ----* rE(x 1 , r), which follows from that of E(x 1 , ) and the crucial estimate ' (x 1 , r) -< c-1 C -< rEE(xt,r) which is readily obtained from the definition of A(x 1 ,r) in (2.2), we find (6.7) C tEt(-X 1s,t) � h(s) - h(t) � C-1 s E(t -x1s, s) . in mind the definition (6.5) of j, from (6.7), and from the fact that E(xKeeping 1 , ) is doubling, we obtain f(x) � C E(x 1 ,r) {f(xt,Xo) - f(x 1 ,x)}, where we have used the hypothesis that X0 E 8B(x 1 , r). The proof of (6.6 ) will be achieved if we show that for x JR.n \ B(x 1 , r) 73
n
=
n
E
=
=
·
,
'
·
E
In view of (2.8), the latter inequality follows immediately from Lemma 6.1. This completes the proof.
0
G(x, y)
Let D C JR.n be a domain. Consider the positive Green function asso ciated to and D. From Theorem 3.2 and from the estimates (2.7) one easily sees that there exists a positive constant such that for every ED
C
CD x, y 2 ' 0 < - G(x, y) � CD I Bd(x,d(x,y) d(x, y))l
(6.8)
x, y E D. Our next task is to obtain more refined estimates for G. the uniform outer X-ball condi tion.THEOREM There exists6.4.a Suppose constant that C DC(X,JR.nD) satisfy > 0 such that d(x,y) d(y, 8D) G( x,y) 1 as in (2.8), and let Ro be the constant in Definition 6. 2 of uniform outer X-ball condition. The estimate that we want to prove is immediate if one of the points is away from the boundary. In fact, if either d(y, aD) ad(��� , or d(y, aD) R0, then the conclusion follows from (6.8). We may thus assume that a d(y, aD) < a(d(x,a +y)3) , and d(y, aD) < R0• (6.9) We now choose aRo ) . r - ( 2ad(x,y) , (a + 3) 2 One easily verifies from (6. 9 ) that ad(y, aD) < 2r. Let Xo be the point in aD such that d(y,aD) d(y,x0) and consider the outer X-ball B(x 1 ,rfa) tangent to the boundary of D in X0• We claim that y E D n B(x1 , (a + 3)r). To see this observe that by (2. 8 ) X0 E B(x 1 , � ) Bd(x 1 , r), and therefore a + 2 < -a + 3 r. d(y, Xl) :::; d(y, X0) + d(x0, X1 ) d(y, aD) + d(x0, Xl ) :::; --r a a This shows y E Bd(x 1 , a - 1 (a + 3)r). Another application of (2. 8 ) implies the claim. Next, the triangle inequality gives d(x,x1 ) d(x, y) - d(x1 , y) > d(x, y) - a-+a-3 r > d(x, y)(1 - 2a21 ), and consequently x E !Rn \ Bd(x 1 , ( 1 - 2�2 )d(x, y)). On the other hand (2. 8 ) implies IRn \Bd(xl , (1 - _2_2 )d(x, y)) IRn \B(xl , a� (1 - 2a1 2 )d(x, y)) IRn\B(x l , (a+3)r), the last inclusion being true since a > 1. now consider the Perron-Wiener-Brelot solution v to the Dirichlet problem .CvsuchWethat 0 in B(x 1 , (a+3)r)nD, with boundary datum a function ¢ E C(a(B(x 1 , (a+ 3)r) n D)), 0 :::; ¢ :::; 1, ¢ 1 on aB(x 1 , (3+a)r)nD, and ¢ = 0 onweaDnB(x 1 , (1+a)r). can only say that We observe in passing that, thanks to the assumptions on v is continuous up to the boundary in that portion of a(B(x 1 , (a + 3)r) n D) that is common to aD. However such continuity is not needed to implement Theorem 3.outer 2 and.C-ball deduce that 0 1. We observe that D n B(x1 , (a+ 3)r) satisfies the condition at the point Xo E aD. Applying Theorem 6.3 one infers for every y E D n B(x 1 , (a+ 3)r) (6.10) jv (y) j :::; C d(y,aD) r . Let CD be as in (6. 8 ) and define w(z) Ci/ E(x, {3d(x, y))G(x, z), where {3 (1Observe - 1a ). Since X tt B(x l , (a + 3)r), then .Cw 0 in B(x l , (a + 3)r) n D. - b that 2 if z E aB(x 1 , (a + 3)r), then d(x, z) d(x, xl ) - d(z , x1 ) (1 - 2a21 ) - (a+ 3)r {3d(x, y), 74
?:
?:
_
.
mtn
=
C
=
?:
C
�
C
=
=
:::;
D,
v
:::;
=
=
=
?:
?:
?:
MUTUAL ABSOLUTE CONTINUITY r
75
r E (x, r) , (a+ w :::; Ci/ E(x,w(y)d(x,:::;z))G(x, z) :::; &(B(x 1 v(y) DnB(x1 , (a + 3)r). 3)r) n D).
from our choice of and (3. Consequently, in view of the monotonicity of ---> and (6.8) , we have that 1 on By Theorem 3.2 one concludes that The estimate in of established above, along with (2. 1 ) , completes the proof. 0
v
in a Carnot group, by exploiting G(y,50]x)that= G(x, y), one can actually improve G(x, y) :::; C d(x, y) -Qd(x, &D)d(y, &D) , x, y E D , x =I y ,
It was observed in [LU2, Theorem the symmetry of the Green function the estimate in Theorem 6.4 as follows
where Q represents the homogeneous dimension of the group. An analogous im provement can be obtained in the more general setting of this paper. To see this, note that the symmetry of and the estimate in Theorem 6.4 give for every
(6 . 1 1 )
G G(y, x) - G(x, y) uni0 fsuch that X -ball condition for D �n . d(x,y) I XG(x,y) i :::; C I Bd(x,d(x,y)) i' for each x, y E D, with x =I y. Proof. Let Ra b e as in Definition 6.2. Fix x, y E D and choose 0 < r < Ra such that x rf:. Bd(y, ar) D. Applying Corollary 5. 3 and ( 2 . 8 ) to G(x, ·) we obtain for every z E B(y, r) IXG(x, z) i :::; -cr G(x, z). C
C
LUCA CAPOGNA, NICOLA GAROFALO, AND DUY-MINH NHIEU d If d(y,aD) � 2ad(x,y), we choose r = min ( ( y2�D), lf ) and then the latter inequality implies the conclusion via Theorem 6.4. If d(y, aD) > 2ad(x, y), then r(x, ) - hx , we use (2.7) to bound j X rj , and, keeping in mind that G(x, ·) d with r min ( ( y2�D ), If), we apply Corollary 5.3 and the maximum principle to 76
·
=
obtain
=
I X hx (Y) i � -cr hx (Y) -cr hy (x) � -cr sup r(y, w) -cr r(y, z) for some z E aD. On the other hand, one has d(x, y) d(y,2aaD) d(y,2az) so that using ( 2. 7) one more time 1 z)) - c E(y,2ad(x,y)) 1 - c r(y, x) - c I Bd(x,d(x,y))i d(x, y) 2 . r(y,z) - c E(y,d(y, Replacing this inequality in the estimate for I X hx ( Y)I we reach the desired conclusion. =
wE8D
, . . . , < v(y), Xm ( Y ) > ) , Nx (y) where v(y) is the outer unit normal to n in y. We also set W(y) I Nx (y) l L j=l < v(y),XJ (Y) >2 • lfy E an \ I:, we set vx (y) INNXx (y)(y) l One has lvx (y)l 1 for every y E aD \ I: . We note explicitly from Definitions 3 . 1 1 and 7.1 that one has for the charac teristic set I: of n I: {y E an 1 W(y) o} . D FI
c
n
c
]Rn
=
m
=
=
=
=
=
=
Using the quantities introduced in this definition we can express (7.4) in the following suggestive way.
LUCA CAPOGNA, NICOLA GAROFALO, AND DUY-MINH NHIEU PROPOSITION 7.2. Let D C JRn be a bounded open2 set with (positive) Green function and consider a C domain n 0 D. For any u E CG00of(D)theandsub-Laplacian every x E 0(1one.2) has u(x) lao{ G(x, y) < Xu(y), Nx (y) > da(y) - lao{ u(y) < XG(x, y), Nx (y) > da(y) + in G(x, y) Cu(y) dy . If moreover Cu=O in D, then u(x) lao{ G(x, y) < Xu(y), Nx (y) > da(y) - lao{ u(y) < XG(x, y), Nx (y) > da(y) . In particular, the latter equality gives for every E n < XG(x, y), Nx (y) > da(y) 1 . { lao REMARK 7.3. If u E c=(D), then we can weaken the hypothesis on n and require only 0 D rather than 0 D. 80
c
c
=
X
-
=
c
c
We consider next a coo domain D C ]Rn satisfying the uniform outer X-ball condition in a neighborhood of I;. Our purpose is to pass from the interior repre sentation formula in Proposition 7.2 to one on the boundary of aD. The presence of characteristic points becomes important now. The following result due to Derridj [Del, Theorem 1 will be important in the sequel.
]
THEOREM 7.4. Let D C ]Rn be a coo domain. If I; denotes its characteristic set, then a('E) 0. =
We now define two functions on D x (aD \ I;) which play a central role in the results of this paper. They constitutes subelliptic versions of the Poisson kernel is the Poisson kernel from classical potential theory. The former function The latter for D and the sub-Laplacian (1.2) with respect to surface measure is instead the Poisson kernel with respect to the perimeter measure This comment will be clear after we prove Theorem 7.10 below.
P(x, y)
a. ax. K(x, y) Poisson kernels) . With the notation of Definition 7.1, DEFINITION for every (x,7.5y) E(Subelliptic D (aD \ E) we let P(x, y) < XG(x, y), Nx (y) > (7.5) We also define (7.6) K(x, y) P(x, < XG(x, y), x (y) > . W (y)y) extend the definition of P and K to all D fJD by letting P(x, y) K(x, y) 0 forWe-a.any x E D andy E E. According to Theorem 7. 4 the extended functions coincide a e. with the kernels in (7.5) , (7.6) . It is important to note that if we fix x E D, then in view of Theorem 3.12 the functions y P(x, y) and y K(x, y) are coo up to fJD \ I;, The following estimates, which follow immediately from (7.5) and (7.6) , will play an important role in the sequel. For (x, y) E D (aD \ E) we have (7.7) P(x, y) ::; W (y) IXG(x,y) i , K(x,y) ::; I XG(x, y)i . x
=
-
=
=
-
v
x
-->
-->
x
=
=
MUTUAL ABSOLUTE CONTINUITY
81
We now introduce a new measure on aD by letting
(7.8)
dax
W da . We observe that since we are assuming that D coo the density W is smooth and bounded on aD and therefore implies that dax « In view of this observation Theorem implies that also ax (E) =
E (7. 8 ) da. 7.4 0. REMARKmeasure 7.6. PxWe(Dmention explicitly that the measure dax in (7.8) is the X perimeter (following concentrated on &D. To explain this point we recall that for; ) any open setDenGiorgi) ]Rn Varx (xv ; O ) , (7.9) Px (D; O) where Varx indicates the sub-Riemannian X -variation introduced in [CDG2] and also developed in [GNl] . Given a bounded C2 domain D C ffi.n one obtains from [CDG2] that =
·
c
=
Px (D; O) (7.10) ( W da . lavno From (7.10) one concludes that for every y E aD and every r > 0 Px (D ; Bd(y, r)) (7.11) ax(&D Bd(y, r)) which explains the remark.and geometry The measure ax Px (D; ) on aD plays a perva sive role in the analysis of sub-Riemannian spaces, andinitsgeometric intrinsic properties have many deep implications both in subelliptic pde ' s and measure theory. For an account of some of these aspects we refer the reader to [DGN2] . PROPOSITION 7. 7. Let D C JRn be a bounded coo domain satisfying the uniform outer we haveX -ball condition in a neighborhood of its characteristic set E . For every x E D f P(x, y)da(y) 1 lav ( K(x, y)dax (Y ) . lav Proof. We x E D and recall that E is a compact set. In view of Theorem 7.n 4 weOkcan D,choose an exhaustion of D with a family of coo connected open sets with n k / D as k _...., oo, such that &n k r� ur�, with rk &D\E, k rk / aD, a(r0 _...., o. B y Proposition 7.2 (and the remark following it ) we obtain for every k E N (7.12) XG(x, y), Nx (y) > da (y) ( - 1 lank x x f . f lar� XG(x, y), N (y) > da(y) + lar� XG(x, y), N (y) > da (y) We now pass to the limit as k oo in the above integrals. Using Corollary 6.11 and a(r%} 0, we infer that x (y) > da(y) 0 . XG(x, y), lim ( N -+ k oo Jar� =
n
,
=
·
=
=
=
fix
c
c
=
=
c
=
da(y) . { K(x, y) dax(y) - lor� lark Since as we have observed dax da, in view of the second estimate K (x, y) ::::; (7.7),respect we can again use Theorem 3.12, Corollary 6.11 and dominated Iconvergence XG(x,y) l in(with to ax ) to conclude that lim f K(x, y) dax (Y ) { K(x, y) dax(Y) . lav laq =
C* ! Bd(y,r r) !
This proves the theorem.
D
8.4. Let D c JRn be a NTAx domain of class C2 satisfying the upperCOROLLARY 1-Ahlfors assumption in iv) of Definition 1 . 1 . Then the measure ax is 1fors, in the sense that there exist A , R1 > 0 depending on the NTA x parameters ofAhl D and on A > 0 in iv), such that for every y E aD, and every 0 < r < R1 , one has (8. 1)
Inandparticular, the measure ax is doubling, i.e. , there exists C > 0 depending on A on the constant C1 in (2.5), such that C ax (tl(y, r)) . (8.2) ax ( ll ( y, 2r ) ) for every y E aD and 0 < r < Proof. According to Theorem 8.3 the measure ax is lower 1-Ahlfors. Since by Definition 1 . 1 it is also upper 1-Ahlfors, the conclusion ( 8 1 ) follows. From iv)the oflatter and the doubling condition (2.5) for the metric balls, we reach the desired R1 .
::;
.
conclusion (8.2).
D
The following results from [CGl] play a fundamental role in this paper. THEOREM 8.5. Let D C JRn be a NTAx domain with relative parameters M, r0• There C > 0, depending only on X and on the NTAx parameters of D, Mexistsanda constant such that for every Xo E aD one has W Ar (x o ) (6.(x0, r)) 2: C . THEOREM 8.6 (Doubling condition for .C-harmonic measure) . Consider a NTAx domain ]Rn with relative parameters M, To . Let Xo E aD and r To· There exist C >D0,c depending on X, M and r0, such that wx(tl(xo, 2r)) ::; cw x (tl(xo, r)) for any x E D \ Bd(X0, Mr). To ,
::;
87
MUTUAL ABSOLUTE CONTINUITY
THEOREM 8.7 ( Comparison theorem) . Let D C ]Rn be a X - NTA domain with relativefunctions parameters r0. Let E {)D and 0 on aDr \ �(x0, ii · If u , v are £ harmonic in D,M,which vanishX0 continuously 2r), then for every x E D \ Bd(x0, Mr) one has u (x) 0 _ 1 u(A,. (xo)) u(A,. (xo)) C v(A,. (xo)) - v(x) v(A,.(x0)) for some constant C > 0 depending only on X, M and r0• For any y E an and a > 0 a nontangential region at y is defined by ra ( Y) = {x E n I d(x , y) :::; (1 + a)d(x , en)} . Given a function u the cx-nontangential maximal function of u at y E aD is defined by Na (u)(y) = xEf',sup(y) i u (x) l . THEOREM 8.8. Let D ]Rn be a NTA x domain. Given a point x 1 E D, let f E L1 (8D, dwx1 ) and define u(x) laD { f(y)dwx (y) , x E D . Then, u is £-harmonic in D, and: ( i) Na (u)(y) :::; CMwxl (f)(y), y E aD ; (ii) u converges non-tangentially a. e. ( dwx1 ) to f. Theorem 8. 7 has the following important consequence. THEOREM 8.9. Let D ]Rn be a ADPx domain, and let K(·, ·) be the Pois son in (7.6). There exists r1 > 0, depending on M and r0 , and Kernel a constantdefined C = C(X, M, r0, R0) > 0, such that given X0 E aD, for every X E D \ Bd(X0, Mr) and every 0 r r 1 one can find Ex 0 ,x , r C �(xo, r), with <Jx (Ex0, x ,r ) = 0, for which K(x,y) C K(Ar(X0),y) wx (�(x0, r)) for every y E �(X0, r) \ Ex0,x,r · Proof. Let Xo E aD. For each y E �(xo, r) and 0 s r/2 set
0 depending on the a - ADPx parameters of D such that for every y E aD and 0 < r < R1 , a(�(y, 2r)) � C a(�(y, r)) . Proof. Applying Theorem 1.5 with p = 2 , we find 1 { P(x1, y)2 da(y) � ( (�rXo, r)) J{L\. (xo ,r) P(x 1 , y) da(y) ) (�( Xo, r )) JL\.(xo ,r) x1(�(xo,r)) ) 2 = C ( wa(�(x0, r)) a
(]"
This gives
2
9I
MUTUAL ABSOLUTE CONTINUITY
0 Our final goal in this section is to study the Dirichlet problem for sub-Laplacians when the boundary data are in LP with respect to either the measure or the surface measure We thus turn to the
ax a. Proof of Theorem 1.4. The first step in the proof consists of showing that functions f E LP(8D, dax) are resolutive for the Dirichlet problem (1. 3 ). In view of Theorem 3. 3 it is enough to show that f E £ I (8D, dwx 1 ) for some fixed x i E D. This follows from (7.16) and Proposition 7. 9 , based on the following estimates { l f (y) l dwx1 (Y) laD { l f (y)I K (x i ,y) dax(y) laD 1 1 ' ::; (� l f (y)I P dax ( Y) ) (� K(x i , y) P dax (y) ) D 1 D ::; C (� l f (y)I P dax(y) ) D This shows that LP(8D, dax) C LI(8D, dw x1 ) and therefore, in view of Theo rem 3. 3 , for each f E LP(8D, dax ) the generalized solution solution Hf exists and it is represented by Hf (x ) { f(y) dwx (y) . laD At this point we invoke Theorem 8. 8 and obtain for every y E 8D (8.8) Na (Hf ) (y) ::; C Mwx1 (f)(y) . Moreover, Hf converges non-tangentially dwx1-a. e . to f. By virtue of Theo rems 1.3 and 1. 5 , we also have that Hf converges dax-a.e. to f. To conclude the proof, we need to show that there exists a constant C depending on 1 < < D and X such that IINa (Hf ) IILP (aD,d - 3/2 we may then attempt to define (2.15) by setting (2. 16)
ROBIN-TO-DIRICHLET MAPS AND KREIN-TYPE RESOLVENT FORMULAS
1 09
A moment's reflection shows that, in order to establish that the mapping (2.15), (2.16) is well-defined, it suffices to prove that (2. 1 7) in the case when n is a bounded Lipschitz domain which is star-like with respect to the origin in �n (cf. (A.6)) . Assuming that this is the case, pick u E H (1 /2l+s (D) for some E > 0, and for each E (0, 1 ) set Ut(x) = u ( tx ) , X E D. We claim that
t
(2. 18) To justify this, it suffices to prove that this is the case when u E coo (0) as the result in its full generality then follows from a standard density argument. However, for every u E C00 (0) one trivially has Ut __... u as t __... 1 in H 1 (D), hence in H(1 / 2 ) +s (n) . Having disposed of (2.18), we may then conclude that "fDUt --> "fDU in L 2 (aD; dn- lw) as t --. 1. Since for each E (0, 1) we have Ut E C(O) , it follows that "fDUt = 1fjy ut = Ut lan . Thus, altogether,
t
t -->
(2. 19) 1. Ut l an --> "fDU in L 2 (an; dn -lw) as On the other hand, for almost every X E an, and every t E (0, 1), we have that 1 , and lx - Yl :=::; ( 1 + ) dist ( y , an) for y = tx belongs to n , converges to X as some sufficiently large = ��:(D) > 0 (independent of x and This implies that
t
11:
__...
t).
��:
Ut(X) --> ('Yn.t. u) (x) pointwise, for a.e. X E an, as t --> 1 .
(2.20)
Combining (2.19), (2.20) we therefore conclude that the functions 'Yn.t . u , "fDU E L2 (aD; dn- 1 w) coincide pointwise a.e. on an. This proves (2.17) and finishes the justification of the fact that the mapping (2.15), (2.16) is well-defined. Granted (2.15), (2. 16) is well-defined, it is implicit in its own definition that the mapping (2.15), (2.16) is also bounded when we equip H;{ 2 (D) + H5+ 2 (D) with the canonical norm (2.21) JJ u JJ H,;; 2 (!1 ) + ll v i i H•+2 (!1) · w��t uE H,;; 2 (!1 ) , vE H•+2(!1) The same type of argument as above (i.e., restricting attention to pieces of W 1-+
n which are star-like Lipschitz domains, and using dilations with respect to the respective center of star-likeness) shows the following: If w E C (O) can be de composed as u + v with u E H;!2 (D) and v E H8+2 (D) for some s > -3/2, then w l an = /n.t.U "(vv. In other words, the action of the trace operator '1v in (2.15), (2.16) is compatible with that of (2.6). This completes the study of the nature and properties of '1v in (2.15), (2.16). Consider next the claim made about (2.10). As regards its boundedness and the fact that this acts in a compatible fashion with (2.7) , it suffices to prove that
+
(2.22) continuously. To see this, pick u E H11 2 (D) such that t:. u E H8(D) and extend (cf. t:.u to a compactly supported distribution w E H8 (�n). Next, set
[87])
v ( x)
=
r � y En (X - y )w(y ) , X E n, }JF. n
(2.23)
110
F.
GESZTESY AND M. MITREA
where
E
n
(X ) -
_
{
},. ln(l x l ) , 1 ( 2 -n)wn -l l x l 2 n
n = 2, n 2: 3,
'
(2.24)
is the standard fundamental solution for the Laplacian in JRn (cf. (C.1) for z = 0). Here Wn-l = 2nn/2 /f(n/2) (f( · ) the Gamma function, cf. [1, Sect. 6.1]) represents the area of the unit sphere in !R.n. Then v E Hs+2 (�) and .6-v = .6-u in 0. As a consequence, the function w = u - v is harmonic and belongs to H112(0), that is, u = w + v with w E H)j2(0), v E H8+2 (0) . Furthermore, the estimate
sn-l
(2.25)
for some C = C(D, s) > 0 is implicit in the above construction. Thus, the inclusion (2.22) is well-defined and continuous, so that the claims ahout the boundedness of (2.10), as well as the fact that this acts in a compatible fashion with (2. 7) , follow from this and the fact that 1v in (2. 15) , (2. 16) is well-defined and bounded. As far as the existence of a linear, bounded, right-inverse is concerned, it suffices to point out (2. 12) and recall that the mapping (2. 14) is onto (cf. [28] ) . \Ve now digress momentarily for the purpose of developing an integration by parts formula which will play a significant role shortly. First, if 0 is a bounded star like Lipschitz domain in JRn and G is a vector field with components in H)j 2 (D) + H8+2(�), s > -3/2, such that div(G) E L1 (0), then =
ln{ dxn div(C) lau{ d"- 1w
v
· 1v G .
(2.26)
Indeed, if as before Gt (x) = G(tx), x E D, t E (0, 1), then (2.2 7) div(C1) t(div(C))t in the sense of distributions in n. Writing (2.26) for Gt in place of G, with 0 < t < 1, and then passing to the limit t -+ 1 yields the desired result . As a corollary of (2.26) and (2.22), we also have that (2.26) holds if n is a bounded star-like Lipschitz domain in !Rn and G is a vector field with components in {u E H112(�) l 6u E H·'(D)}, s > -3/2, such that div(C) E L1 (D). Since the latter space is a module over C0 (JRn) and any LipHchitz domain is locally star-like, a simple argument based Oil a smooth partition of unity shows that the star-likeness condition on 0 can be eliminated. More precisely, Hypothesis 2.1, (2.28) G E { u E H11 2 (D) i 6u E H"(O)} n , s > -3/2, ===> (2.26) holds. =
}
div(G) E L1 (0; dnx )
Moving on, consider the operator (2. 1 1 ) . To get started, we fix s > -3/2 and ::u,;,;ume that the function u E H31 2 (0) is such that 6u E H 1 +8 (0). Then, by the . second line in (2.7),
/DU E H1 -"(0D.) for every
c
> 0.
(2.29)
To continue, we recall the discussion (results and notation) in the paragraph containing (A.l l )-(A.16) in Appendix A. For every j, k E { 1 , . . . , n}, we now claim that (2.30)
ROBIN-TO-DIRICHLET MAPS AND KREIN-TYPE RESOLVENT FORMULAS Since the functions Oju, /A u belong to the space {w E H 1 1 2 (0) may then conclude from (2.30) and (2.10) that
o('YD u) OTj,k
111
J tl.w E H8(0) } , we
E L 2 (80 ; dn-lw),
(2.31)
and, in addition, (2.32) for every j, k E { 1 , ... , n }. In concert with (2.32) and (2.29), the characterization in (A.16) then entails that /D U E H 1 (80) and II !D u llH l (an) ::; C( J i u iiH3/2(!1 ) IJCl.uJI H l +s (n )) · In summary, the proof of the claims made about (2. 1 1) is finished, modulo establishing (2.30) . To deal with (2.30) , let '¢ E C0(JR.n) and fix j, k E { 1 , ... , n}. Consider next the vector fields Fj,k = ( o, . . . , o , u k'¢ , 0 . . . , 0, -u j'¢ , 0 ... , o) , (2.33)
+
Gj,k
a
=
a
( o, ... , 0, '¢8ku, O ... , o, -'¢8ju, O ... , o) ,
with the nonzero components on the j-th and k-th slots. Then Fj,k, Gj,k have components in the space { u E H 1 12 (0) J tl.u E H8 (0)} with > -3/2 and satisfy div ( Fj,k )
=
s
- di ( Gj,k ) = (ojuok'¢ - OkUOj'¢) E L2 (0; dn x),
v
(2.34)
in the sense of distributions. Also,
v '1D (Fj,k) ('YD u) ( vkoj 'lj; - lljOk'¢) , v '1D (Gj,k) = 'l/J ( vk !D (OjU) - llj/D (oku)) . ·
=
(2.35)
·
Hence, using (2.28), we obtain
{
lan
dn-lw ('YD u) ( vkoj'¢ - vjok'¢) = =
{
lan
dn- 1 w v '1D (Fj,k) ·
In dnx div(Fj,k)
=-
{
lan
=
-
In dnx div(Gj,k)
dn -l w 'l/J (vk/D ( Oju) - Vj !D (oku)) .
(2.36)
This justifies (2.30) and shows that the operator (2. 1 1 ) is well-defined and bounded. Clearly, this acts in a compatible fashion with (2.7) and (2.10). To finish the proof of Lemma 2.3, there remains to show that this operator also has a bounded, linear, right-inverse. This, however, is a consequence of the well-posedness of the boundary value problem u
E H 31 2 (0),
Cl.u = 0 in 0, !D (u) = E H 1 (80),
f
a result which appears in [101] . Next, we introduce the operator
/N
(2.37)
D
(the strong Neumann trace) by
/N = v " /D '\1 : Hs +l (O) -+ L2 (80; dn- 1 w),
1/2
< s < 3/2,
(2.38)
where v denotes the outward pointing normal unit vector to 80. It follows from (2.7) that IN is also a bounded operator. We seek to extend the action of the
112
F. GESZTESY AND M. MITREA
Neumann trace operator (2.38) to other (related) settings. To set the stage, assume Hypothesis 2.1 and recall that the inclusion (2.39) t : H8 (D.) '--7 (H1 (D) ) * , s > -1/2, is well-defined and bounded. We then introduce the weak Neumann trace operator :YN : { u E H1 (D) I �u E H 8 (D) } ___. H-1 12 (80.) , s > -1/2, (2.40) as follows: Given u E H1 (0.) with �u E H8 (0.) for some s > -1/2, we set (with L as in (2.39)) ( ¢, :YNuh;z
=
l dnx V7
( x )
·
V7u(x) + Hl ( f!j ( , L (�u) )(Hl (rl ) ) • ,
(2.41)
for all ¢ E H112 (80.) and E H1 (0.) such that 1o = ¢. We note that this definition is independent of the particular extension of ¢ , and that :YN is a bounded extension of the Neumann trace operator IN defined in (2.38). The end-point case s = 1/2 of (2.38) is discussed separately below. LEMMA 2.4. Assume Hypothesis 2.1. Then the Neumann trar-e operator (2.38) also extends to (2.42) :YN : { u E H312 (0.) I �u E L2 ( D ; dnx) } ___. L2 (8r!; dn-1w) in a bounded fashion when the space { u E H312 (0.) I �u E L2 (0.; dnx) } is equipped with the natural graph norm u f--l lluiiH3/ 2( r!) + ll�ull ucn ; dn x) · This extension is compatible with (2.40) and has a linear, bounded, right-inverse (hence, as a conse quence, it is onto) . Moreover, the Neumann trace operator (2.38) further extends to :YN : { u E H112 (r!) I �u E L2 (0. ; dnx ) } ___. H- 1 (80.) (2.43) in a bounded fashion when the space { u E H112 (0.) I �u E L2(r!; dnx) } is equipped with the natural graph norm u ,..... llui1 H l/ 2 (f!) + ll�uiiP ( rl ;d"x) · Once again, this extension is compatible with (2.40) and has a linear, bounded, right-inverse (thus, in particular, it is onto) . PROOF.
Fix 'l/J E c=(fi). Applying (2.28) to the vector field G = "1f;V7u yields
r dn-lw 1f v · -w('lu) = r dxn V7lf; · V7u + r dxn 1f; �u. Jn Jn �n
(2.44)
f dn-1w "¢ v · !o ('lu) = f dxn 'l · Vu + f dxn � �u. lao ln ln
(2.45)
Consider now ¢ E H112 (80.) and E H1 (0.) such that 1o = ¢. Since c= (n) '--7 H 1 (0.) is dense, it is possible to select a sequence 'l/Jj E c=(f!), j E N, such that and 'l/;j l ao ___. ¢ '1/Jj ___. 4> in H 1 (r!) as j ___. oo. This entails V'l/;j ___. 'l in L2 (!1; dnx) · in H 1 12( 8!1) as j ___. oo. Writing (2.44) for 'l/Ji in place of '1/) and passing to the limit j ___. oo then yields This shows that the Neumann trace of u in the sense of (2.40), (2.41) is actually v · !o ('lu). In addition, II:YNull £2 (8!1; dn -1w) = ll v · !o ( V7u ) IIL2 (ofl;dn-lw) ::; llfn ( Vu) II£2(Di1;dn- lw)n ::; C(IIV7 ui 1Hl/2 (f!)" II �( Vu) IIH-l(f!)n.)
+
ROBIN-TO-DIRICHLET MAPS AND KREIN-TYPE RESOLVENT FORMULAS C ( ll\7ui !Htl2(n)n + ll\7(6-u) IIH-t (n)n) :::; C ( ll u iiH3/2(!1) + ll 6.u ii L2(!1;dnx)) ,
113
=
(2.46)
where we have used the boundedness of the Dirichlet trace operator in (2. 10) with s = - 1 . This shows that, in the context of (2.42) , the Neumann trace operator
;:;Nu
= v
· 'Yv (\7u)
(2.47)
has is well-defined, linear, bounded and is compatible with (2.40) . The fact that this has a linear, bounded, right-inverse is a consequence of the well-posedness result in Theorem 3.2, proved later. As far as (2.43) is concerned, let us temporarily introduce
;yn { u E H 112 (0.) l 6.u E L 2 (0.; dnx) } :
by setting
____.
H - 1 (80.)
=
(H1 (80.) )
*,
(2.48)
(2.49) (¢, ;:;n u h (;:;N( IP ) , -yv u)o + ( IP , 6.u)L2(!1;dnx) - (6. 1P , u)p(n;dnx) > for all ¢ E H1 (80.), where IP E H312 (0.) is such that 'Yvlfl = ¢ and 6.1P E L2(0.; dnx). That such a IP can be found (with the additional properties that the dependence ¢ IP linear, and that IP satisfies a natural estimate) is a consequence of the fact =
f---+
that the mapping (2. 1 1) has a linear, bounded, right-inverse. Let us also note that the first pairing in the right hand-side of (2.49) is meaningful, thanks to the first part of Lemma 2.3 and what we have established in connection with (2.42). We now wish to show that the definition (2.49) is independent of the particular choice of IP. For this purpose, we recall the following useful approximation result: (2.50) where the latter space is equipped with the natural graph norm u f---+ ll u ii H • ( O ) + ll 6.u iiL2(!1;dnx) · When s = 1 this appears as Lemma 1 .5.3.9 on p. 60 of [48] , and the extension to s < 2 has been worked out, along similar lines, in [26]. Returning to the task ast hand, by linearity and density is suffices to show that (2.51) whenever IP E H312 (0.) is such that 'Yvlfl = 0, 6.1P E L2 (0.; dn x) , and u E c= (fi") . Note, however, that by (2.41) with the roles of IP and u reversed we have
(;yN( IP) , -yvu)o = /{ dnx \71P(x) · \7u(x) + (6.1P, u )L2(!1 ;dnx) > .n
(2.52)
so matters are reduce to showing that
in �X \71P(x) · \7u(x)
=
- ( IP , 6.u)L2(!1;dnx) ·
(2.53)
Nonetheless, this is a consequence of Green's formula (2.28) written for the vector field G = �\7u (which has the property that 'YvG = 0). In summary, the operator (2.48), (2.49) is well-defined, linear and bounded. Next, we will show that this operator is compatible with (2.40), (2.4 1 ) . After re-denoting ;y by ;:;N, then this becomes the extension of the weak Neumann trace n operator, considered in (2.43). To this end, assume that u E H1 (0.) has 6.u E L 2 (0.; dn x). Our goal is to show that (2.54)
F.
114
GESZTESY AND M. MITREA
for every ¢ E H1 (80.) or, equivalently,
for
E
ln cf'x 'V(x)
·
'Vu(x)
=
(7N(iP ) , �fDu)o - ( A
0 there exists a > 0 ({3(t:2.5. {3(10)LEMMA ) Assume ( 1/t:)) such that 0 ll fvu lli2(8!1;dn-lw ) ciiVu lli2(!1; dnx ) n + {3(t:) llu111, 2(0 ; dnx) for all E H 1 (0). ( 2.65 ) PROOF. Since n is a bounded Lipschitz domain, there exists an h E CO' (�n)n 40]) with real-valued components and > 0 such that (cf., [48, Lemma 1 5. 1 . 9 ( 2. 66) ( h)cn � a.e. on an. =
dO
�
U
K
.
v .
Thus,
l l !vull i2(8!1;dn-lw)
,
p.
/'\,
ddx � ( in d x
� r o n lw ( h) cn l {vul 2 "' la = � r n div (l u l 2 h) , "' lo = n (V I u l 2 , ) c n lu l 2 div �
v .
(h) ). u E H1 (0), (2.67) using the divergence theorem in the second step. Since for arbitrary t: > 0, l2uVu l � c i Vu l 2 + (1/t:) l ul 2 , u E H 1 (0), ( 2. 6 8 ) and h E C0 (�n)n , one arrives at ( 2. 6 5 ) . Next we describe a family of self-adjoint Laplace operators -L\e,o in L 2 (0 ; dn x) indexed by the boundary operator 8. We will refer to as the generalized h +
0
-L\e,o
Robin Laplacian.
THEOREM defined2.by6 . Assume Hypothesis 2.2. Then the generalized Robin Laplacian, ( 2. 69) { E H 1 (0) I L\u E L2 (0 dnx); (-;:;N + e,D )u 0 in H- 112 (80) } , is self-adjoint and bounded from below in L2(0; dnx). Moreover, dom ( l - L\e, o l 1 /2 ) H 1 (0). (2 . 70) PROOF. We introduce the sesquilinear form ( , ) with domain H1 (D) H1 (0) by ( ) ( ) + ( !vu , e{v v ) , u , v E H 1 (0), ( 2.71 ) 1 12 where ( , ) on H 1 (D) H 1 (D) denotes the Neumann Laplacian form ( ) k dn ( Vu) ( ) ( V v)( ) , u, v E H 1 (0). ( 2.72 ) -L\e, o, - L\e , o -L\, dom( -L\e,o) = u =
;
=
=
a-�e. n u, v
a-�o . n
·
=
a _�9 n
a-�o. n u, v
·
a-�o.n u, v
x
=
x
x
·
x
·
·
x
1 16
F. GESZTESY AND M. MITREA
One verifies that
a-� e r .
J · , · ) is well-defined on H1 (D)
B(H1(D), H1 1 2 (8D))
x
H1 (D) since
1o E , e E B(H1 12 (8D) , H -112 (8D ) ) , (8 + (1 - ce)Ian) 1 12 E B(H1 12 (8D ) , £2(8!1; d"-1w) )
(cf. (B.43)). This also implies that
(8 + ( 1 - ee)Ian)1 121o
E
(2.73) (2.74) (2.75)
B(H 1 (!1) , £2 (8!1 ; dn-1w) ) .
Employing (2.1) and {2.2), a-�e.n is symmetric and bounded from below by ce . Next, we intend to show that a-� e . n is a closed form in L2 ( D; d"x) x £2 (!1; d"x) . For this purpose we rewrite (lo U , elo v)l/ as
( lo U, elvv\/2 =
(2 . 76)
2
(( 8 + ( 1 - ce)Ian)1121ou, (8 + ( 1 - ce)Ian ) 1121ov) £2( an ;d"-1w)
- (1 - ce) (lo u , lvV)£2 (80 ; d" - 'wJ •
u, v E H1 (!1) ,
(2.77)
(cf. (B.31), (B .32) ), and notice that the last form on the right-hand side of (2.77) is nonclosable in L2( D; dnx) since ID is nonclosable as an operator defined on a rlem;e subspace from £2(!1; dnx) into £ 2( 8!1; dn - lw) (cf. the discussion in connection with (B.44) ) . To deal with this noncloseability issue, we now split off the last form on the right-hand side of (2.77) and hence introduce
b_ �e.n (u, v) = (V'u, V'v) £2( 0-; d"x)" + ((8 + (1 - ce)Ian)1121ou, (8 + (1 - ce)Ian) 1121vv) £2(CJO;d"-'w) + db( u , v)L2(0;d"x) • u, v E H1 (D) , (2.78)
for db > 0. Then due to the nonnegativity of the second form on the right-hand side in (2.78) , b_�e.n is H1(!1)-coercive, that is, for some c > 0, where
llull � � (n}
=
1 b_�H u ( u, u) � cl ll u ll�l(O) • IIY'ulll2(!1;d"x}" + llulll2 ( fl; d" x) ' Next, we note that by
I ( (8 + ( 1 - ce)Ian) 1121ou, (8 + ( 1 - ce )Ian ) 112 !'oV) �
£2 (aO;d"-'w)
(2.79) (2.76),
I
li Ce + (1 - ce)Ian) 1 12 1o II �(Hl (n),P (an;d"-' w)) llu iiHl( n) llv i iHl en > , (2.80) u, v E H 1 ( !1). Since trivially, I I V' ull i2(0; d"x) + db ll u ii1,2(0;d"x) � cllu ll��(o) for some c > 0, one infers that b_�e.n i::; also H1 (D ) -bounded, that is, for some c2 > 0, (2.81) b_ �e .o ( u, u) � c2 llull�1 (!1)· Thus, the ::;ymmetric form b_ �e.n is H1 (D ) -bounded and H1 (!1)-coercive and hence densely defined and closed in L 2 ( !1 ; dnx) x L2(D; dnx) by the discussion following
(B.46). To deal with the nonclosable form bvu , /'o v)L2(80 ;d"- l w) > u , v E H1 (D), it suffices to note that by Lemma 2.5 this form is infinitesimally bounded with respect to the Neumann Laplacian form a-�o . o on H1 ( D) x H1 (!1), and since the form
ROBIN-TO-DIRICHLET MAPS AND KREIN-TYPE RESOLVENT FORMULAS
11 7
( ( 8 + 1 - ce) 1 1 2 rDu, ( 8 + 1 - ce ) 1 12 rDV) L 2 ( an;dn - l w) ' u, v E H1 (0) , is nonnegative, the form (/D u, rn v ) u (an dn - l w ) is also infinitesimally bounded with respect to the ;
form L t:. e , n . By the discussion in connection with (B.48) , (B.49) , the form a-t:. e, n (possibly shifted by an irrelevant real constant) defined on H1 (0) x H 1 (0) , is thus densely defined in L2(0; dnx) x L2 (0; dnx), bounded from below, and closed. According to (B.34) we thus introduce the operator - .6. e , n in L2 (0; dnx) by dom( - .6.e ,n)
=
{ v E H 1 (0) I there exists an Wv E L2 (0; dnx) such that
L dn x \i'w \i'v + ( /DW, e/DV )
1 /2
- .6.e , nu = Wu , u E dom( - .6.e,n).
=
}
L � X WWv for all w E H 1 (0) , (2.82)
By the formalism displayed in (B.20)-(B.43) (cf. , in particular (B.27)), self-adjoint in L 2 (0; dnx) and (2. 70) holds. We recall that
- .6. e,n
{u E H1 (0) l rDU = 0 on 80}.
(2.83)
L dnx v .6.u for all v E cgo (O), and hence Wu = - .6.u in
V'(O),
HJ (O)
=
Taking v E C0 (0) '---' H{j (O) '---' H1 (0) , one concludes
L dn X VWu
=
-
is
(2.84) with = C0 (0)' the space of distributions in 0. Next, we suppose that u E dom( -.6. e,n) and v E H1 (0) . We recall that : /D H1 (0) -> H112 (80) and compute
V'(O)
L dnx \i'v \i'u =
= =
-
L dnx v .6.u + (/D v , 1N u h; 2
L dn X VWu + (!DV, (1N + e,D )u \ / 2 - (rDV, e,D u \ / 2 L �x \7v \7u + (rDV, (1 N + EhD ) u) 11 , 2
(2.85)
where we used the second line in (2.82). Hence,
(2.86) Since v E H1 (0) is arbitrary, and the map /D : H1 (0) onto, one concludes that
->
H112 (80) is actually (2.87)
Thus,
dom( - .6.e,n)
�
{v E H1 (0) j .6.v E L2 (0; dnx) ; (1N + SrD ) v = 0 in H - 1 12 (80) } .
Finally, assume that u E {v E H1 (0) j .6.v E L2(0; dnx ) ; (1N + 8-·tn)v w E H1 (0) , and let Wu = �.6.u E L2 (0; dn x ) . Then,
L dnx wwu
= =
-
L dn x w div(\i'u)
L dnx \i'w \i'u - (ro w , 1N uh;z
(2.88) 0} ,
=
F. GESZTBSY AND M. MITREA
118
(2.89)
Thus, applying (2.82), one concludes that u E dom( -Lle , n) and hence dom( -Lle ,n) d { v E H1 (fl) I Llv E L2(fl; cf'x ) ; (::YN + e'Yn ) v = 0 in H-112 (80) } , (2. 90) finishing the proof of Theorem 2.6. 0 COROLLARY
2.7. Assume Hypothesis 2.2. Then the genemlized Robin Lapla
-� e , n , has purely discrete spectrum bounded from below, in particular, aess ( -b..e ,n) 0 . (2.91) PROOF. Since dom (l .6. ,n l 1 12 ) = H1 (n) , by (2.70), and H1 (fl) embeds com
cian,
=
e
pactly into L2 (fl; dnx) (cf. , e.g. , [37, Theorem V.4.17]), one infers that (-.6.e,n + In)-112 E B00 (L2(fl; dnx) ) . Consequently, one obtains ( -Lle,n + In) -1 E B00 (L2 (fl; dnx)), (2.92) 0 which is equivalent to (2 .91). -
The important special case where 8 corm'ipouds to the operator of multipli cation by a real-valued, essentially bounded function () leads to Robin boundary conditions we discuss next: COROLLARY 2.8. In addition to Hypothesis 2.1, assume that e is the oper
l
ator· of multiplication in L 2 (8D.; dn w) by the real-valued f1Lnction () satisfying () E L=(aD.; dn -1w). Then 8 satisfies the conditions in Hypothesis 2.2 resulting in the self-adjoint and bounded from below Laplacian - b.. fl ,n in L2 (0; d" x) with Rubin boundary conditions on 80 in (2.69) given by -
PROOF.
(::YN + B'Yn ) u = 0 in H-112 (8!1).
( 2.93)
H1 (fl),
(2.94)
By Lemma 2.5, the sesquilinear form ('Yo u, B'Yvvh;2 ,
u, v E
is infinitesimally form bounded with respect to the Neumann Laplacian form a -c.o n · By (B.48) and (B .49) this in turn proves that the form a-c.e . n in (2.71 ) is clos�d and one can now follow the proof of Theorem 2.6 from (2.82) on, step by step. 0 REMARK 2.9. ( i) In Lhe case of a smooth boundary afl, the boundary conditions in (2.93) are also called "classical" boundary conditions (d. , e.g., [91] ); in the more general case of bounded Lipschitz domains we also refer to [6] and [102, Ch. 4] in this context. Next, we point out that, in [62] , the authors have dealt with the case of Laplace operators in bounded Lipschitz domains, equipped with local boundary conditions of Robin-type, with boundary daj;a-in LP(8fl; dn 1 w ) and produced nontangential maximal function estimates. r'For the case p = 2, when onr setting agrees with that of [62], some of our rest'l.Jts in this section and the following are a refinement of those in [62] . Maximal ))•.:: regularity and analytic contraction semigroups of Dirichlet and Neumann Laplacians on bounded Lipschitz domains were studied in [106]. Holomorphic C0-semigroups of the Laplacian with Robin boundary conditions on hounded Lipschitz domains have been discussed in [1 03] . Moreover, Robin boundary conditions for elliptic boundary value problems on arbitrary open domains were first studied by Maz'ya [67] , [68 , Sect. 4.11.6], and subsequently in [29 ] (see also [30] which treats the case of the Laplacian). In -
,
ROBIN-TO-DIRICHLET MAPS AND KREIN-TYPE RESOLVENT FORMULAS
1 19
addition, Robin-type boundary conditions involving measures on the boundary for very general domains n were intensively discussed in terms of quadratic forms and capacity methods in the literature, and we refer, for instance, to [6] , [7] , [17] , [102] , and the references therein. In the special case () 0 (resp., § = 0), that is, in the case of the Neumann Laplacian, we will also use the notation (2.95) The case of the Dirichlet Laplacian -D.v,n associated with n formally corre sponds to e = 00 and so we isolate it in the next result: 2.10. 2.1. -D.v,o ,
(ii)
=
THEOREM defined by
Assume Hypothesis
- D.v,o = - D. , dom( - D.v,o) =
Then the Dirichlet Laplacian,
0 in H112 (an) } { u E H1 (D) I D.u E L2 (D; dn x) (2.96) { u E HJ (D) I D.u E L2 (D; �x) } , is self-adjoint and strictly positive in L2(D;dnx). Moreover, ; "fD U
=
=
(2.97)
PROOF. We introduce the sesquilinear form av,o( · , · ) on the domain HJ (D) a v ,o (u , v) l �x ('Vu)(x) ('Vv)(x), u, v E H6 (D). (2.98) x
HJ(D) by
=
Clearly, av ,o is symmetric, nonnegative, and well-defined on HJ (D) x HJ (D). Since n is bounded, that is, /D/ < oo, HJ (D)-coercivity of av,o then immediately follows from Poincare's inequality for HJ (D)-functions (cf., e.g., [105, Theorem !.7.6] ) . Next we introduce the operator -D.v,o in L2(D; by
dnx) E L2 (D;dnx) such that for all w E HJ (D) }•
{ v E HJ (D) I there exists an l dnx'Vw'Vv l (2.99) u E dom(-D.v,o). - D.v , ou = By the formalism displayed in (B.1)-(B.19), - D.v,o is self-adjoint in L2(D; dnx) and (2.97) holds. Taking v E C0 (D) HJ (D), one concludes l In x v D.u in D'(D) and hence -D.u in D'(D) . (2.100) Since v E H{j (D) if and only if v E H1 (D) and -y0v = 0 in H1 12 (8D) (cf., e.g., (48, Corollary 1.5.1.6 ]), and v E dom( - D.v,o) implies D.v E L2(D; dnx), one computes for u E dom( -D.v,n) and v E HJ (D) that (2.101) l dnx'Vv'Vu = - In dnx vD.u = l �xvwu. Thus, - D.u E L2 (D; dnx) and hence dom( -D.v,o) { v E HJ (n) I D.v E L2(D; � x ) } . (2.102) Wv
dom ( - D.v , n ) =
=
Wu ,
� X WWv
S, ,f!> In L2 ( UH j dn W) . •
ROBIN-TO-DIRICHLET MAPS AND KREIN-TYPE RESOLVENT FORMULAS 125 In this section we strengthen Hypothesis 2.2 by adding assumption (3.1) below: HYPOTHESIS 3.1. 2.2 (3.1) We note that (3.1 ) is satisfied whenever there exists some > 0 such that e (3.2) We recall the definition of the weak Neumann trace operator ::YN (2.40), (2.41) and start with the Helmholtz Robin boundary value problems: THEOREM 3.2. 3.1 -D.e )
In addition to Hypothesis suppose that E B(H1-c(an), L2 (an; rr-1 w)).
c
m
Assumedn-Hypothesis and generalized suppose thatRobin z E boundary C\tr ( value n 1w), the following Then for every g E L2(an; problem, { (-D. - _: )u = 0 in n , u E H312(n), (3.3) (::YN + 8rD)u = g on an, has a unique solution u ue. This solution ue satisfies /DUe E H 1 (an), ::YN ue E L2(an;dn- 1 w), (3.4) llrD ue iiH1 (80) + II ::YN ue ll£2(8!1;d"- 1w) • Cllgll£2(8!1;d" - 1w) and (3.5) ll ue ii H3/2(!1) Cllgll£2(8!1;d"- lw) • for some constant constant C = C(8, n, z) > 0. Finally, (3.6) (rD ( -D.e,n - zln) - 1 ) * E B(L 2 (an; rr - 1 w), H 312 (n)), and the solution ue is given by the formula (3.7) ue = (rD (-D.e,n - Z/n)- 1 ) * g. PROOF. It is clear from Lemma 2.3 and Lemma 2.4 that the boundary value problem (3.3) has a meaningful formulation and that any solution satisfies the first line in (3.4). Uniqueness for (3.3) is an immediate consequence of the fact that zcandidate E C\cr( -D.e,n). As for existence, as in the proof of Theorem 2.17, we look for a expressed as (3.8) u(x) = (Szh)(x), x E for some h E L2(an; dn-1w). This ensures that u E H312(n) and ( - D. - z)u 0 in n. Above, the single layer potential Sz has been defined in (2.121). The boundary condition (::YN + SrD )u = g on an is then equivalent to ,
.
=
:=:::
:=:::
f2
=
(3.9)
respectively, to
(3.10) (-� Ian + Kff) h + SrDSzh = g . Here Kff has been defined in (2.122). To obtain unique solvability of equation (3.10) for h E L2(an;dn- 1 w), given n- 1 w), at least when z E C\D, where D C is a discrete set, we gproceed E L2(an; d in a series of steps. The first step is to observe that the operator in question is Fredholm with index zero for every z E C. To see this, we decompose C
(3.11)
126
F.
GESZTESY AND M. MITREA
E 800(L2(8r!;dn-lw)) L2(8fl; dn-lw)
and recall that (Kff - Kt') (cf. Lemma D.3) and that with Fredholm index equal -� Ian + K! is a Fredholm operator in to zero as proven by Verchota [101]. In addition, we note that (3 .12) which follows from Hypothesis 3.1 and the fact that the following operators are bounded L 2 r! ; cf!'-1w) ---+ { u �u E (3.13)
Sz : (8 E H312(r!) I L2(r!; dnx)}, '"'/D : {u H312(n) 1 �u E L2(r!; dnx)} ---+ H1(8n), H312(Q) I �u E L2(r!; dnx)} i n (- �Ian Kff) EhvSz L2(8r!; d"-1w) (-�Ian Kff) + �hvSz L2(8Q; dn-1w). E
(where the space {u E is equipped with the natural graph norm u �---+ llu i H3/2 ( ) + l l �ull£2 (f!;d"x) ) · See Lemma 2.3 and Theorem D.7. + Thus, + is a Fredholm operator in with Fredholm index equal to zero, for every z E C. In particular, it is invertible if and only if it is injective. In the second step, we study the injectivity of on + For this purpose we now suppose that
Introducing w =
one then infers that w satisfies
Szk n {((;:.;-�N -_:)w in
6'"'!D )w
+
=
0 in 0 On of!.
r!, w E H3f2(f!),
(:3.15)
=
Thus one obtains, 0S
1 dnx
l'iwl 2 =
!!
= =
t J dnx Oj WOj'W - r dn x Aww t r cr - 1w ('"'!v 8pD) ln j= 1 lan � in dnx lwl 2 + bvw, '1N w)L2(8D;dn-lw) r d"x lwl2 ('YD'W, ;:.;N wh/2 zjn d"x lwl2 - ('"'!ow, Ehvw)112 . j= l n /-
1/j'"'/DW
+
=Z =
+
(3.16)
At this point we will first conHider the case when z E C\:IR (so that, in particular, Im(z) f. 0). In this scenario, recalling (2 .1) and taking the imaginary parts of the two most extreme sides of (3.16) imply that. fn d"x lwl2 = 0 and, hence, = 0 in
w
n.
To continue, let ;:.;'f;t and '"'lrr denote, respectively, the Neumann and Dirichlet traces for the exterior domain Also, parallel to (2 .121), set
Rn\n.
ROBIN-TO-DIRICHLET MAPS AND KREIN-TYPE RESOLVENT FORMULAS
127
where g is an arbitrary measurable function on 80. Then, due to the weak singu larity in the integral kernel of Sz , ext s x , z9 /D Sz 9 = rD e t
(3.18)
whereas the counterpart of (2.123) becomes �ext s z fN ext , 9 - ( 2l J80 + _
K#z ) 9
(3.19) Compared with (2.123), the change in sign is due to the fact that the outward unit normal to !Rn\f! is Moving on, if we set wext (x ) (Sext , z k ) (x ) for x E !Rn\f!, then from what we have proved so far. -v
.
=
(3.20)
Fix now a sufficiently large R > 0 such that !1 c B (O; R) and write the analogue of (3.16) for the restriction of wext to B (O; R) \D: dnx l wext l 2 (r'ir wext , ;y'fytwext h dnx I Y'wext l 2 = z f f 12 jB(O;R)\0_ jB (O;R)\0_ . V'wext (O. (3.21) f dn -l w(�) wext (�) i_ �� � jl f. I == R _
_
In view of (3.20), the above identity reduces to dnx IY'wext l 2 f jB (O;R)\0_ dnx l wext l 2 z r jB(O;R)\0 =
-1
dn - l w(�) wext (�) i_ V'wext (�) . �� � l f. I = R 0
(3.22)
Recall that we are assuming E C\R Given that, by ( C.17) ( and the comment following right after it ) , the integral kernel of Sext ,z k has exponential decay at infinity, it follows that wex t decays exponentially at infinity. Thus, after passing to limit R oo, we arrive at z
_,
f dnx IY'wext 1 2 = z f dnx l wext 1 2 . }H{n\f! }H{n \f!
(3.23)
-
-
Consequently, taking the imaginary parts of both sides we arrive at the conclusion that wext 0 in !Rn\f!. With this in hand, we may then invoke (2.123), (3.19) to deduce that (3.24) w we t given that , x vanish in 0 and !Rn\f!, respectively. Hence, one concludes that 0 in £2 (80; �-1w). This proves that the operator ( 1Iao + + erDSz is injective, hence, invertible on £2 (80; dn - 1w) whenever E C\R In the third step, the goal is to extend this result to other values of the param eter To this end, fix some z0 E C\IR, and for E C, consider (3.25) =
k
-
=
z
z.
Kff')
z
Observe that the operator-valued mapping Az E B (£2 (80; dn - 1w)) is analytic and, thanks to Lemma D.3, Az E Boo (L2 (80; dn- l w)) . The analytic Fredholm z �----+
128
F. GESZTESY AND M. MITREA
er
theorem then yields inv tibility of Thus,
I + Az
cep for
z
a
e t in discrete x
set D
C
C.
(3.26)
for z E where D is a discrete which, the inv rt bil ty esult roved in the previous paragraph, contained n R The above argument proves unique solvability of (3.3 ) for z E C\D, where D is a. di rete subset of JR. The (3.8) the fact that
isp invertible C\D is seti by e i i r sc representation and "YDSz : L2(80; dn- l w) --> H1 (80 ) then yields rDue H 1 (80) . Moreover, fN 'U.e = fNSzh = (-�Ian + KfF ) h L2 (80; dn-Iw) B(L2 (80; d"'- 1 w)). This ov z C\D , n t al estimate representati o m a om le b C\(D U a(-�e,n)) along two functions,on L2 (80; dn-lw ) . ue l v L2 ( D; dnx) and boundedly,
(3.27)
by (2.123) and (3.8) ,
E
E
(3.28)
sinee by (2. 124) , K'ff E pr es (3.4) when E C\D. For z E the a ur (3.5) is a consequence of the integral f r ula (3.8) and (D.28) . Next, ftx c p x num er z E with Also, let E so ve (3.3) . One computes g E
(ue , v ) D2 (D;dnx) = (ue , (- � - z) ( - �e .o - zln)-Iv) £2(!1;d" x) = (( -� - z)ue , ( - �e. o - Zlo) - 1 v) £2(1l;d"x)
+ (::YNue, /D(-�e,n - Zfn)-1v) £2(a!1 ;dn-Iw)
- (!Due, ::YN ( - �e . n - Zior 1 v) 1 12 (::YN ue, ID ( -�e,n - Zln) -1v ) L2(&!1;dn-Iw) + (rDue, erD ( -�e,!1 - zlo )- 1 v) l/2 = (::YNue, ID ( -�e,n - Zln)-1v) L2(&!1;dn -lw) =
-_t.kfD ( -�e ,!1 - zJn) -l v , G/D Ue )112
/
(= (1Nue, !D ( -�8,!1 - zfo)-1v) L2(8!1;dn- lw) \
"'-+-(EhDue , rD ( -�e.n - zln)- 1 v)112
( (::YN + GrD)ue, /D( -�e.n - zln)-1'v) 1 12
= (('1N + EhD )ne, ID ( -�e.o - Zin)-1v ) L2 (o!1:dn-'w) = (9, !D( -�e.n - Zln) -1v)L2(o!1 ;dn-lw) (3.29) = ((rv (-�e.n - zln) -1rg , v)L2(n;dnx) ' =
Since v
E
L2(0; �x) wa
s
arbitrary,
ue = (rD ( -�e. o - Zlo)-1 r g
this yields in L2(0; �x),
for
z E C\(D U a( -�e.n)),
(3.30)
which proves (3.7) for z E C\(D ua( - � e .n)). From this and (3. 5), the membership (3.6) also follows when z E C\(D U a( -�e.n)). The extension the more general case when z E C\a( -�e . n) then done by resorting to analytic continuation with respect to z. specifically, fix Zo E
to
is
More
ROBIN-TO-DIRICHLET MAPS AND KREIN-TYPE RESOLVENT FORMULAS 129 C\a(Ae,o) . Then there exists r > 0 such that B ( zo, r) n a( -Ae,o) (3.31) ( B(zo , r) \ { zo } ) D n
=
0,
=
0,
since D is discrete and a( -Ae,o) is closed. We may then write
hv ( - Ae,o - zo lo)- 1 ]* = � J 2 rrt Jc(zo;r)
dz (z - zo)- 1 hv ( -Ae,o - Zlo)- 1 ] * (3.32) as operators in B(H - 1 12(80), L2(D; dnx)), where C(z0; r) C denotes the coun terclockwise oriented circle with center zo and radius r. (This follows from du alizing the fact 'YD( - Ae , o - zo lo ) - 1 E B(L2(D; dnx), H 1 12(8D)), which in turn follows from the mapping properties (-Ae,o - z0 Io) - 1 E B(L2(D;dnx), H 1 (8D)) and 'YD E B(H 1 (8D),H 1 12(8D)). ) However, granted (3. 3 1), what we have shown so far yields that hv ( - Ae,o - zlo)- 1 ] * E B(L2(8D; � - 1 w), H312(D)) whenever lz - zo l r, with a bound I hv ( -Ae,o - zlo)- 1 ]* II B(L 2 (80;dn - l w),H3/2(f!)) c C(D, Zo, r) (3. 3 3) independent of the complex parameter z E 8B (z0 , r). This estimate and Cauchy's representation formula (3. 3 2) then imply that hv(-Ae , o - zolo)- 1 ]* E B(L2 (8D;dn - 1 w),H312 (D)). (3. 34) This further entails that u hv ( -Ae,o - zo lo ) - 1 ]* g solves (3. 3 ), written with zo in place of z, and satisfies (3. 5 ). Finally, the memberships in (3. 4 ) ( along with natu rally accompanying estimates) follow from Lemma 2. 3 and Lemma 2. 4 . This shows that (3. 6 ), along with the well-posedness of (3. 3 ) and all the desired properties of the solution, hold whenever z E C\a(Ae,o). The special case 8 0 of Theorem 3.2, corresponding to the Neumann Lapla cian, deserves to be mentioned separately. OROLLARY 3 . Assume 2.1 andNeumann supposeboundary that z E value C\a( -AN,n). ThenCfor every g E3.L2(8D; dn- 1Hypothesis problem, w), the following { ;;;(-Au =- gz)uon 80,0 in D, u E H312(D), (3.35) N has a unique solution u UN . This solution UN satisfies 'YDUN E H 1 (8D) and lbv uN I I H'(80) + ll;yN uN ii £2(80;dn-lw) C II 9 II L2(80;dn - 1 w) (3.36) as well as l i uN II H3f2(f!) Cllg l i £2(80;dn-1w) > (3.37) for some constant constant C C(8, n, z) > 0. Finally, ['Yv ( -AN,o - zlo)- 1 ] * E B(L 2 (8D; dn - 1 w), H 312 (D) ) , (3.38) and the solution UN is given by the formula (3. 39) ue ('Yv ( -AN,O - zlo)- 1 ) * g. Next, we turn to the Dirichlet case originally treated in [46, Theorem 3.1] but under stronger regularity conditions on D. In order to facilitate the subsequent considerations, we isolate a useful technical result in the lemma below. c
=
:::;
e,
=
=
0
=
=
=
::=;
::=;
=
=
1 30
F.
LEMMA
GESZTESY AND M. MITREA
3.4. Assume Hypothesis 2.1 and suppose that z
( -.6-D, n - zln)-1 : L2(D ; dnx)
-.
E
C\a( -.6-D,n). Then
{ u E H312 (D ) 1 .6-u E L2 (D ; �x) }
(3.40)
is a well-defined bounded operator, where the space {u E H312 (D) 1 .6-u E L2(D; dnx)} is equipped with the natural graph norm u r--> ll u iiH i 2 (fl) + ll.6.u ll£2 (f!;d"x) · PROOF.
Consider
z
zln ) -1 f. It follows that
E ·u
3
C\a ( -.6-D,n ), f E L2 (D; dnx) and set w = ( -.6-D,n is the unique solution of the problem
(-.6. - z)w = f in D,
w E HJ (D).
(3.41)
The strategy is to devise an alternative representation for w from which it is clear that w has the claimed regularity in D. To this end, let j denote the extension of f by zero to IRn and denote by E the operator of convolution by En (z; ) . Since the latter is smoothing of order 2, it follows that v = (Ei) I n E H2 (D) and ( -.6. - z ) v = f in !'2. In particular, g = -"(DV E H1 (aD). We now claim that the problem ·
(-.6. - z )u
=
0 in D,
u
E
H312 (D) ,
rDU = g on a!t,
(3.42)
has a ::;olution (satisfying natural estimates). To see this, we look for a solution in the form (3.8) for some h E L2 (aD; dn- lw) . This guarantees that u E H312 (D) by Theorem D.7, and (-.6. - z)u = 0 in D. Ensuring that the boundary condition holds comC'.s down to solving rDSzh = g. In this regard, we recall that (3.43)
(cf. [101]) . With this in hand, by relying on Theorem D.7 and arguing as in the proof of Theorem 3. e can show that there exists a discrete set D C C such that
21£u \
rDSz : L2 (a!t;'cr+-'1 w) -. H1 (aD) is invertible for z E C\D.
(3.44)
Thus, a solution of (3.42) is given by (3.45) Moreover, by Theorem D. 7, this satisfies ll u iiH3/2 (f!J � C (D , z ) ll g iiHl(an ) � C (n , z ) II ! IIP(n; d"xl •
z
E
C\D.
(3.46)
Consequently, if z E C\(D U a( - .6- D,!J ) ) , then u + v solves (3.41 ). Hence, by uniqueness, w = u + v in this case. This shows that w = ( -.6-D ,n - zln)- 1 f belongs to H312 (n) and satisfies .6.w E £2 (.12; d" x) with
ll w ii H3/2 (0) + ll.6.w ll £2co;d...
x
)
� C(D, z ) II ! II L2(fl;dnx) >
z E C\(D U
a
( - .6- D ,n ) ) .
(3.4 7) In summary, the above argument shows that the operator (3.40) is well-defined and bounded whenever z E C.\(D U a(-.6-D.n)). The extension to z E C.\a( - .6.D , n ) is then achieved via analytic continuation (as in the last part of the proof of Theorem 0 3.2). Having established Lemma 3.4, we can now readily prove the following result.
ROBIN-TO-DIRICHLET MAPS AND KREIN-TYPE RESOLVENT FORMULAS 131 LEMMA 3.5. Assume Hypothesis 2.1 and suppose that z E C\a( -�v.n) . Then (3.48) :YN ( -�v,n - zln) - 1 E B(L 2 (n; dn x), L 2 (an; dn- 1 w) ) , and [:YN ( -�v,n - zin)- 1 ] * E B(L 2 (an;dn - 1 w),L 2 (n;dn x) ) . (3.49) PROOF. Obviously, it suffices to only prove (3.48) . However, this is an imme0
diate consequence of Lemma 3.4 and Lemma 2.4.
We note that Lemma 3.5 corrects an inaccuracy in the proof of [46, Theorem 3.1] in the following sense: The proof of (3.20) and (3.21) in [46] relies on [46, Lemma 2.4] , which in turn requires the stronger assumptions [46, Hypothesis 2.1] on than merely the Lipschitz assumption on However, the current Lemmas 2.15 and 3.5 (and the subsequent Theorem 3.6) show that (3.20) and (3.21) in [46], as well as the results stated in [46, Theorem 3.1] , are actually all correct. After this preamble, we are ready to state the result about the well-posedness of the Dirichlet problem, alluded to above. 3.6. 2.1 z C\a(-�v,n).
n
n.
and suppose 1 (an), theHypothesis ThenTHEOREM for every f E HAssume following Dirichlet boundarythatvalueE problem, { (-� =- fz)uon an,0 in n, u E H3f2(n), (3.50) /DU has a unique solution u = uv . This solution uv satisfies :YN uD E L2 (an; � - 1w) and I !:YN uv ll u( an;d"-lw) Cv ll f i!Hl(Bf!) , (3.51) for some constant Cv Cv (n , z) > 0. Moreover, (3.52) llu v ii Ha/2 ( !1) :::; Cv llf ii H1 ( &n )· Finally, (3.53) [:YN ( -�v.n - zin)- 1 ] * E B(H 1 (an), H312 (n) ) , and the solution uv is given by the formula (3.54) uv [:YN ( -�v,n - zin)- 1 ] * f. PROOF. Uniqueness for (3.50) is a direct consequence of the fact that z E Existence, at least when E C\D for a discrete set D C, is implicit =
:::=:
=
=
-
C\a( -� v, n).
z
c
in the proof of Lemma 3.4 ( cf. (3.42)) . Note that a solution thus constructed obeys (3.52) and satisfies (3.51) (cf. Lemmas 2.3 and 2.4). Next, we turn to the proof of (3.54). Assume that z E C\(D U a( -� v , n)) and denote by uv the unique solution of (3.50). Also, recall (3.48)-(3.49). Based on these and Green's formula, one computes
(uv , V) £2(!1;d"x)
= =
= =
(uv, ( -� - z) ( -�D,f! - zfn)- 1 v) L2 (f!;dnx ) (( - � - z)uv, ( -�v,n - zin)- 1 v) L2(f!;dnx) (:YN uD, /D ( -� D,f! - zfn )- 1 v)U( 8!1;dn-1w) - ( !v uv, :YN (-�v,n - Zin)- 1 vh; 2 :YN ( -�D,O - ( (:YN ( -�D ,f! - Zfn) - 1 r V) L2(f!; dnx)
+ -(!,
zfn)- 1 v) l/2 f,
(3.55)
1 32
F. GESZTESY AND M. MITREA
for any v E L2 (n; d"x) . This proves (3.54) with the operators involved understood in the sense of (3.49) . Given (3.52), one obtains (3.53) granted that z E C\(D U
a ( -�o . n )) .
Finally, the extension of the above results to the more general case in which C\a( -�o .o ) is done using analytic continuation, as in the last part of the proof of Theorem 3.2. 0 z
E
Assuming Hypothesis 3.1, we introduce the Dirichlet-to-Robin map associated with ( -� - z) on n, as follows,
coJ n ( z) .. MD,e,
{
H 1 (an) __, L2 (an; d" -1w) , f ...... - (:YN e'YD ) uD,
where uo is the unique ::;olut.ion of
( -� - z ) u
=
0 in 0,
+
M�?e , o (z)
z E C\a ( -� o.o ) ,
u E H312 (n), /D 'U
=
f on
an.
(3.56)
(3.57)
Continuing to assume Hypothesis 3.1, we next introduce the Robin-to-Dirichlet map M�� b . o (z) associated with ( - � - z) on n, as follows, Af(o) "�
e , D,n (z)
where
ue
· .
{
£2 (an ; dn-lw) --> Hl (an) , 9 ...... , o u e ,
z E \1...""'\a ( -.u.e ,n ) ,
(3.5 8)
n, u E H312 (n), (:YN + 8--yo ) u = g on an.
(3. 59)
�
"
is the unique solution of
( -� - z)u
0 in
We note that Robin-to-Dirichlet maps h_51;ve abo been studied in [10] . We conclude with the following th rem, one of the main results of this paper: =
,O
THEOREM 3.7. Assume Hypothesis\i.J.:_/Then
Mg�,0(z) E B(H1 (an) , L2 (an; dn-lw)) , z E C\a(- �D .n) ,
(3.60)
and M�.� .o ( z ) = (:YN + e'i'D ) [ (:YN + EhD ). ( -�D.n - Zlo )-1] * ' z E C\a( - � , o ) . o
(3.61)
Moreover,
M��b, n Cz) E B(L2(an; dn - 1w) , H1 (an)) , z E C\a ( - �e .o ) ,
(3.62)
and, in fact, In addition,
M�b .n Cz) = rD ['i'D (-�e.o - Zlo ) - 1 r, z E C\a( -�e.n). Finally, let z E C\(a( -� D ,n) U a( -�e,n)). Then MS(O)D O (Z ) = - MD(O)S :O (Z ) - 1
(3.64) (3.65)
PROOF. The membership in (3.60) is a consequence of Theorem 3.6. In this context we note that by the first line in (2.96) , ID ( � o n - zln ) - l = 0, and hence ,
J
'
·
- .
'UD = - [:YN ( -�D,O - zJn)-l r f = - [ (:YN + e')'D ) ( - �D,f! - zfo ) - 1 ] * f
(3.66)
ROBIN-TO-DIRICHLET MAPS AND KREIN-TYPE RESOLVENT FORMULAS 133 by (3.54). Moreover, applying - (;yN + e"Yv ) to uv in (3.54) implies formula (3.61) . Likewise, (3.62) follows from Theorem 3.2. In addition, since H 1 (80) embeds compactly into L2(80;dn- 1 w) (cf. (A.10) and [72, Proposition 2.4] ) , M��b,n (z), z E C\a( - �e.n), are compact operators in £2(80; dn-lw), justifying (3.63). Ap plying "YD to ue in (3.7) implies formula (3.64). There remains to justify (3.65). To this end, let g E £2(80; dn- 1 w) be arbitrary. Then
M�_le , n (z) M��b .n (z) g = M�_le,n (z)"Yvue = - (;yN + B"Yv ) uv, (3.67) f = "Yvue E Here ue is the unique solution of ( -� - z)u = 0 with u E and (;yN + Ehv)u = g, and uv is the unique solution of (-� - z)u = 0 with u E and "YDU = f E Since (uv - ue) E and "YDUD = f = "Yvue , one
H1 (80). H312(0)
H312(0)
H1 (80).
concludes
H312(0)
"Yv (uv - ue) = 0 and ( - � - z)(uv - ue) = 0.
(3.68)
Uniqueness of the Dirichlet problem proved in Theorem 3.6 then yields uv = ue which further entails that - (;yN + e"YD ) uv = - (;yN + e"YD )ue = - g . Thus, (3.69) M�.)e,n (z) M��b .n (z) g = - (;yN Ehv) uv = - g , implying M�_le , n (z)M��b.n (z) = -Ian · Conversely, let f E Then (3.70) M��b .n (z)M�_le ,n (z)f = M��b .n (z) ( - (;yN + Ehv)uv) "Yvue,
+
H1 (80). =
and we set
(3.71 ) Here uv, ue E are such that -� - z)ue = ( -� - z)uv = 0 in and "YDUD = (;yN e"Yv)ue = g . Thus (;yN Ehv) (ue + uv) = 0 , ( - � - z) (ue + Uniqueness of the generalized Robin problem uv) = 0 and (uv + ue) E proved in Theorem 3.2 then yields ue = -uv and hence "Yvue = -"Yvuv = -f . Thus, (3.72) Me( 0,)v , n z ) MD(0),e ,n z ) f - "Yvue -
( H312(0) J, + + H312(0). (
0
- J,
(
implying M�0b , n (z) M�� n (z) � -Ian · The desired conclusion now follows. '
'
'
0
3.8. In the above considerations, the special case e = 0 repre sents the frequently studied Neumann-to-Dirichlet and Dirichlet-to-Neumann maps . (0)N, n (z), respectively. That . MN(0,)D , n (z) = Mo(0,D) , n z) and MN( 0,)D ,n z) and Mv,
REMARK (
lS,
M�)N n (z) = M�� n (z). Thus, as a corollary of Theorem 3.7 we have (O ) n ( ) = - MD(O),N,n MN,D, whenever Hypothesis 2.1 holds and z E C\(a( -�v.n) U a ( -�N.n)). '
'
'
'
( Z) - 1 '
Z
(
(3. 73)
REMARK 3.9. We emphasize again that all results in this section extend to Schrodinger operators e , n = -�e.n + dom ( e ,n) = dom ( - �e.n) in for (not necessarily real-valued) potentials satisfying E or more generally, for potentials which are Kato-Rellich bounded with respect to -�e . n with bound less than one. Denoting the corresponding M-operators by
£2(0; dnx)
H
V,
V
H V
V L00(0; dn x),
F. GESZTESY AND M. MITREA
1 34
Mv,N,n (z) and Me , n , n (z), respectively, we note, in particular, that (3.56) -(3.65) extend replacing - � by - � + V and restricting z E C appropriately.
discussion of Weyl-Titchmarsh operators follows the earlier papers [43] and [46] . For related literature on Weyl-Titchmarsh operators, relevant in the context of boundary value spaces (boundary triples, etc. ) , we refer, for instance, to [3], [5] , [1 2] , [13] , [18]-[22] , [32]- [35] , [42] , [44] , [47 , Ch. 3], [49, Ch . 1 3] , [65] , [66] , [11 ] , [80] , [81] , [84], [85] , (88] , [89] , [100] .
Our
4. Some Variants of Krein's Resolvent Formula In this section we present our principal new results, variants of Krein's for mula for the difference of resolvents of generalized Robin Laplacians and Dirichlet Laplacians on bounded Lipschitz domains. We start by weakening Hypothesis 3.1 by using assumption (4. 1) below:
HYPOTHESIS 4. 1 . Jn addition to Hypothesis 2.2 SUPJ!PSe that
e E E= (H 112 (&n ) , H- 112 ( an r) .
We note that condition (4. 1 ) is satisfied if there exi�s. some e E E (H112-" ( 80) , H - 1 12 ( 80) ) .
E
(4. 1 ) > 0 such that
(4.2)
Defore proceeding with the main topic of this section, we will comment to the effect that Hypothcsil:i 3.1 is indeed stronger than Hypothesis 4.1, as the latter follows from the former via duality and interpolation, implying
e E E= (H8 ( 80), H 8 - 1 (80)) , To see this, one first employs the
fact
0 -::; s -::; 1 .
(4.3)
that
for s = ( 1 - B)so + Os 1 , 0 < (} < 1, 0 -::; so, s1 -::; 1, and s0 =/= s 1 1 where ( · )o,q denotes the real interpolation method. Second, one uses the fact that if T : Xj � Yj , j = 0, 1, is a linear and bounded operator between two pairs of compatible Banach spaces, which is = 0, then T E E= ( (Xo, Xl)o,p, ( Yo , Y1 )o,p) every (} E (0, 1). This is a result due to Cwikel [2 7] :
( 4.4)
·,
compact for
for
j
THEOREM 4.2 ( [2 7] ). Let Xj , Yj , j = 0, 1, be two compatible Banach space couples and suppose that the linear operator T : Xi � }j is bounded for j = 0 and compact for j = 1. Then T : (Xo, Xl )o,p � (Yo , Y1 )e,p is compact for all (} E (0, 1) and p E [1, oo] .
(Interestingly, the corresponding result for the complex method of interpolation remains open. ) In our next two results below (Theorems 4.3-4.5) we discuss the solvability of the Di ichlet and Robin boundary value problems with solution in the energy space H1 (0) .
r
THEOREM 4.3. A ssume Hypothesis 4.1 and suppose that z E C\a ( - �e .n) .
Then for every g problem,
E H- 1 12 (8fl), the following generalized Robin boundary value
{
( -� - � )u = 0 in 0, u E (;yN + 8·YD ) u = g on 80,
H 1 (U),
(4.5)
ROBIN-TO-DIRICHLET MAPS AND KREIN-TYPE RESOLVENT FORMULAS 1 35 has a that unique solutionu ue. Moreover, there exists a constantC C ( , n , z) > 0 such (4.6) In particular, (4.7) ['yv( -�e.n - zin)-1 ] * E B(H - 1 12 (8!1), H 1 (D)), and the solution ue of (4.5) is once again given by formula (3. 7) . PROOF. The argument follows a pattern similar to that in the proof of Theorem 2.17. In a first stage, we look for a solution for (4.5) in the form (4.8) u(x) (Sz h ) (x) , E n, for some h E H - 1 12(8!1). Here the single layer potential Sz has been defined in (2.121), and the fundamental Helmholtz solution En is given by (2.120) (cf. also ( C .1)). Any such choice of h guarantees that u belongs to H 1(D) and satisfies (-� - z)u 0 in n . See (D.29). To ensure that the boundary condition in (4.5) is verified, one then takes =
e
=
=
X
=
(4.9 ) That the operator
(- � Ian + K'ff ) + G"(v Sz
:
H - 1 12 (8!1) --> H- 112 (8!1)
(4.10)
z,
is invertible for all but a discreet set of real values of the parameter can be established based on Hypothesis 4.1 by reasoning as before. The key result in this context is that the operator - �Ian + Kff E is Fredholm, with 0 Fredholm index zero for every E C.
z
B(H- 1 12(8!1))
The special case 8 = 0, corresponding to the Neumann Laplacian, is singled out below.
COROLLARY 4.4. Assume Hypothesis 2.1 and suppose that z E C\a(-�N.n) . 1 Then for every g E H- 12(8!1), the following Neumann boundary value problem, { '1(-�u - gz)uon=ail,O in n , (4.11) N has unique solution u = UN . Moreover, there exists a constant C C(D, z) > 0 suchathat (4.12) In particular, (4.13) ['yv ( -� N,n - :Zin)-1) * E B(H - 1 12 (8!1), H 1 (D)), and the solution u9 of (4.5) is given by the formula (4.14) uN ('Yv ( -�N,n - zin)- 1 ) * g . Finally, as a byproduct of the well-posedness of (4.11), the weak Neumann trace '1N in (2.40), (2.41) is onto. =
=
=
136
F. GESZTESY AND M. MITREA
In the following we denote the inclusion (embedding ) map of into a slight abUBe notation, denote the con tinuous inclusion map of into (HJ (n))* by the same symbol fn. We recall the ultra weak Neumann trace operator ;yN in Finally, assuming we
by fn contiof nuous we also H1(0) (H1(0)) *. HJ(O) By (2.60), (2.61). Hypothesis 4.1, denote by (4.15) -Lie,o B (H 1 (fl ) , (H1(0))*) the e o - t..e,n in (B.26). \i'v(x) ('YDU, EhDv)112 , H'(n) {u, - Li8,nv) (H'(fl))• = k u, E H1(0), (4.16) and -t..e,n is the restriction o -Li e,n L2(0: dnx) (B.27)). 4.5. 4.1 z E -t..e,n) . E (H1 (fl.)) * , ln inOV'(fl) , u E H1(fl), w { ;y(-t.-z)u =:_ (4.17) u, w) + erDu = N( = 'Ue,w· = z) > 0 (4.18) (L2(fl;-t..e,n -zlo)-1, z E -t..e,n), (-.6e,n - zln)-1 E B (L2(fl; (4.19) B ( (H1 ( )*, H1 (fl.) ), (4.20) ( - Lie ,n - zfn r1 E B( (H 1 (fl. ) )*, H 1 ( fl ) ) . (2.57). Hence, w E (H1(0.) )*, w (C.1) H1(fl) (-t.. i n V'(fl). sol u ti o n (4.17) s z)u0 w 0 u , u = 1 {(-t.. -B1v)u z)ul =1 O=in- ;y_,v (uo, Ew)H1(0), (4.21) e/ouo) E ( u o, w ) + ;yN(u, w ) = ;yN(( Uo, w ) (ul, 0) ) = N (u o , w ) U1 , 0) ho u l -B'Yo uo = -Ehou, (4.22) (2.64). solvable (4.18). n s for 4.3. (4.17) follows from 4.3. s operator (4.20) ( H 1 ( fl. ) ) * , (4.23) -.6 e , n), -t.. e ,o -zln H E
ext
ns i n of
accordance with
In particular,
cfl x \i'u(x) ·
f
THEOREM
Then for every w lem,
+
(cf.
to
v
Assume Hypothesis and supplse that C\a( the following generalized_ inhomogeneous Robin prob\,,.,
on 80. ,
has a un·ique solution ·u such that
Moreover, there exists a constant c
C(e, n,
llue,w iiHl (fl) .::::; Cll w llcH1(8fl))• ·
In particular, the operato1· as a bounded operator on
C\a(
dnx),
originally defined
dnx)) ,
n)
can be extended to a mapping in
which in fact coincides with
PROOF. We recall if taking the convolution of with En ( z; · ) in and then restricting back to fl. yields a function u0 E = for which in A of i then given by u where u1 satisfies +
fl.,
(;yN +
u1
+
Indeed, we have =
-€
+
;y
H-112 (80.) on {)fl.
+ ;yN(
=
;yN (
;yN u l
That the latter boundary problem is is guaranteed by Theorem We note that the solution thus constructed satisfies U nique es the corresponding uniqueness statement in Theorem Next, we observe that the inver e in is well-defined. To prove that 1 z E C\u( : (Q) �
by
ROBIN-TO-DIRJCHLET MAPS AND KREIN-TYPE RESOLVENT FORMULAS
137
is onto, assume that w E (H1 (0)) * is arbitrary and that u solves (4. 17) . Then, for every v E H1 (0) we have
H' (!1 ) (v , ( -Lie,n - zfn)u) (H'(!1))•
l dnx V'v(x) = l dnx V'v(x) =
=
·
·
l dnx v(x)u(x) + (!DV, e/Du) 112 V'u(x) - z l dnx v(x)u(x) - (/D v , -;:;N (u , w) ) I ;2
V'u(x) -
Z
H' (!1) (v, w) (H'(!1))• ,
(4.24)
on account of (2.61) , (4. 16), and (4. 17) . Since the element v E H1 (0) was arbitrary, this proves that ( -Lie ,n - zfn )u w , hence the operator (4.23) is onto. In fact, this operator is also one-to-one. Indeed, assume that u E H1 (0) is such that (-Lie ,n - zln)u = 0. Then, for every v E H1 (0) , formula (4. 16) yields =
0= =
H'(n) ( v, ( - Lie , n - zfn)u ) (H'(!1) )•
l dnx 'Vv(x)
·
V'u(x) - z
l dnx v(x)u(x) + (!DV, e/Du) 112 .
(4.25)
Specializing (4.25) to the case when E C0(0) shows that (-� - z)u = 0 in the sense of distributions in 0. Returning with this into (4.25) we then obtain (!DV, (-;:;N + e,D )u ) / 2 = 0 for every v E H1 (0). Given that the Dirichlet trace 1 /D maps H1 (0) onto H1 1 2 (80) , this proves that (-;:;N + e,D )u = 0 in H-1 1 2 (80) so that ultimately u = 0, since z E C\a( -�e ,n) . In summary, the operator (4.23) is an isomorphism. Finally, there remains to show that the operators (4. 19) , (4.20) act in a compat ible fashion. To see this, fix z E C\a( -�e ,n) and assume that w E L2 (0 ; dnx) (H1 (0)) * . If we then set u = (-Lie,n - zln)- 1 w E H1 (0) , it follows from (4. 16) that v
'---->
H' (!1) (v, w) (H'(!1))• =
l dnx 'Vv(x)
= ·
H'(n) ( v, ( - Lie ,n - zln)u \ H ' (!1)) •
V'u(x) - z
l dnx v(x)u(x) + (!DV, e/D u\12 ,
(4.26)
for every v E H1 (0). Specializing this identity to the case when v E C0 (0) yields ( -� - z)u = w E L 2 (0; dnx). When used back in (4.26) , this observation and (2.41) permit us to conclude that (/D v , (-;yN
+ Ehv)uh;2 = l �x V'v(x)
l dnx v(x )u(x) - H' (!1) (v, ( -�u - zu) ) (H' (!1))• + (!DV, E:),D u ) 1 12 = l dnx 'Vv(x) V'u(x) - z l dnx v(x)u(x) - H' (n ) (v, w )(H'( n ))• + (!D v , e,v u ) 1 1 2 ·
V'u(x) - z
�
·
= 0,
( 4.27)
138
F. GESZTESY AND M. MITREA
for every v E H1 (fl). Upon recalling that the Dirichlet trace /D maps H 1 (fl) onto H112 (8f!), this shows that C.:YN + EhD )u = 0 in H- 112 (80). Thus, u = ( -� e. n - z ln) -1 w , as desired. 0
REMARK 4.6. Similar (yet simpler ) considerations also show that the operator ( -�D.n - z ln ) - 1 , z E ( (( - Ae ,n - zln)- 1 ) * 0 =j:. 0 implies ue
=j:.
(4.73)
0. This proves (4.70). Restriction of (4.70) to
L2 (ofl; dn-1w) then yields (4.72) .
g E
0
Returning to the principal goal of this section, we now prove the following variant of a Krein-type resolvent formula relating Lie , n and Lin ,n: THEOREM 4.15. Assume Hypothesis 4.1 and suppose that z E manuscript we use the fol1owing notation for the standard Sobolev Hilbert spaces (s E IR) , H S (JRn )
=
{
U E S (!R n )'
H s (n ) = {u
E D'(O)
I
I I U I �•(JR" )
u =
=
i_, dn� ttJ (�) I 2 (1 + I�J2s)
Uln for some U
H0 (0 ) = { u E H8 (IRn ) supp (u ) s::; 0}.
I
E
H8 (!Rn) } ,
< 00
},
(A. I)
(A.2) (A.3)
Here D' (D) denotes the usual set of distributions on n C (antilinear in the first and
linear in the second argument) be V- bounded, that is, there exists a that l a(u, v ) l � ca ll u llv ll v llv , u, v E V. ·
·
ca
> 0 such (B. lO)
ROBIN-TO-DIRICHLET MAPS AND KREIN-TYPE RESOLVENT FORMULAS 153 Then A defined by -> V-* ' (B.ll) A:- {v--V f---t AV = a(· ,v), satisfies A E B(V, V * ) and v ( u,Av ) v • = a(u,v), u,v E V. (B.12) Assuming further that a( , · ) is symmetric, that is, (B.13) a(u,v) = a(v,u), u,v E V, and that a is V-coercive, that is, there exists a constant C0 > 0 such that (B.14) a(u, u) � Co ll u ll �, u E V, respectively, then, A: V ----> V * is bounded, self-adjoint, and boundedly invertible. (B.15) Moreover, denoting by A the part of A in H defined by dom( A) = { u E V I Au E H } H, A = A idom(A) : dom(A) ----> H, (B.16) then A is a (possibly unbounded) self-adjoint operator in H satisfying A � Colrt , (B.17) dom (A 1 1 2 ) = V. (B.18) In particular, A - 1 E B(H). (B.19) The facts (B.1)- (B.19) are a consequence of the Lax-Milgram theorem and the second representation theorem for symmetric sesquilinear forms. Details can be found, for instance, in [31, §VI.3, §VII.l] , [37, Ch. IV] , and [63] . Next, consider a symmetric form b( · , ) V V C and assume that b is bounded from below by cb E JR, that is, (B.20) b(u, u) � cbl l u l � , u E V. Introducing the scalar product ( · , · )v ( b) : V x V ----> C (with associated norm denoted by I l v < bJ ), (u,v)v(b) = b(u,v) + (1 - cb)(u,v) H , u,v E V , (B. 2 1) turns V into a pre-Hilbert space (V; ( · , ) v ( b) ), which we denote by V(b) . The form bb isis called closed if V(b) is actually complete, and hence a Hilbert space. The form called closable if it has a closed extension. If b is closed, then l b(u, v) + (1 - cb )(u, v)rt l l u l v(b) l v l v(b) , u, v E V, (B.22) and (B .23) l b(u,u) + (1 - cb) l u l � l = l u l �(b) ' u E V, show that the form b( · , · ) + (1 - cb )( · , · ) H is a symmetric, V-bounded, and V coercive sesquilinear form. Hence, by (B.ll) and (B.12), there exists a linear map { V(b) f---t V(b)*, Bcb .· (B.24) V Bcb v = b( · ,v) + (1 - cb )( · ,v) H , ·
F. GBSZTESY AND M. MITR.EA
154 with
Bch E B(V(b) , V(b) *) and V (u) (u , Bcb v ) V (b) + = b(u, v) + ( 1 - cb)(u, v)1-t , u, v E V. (B.25)
Introducing the linear map
B = Bcb + (cb - 1 )1: V(b) ---+ V(b)*,
(B.26)
where Y: V(b) ----+ ( a 1 + a2 )
(u,
1 (u, v) (u,
-t
C,
(B.47)
is bounded from below and closed (cf. , e.g., [53, Sect. VI. 1 .6] ) . Finally, we also recall the following perturbation theoretic fact: Suppose a is a sesquilinear form defined on V x V, bounded from below and closed, and let b be a symmetric sesquilinear form bounded with respect to a with bound less than one, that is, dom(b) :;2 V x V, and that there exist 0 � a < 1 and f3 ;;::: 0 such that Then
l b(u, u) l � a ! a(u, u) l + fJI I ul � , u E V. (a + b) :
{
V x V -> C, (u, >----+ (a + b) (u,
v)
v) a(u, v) + b(u, v)
(B.48)
(B.49)
=
defines a sesquilinear form that is bounded from below and closed (cf., e.g., [53, Sect. VI. 1 .6] ) . In the special case where a can be chosen arbitrarily small, the form b is called infinitesimally form bounded with respect to a.
1 56
F. GESZTESY
AND M. MITREA
Appendix C. Estimates for the Fundamental Solution of the Helmholtz Equation
The principal aim of this appendix is to recall and prove some estimates for the fundamental solution (i.e., the Green's function) of the Helmholtz equation and its x-derivatives up to the second order. Let En ( z; x) be the fundamental solution of the Helmholtz equation ( - L\. z)'!j!(z; ) = 0 in �n, n E N , n � 2, already introduced in (2. 1 20) , and reproduced for convenience below: (2- n)/2 (1 ) i 21T/xl H(n- 2 );2 (z 1/2 lxl) , n � 2, z E C\{0}, � zi/2 _ . En (z , x) - __!ln(lxl ( C.1) n = 2' z = O' ), 21r
{(
·
)
(n 2�Wn-l l x l2-n ,
n ;?: 3, = 0, Im ( z112) 2: 0, X E �n\{0} , Z
where H�1 ) ( ) denotes the Hankel function of the first kind with index v 2: 0 (cf. [1 , Sect. 9.1]) and Wn - l = 2rrn1 2 jr(n/2) (r( · ) the Gamma function, cf. [1, Sect. (l.l]) represents the area of the unit sphere sn- 1 in �n. As z ----; 0, En (z , x ) , x E lRn\{0} is continuous for n 2: 3, x 2 -n En (z, x) z-->0 = En(O, x) = (n 2)1 (C . 2 ) - Wn-l l l , X E R." \ { 0} , n � 3 , ·
but discontinuous for n = 2
as
-1 1 2 ln(z 112 lxl/ 2) [1 + O (z lxl 2 ) ] + E2 (z, x) z-+0 = 27r 'I/J(1) + O( zlxl ), 27r (C.3) X E R.2 \{0}, n = 2. Here '1/J (w) = r'(w)jr(w) denotes the digamma function (cf. [1, Sect. 6.3]). Thus, we simply define E2 (0; x) = ;-;ln( lxl), x E R.2\{0} as in (C.1). To estimate En we recall that (cf. (1 , Sect. 9.1)) (C.4) Hc(!� 2)/2 ( · ) = Jcn-2)/2( · ) + iY(n - 2)/2 ( · )
with J�.� and Yv the regular and irregular Bessel funcLions, respectively. We start considering small values of lxl and for this purpose recall the following absolutely convergent expansions (cf. (1, Sect. 9 . 1]) : ,
J�.� (( ) =
( ( ) t; 2
" 00
( - 1 ) k (2k + k + 1) ' k!r(v k 4
( E C\( - oo, 0] ,
Lm (() = ( - 1) m Jm ((), ( E C, m E No , Y.l/ (1') __lv ( () cos(znr) - L., (()
v E R.\( -N) ,
]
)\ . , ( E C \( -oo, 0 , v E ( 0, oo N, sm(vtr) m (- m - 1 (m - k - 1) ! (2k 2 4k + ; Jm (( ) ln((/2) k! Ym (( ) = - 2m 7r L k=O (- 1 ) k (2k (m oo , -,
- 2m L['lp ( k + 1) + '!j!(m + k 1)] 4k k!(m + k )! ' tr k=O ( E C\( OJ , m E No. =
- oo ,
(C.5) (C.6) (C.7)
(C.8)
ROBIN-TO-DIRICHLET MAPS AND KREIN-TYPE RESOLVENT FORMULAS 157 We note that all functions in (C. 5 ), (C. 7), and (C. 8 ) are analytic in C\ ( -oo, 0] and that Jm ( ) is entire for m E In addition, all functions in (C. 5 )- (C. 8 ) have continuous nontangential limits as ( < 0, with generally different values on either side of the cut ( -oo, 0] due to the presence of the functions (v and ln( ( ). (We chose E lR and subsequently usually � 0 for simplicity only; complex values of are discussed in [1, Ch. 9]. ) Due to the presence of the logarithmic term for even dimensions we next dis tinguish even and odd space dimensions n: (i) n = 2m + 2, m E No, and z E C\{0} fixed: m E2m+2 (z,. x) - !:_4 ( 2z7T1/ix2 l ) - Hm(1 ) (z 1/2 i x i ) = � ( 2z��� ) -m [Jm (z 1 !2 l x l ) + iYm (z 112 l x l ) ] = � ( 2z���l ) -m { O( lx l m) � ln ( z 1/;lx l ) o ( lx l m) (C.9) m 1 1 12 i z 1 x ) ( - ;: -2- (1 - 0, X E !Rn \{ 0 }. Define the singular integral operator
( Tf)(x)
=
{
Jan
d n-1w (y) k(.r- - y) f( y ) ,
x E IR"'\an.
(D.12)
(D.13)
Then for each p E { 1 , oo) there exists a finite constant C = C(p, n, on) > 0 snch that (D.14) IIM(Tf)II LP(8!1;d"-lw) S Cllklsn-t l l cN IIJIILP(8!1;dn-lw) · Furthermore, for· each p E (1, oo ) , f E LP (an; dn - lw), the limit
(Tf)(x)
= p .v.
{
Jan
dn -l w (y) k(x - y)j(y) = lim
exists faT a. e. X E an' and the jump-formula
"�o+
1
lx-yl >" yEan
dn- lw(y) k (x - y)f(y)
"fn.t. (Tf)(x) = !� (Tf) (z ) = ±-1;k (v(x))f(x) + ( Tf ) (x)
z Er;!' (x)
(D . 15) (D.l6)
is valid at a. e. X E on, where v denotes the unit normal pointing outwardly relative to n (recall that 'hat ' denotes the Fourier transform in !Rn). Finally,
liT!II H•I2(n ) See the discussion in [24] , [25] , [73]. LEMMA
and
S
Cllf iiP can;dn-lwJ·
(D. l7)
D.3. Whenever n is a Lipschitz domain with compact boundar·y in IR.71 , (D. l8) Kf' E B (L2 (8n; d71-1w)) , z E C,
(K"ff: - K�)
"YDSz
E
E
B00 (L2(an; dn- l w)) ,
B (L2 (8n; dn- 1 w ) , H 1 (8D) ) ,
z1 , z2 E C,
z E C.
(D.l9) (D.20)
PROOF. We recall the fundamental solution En (z; · ) for the Helmholtz equa = 0 in IR71 introduced in (2. 120). Then the integral kernel of the operator Kf - K/! is given by (D.21) k(x, y) = v(x) . (� En ( z ; X - y) - �En (O ; X - y)) , x, 'Y E an. By (C .l2) we therefore have l k(x, y) l S C lx - y l 2 - n , henec (D.l) holds with '1/J (t ) = t. �otc that (D.2) is satisfied for this choice of 1/J, so (D. 19) is a consequence of Lemma D.l. In addition, (D.18) follows from (D. 19) and Theorem D.2, according to which K! E B (L2 (80.; d"-1w)) . Finally, the reasoning for (D.20) is similar (here (A. l5) is useful) , 0 tion ( -ll - z)�(z; · )
LEMMA
then
D.4, If n is a Cl,r,
r
> 1/2, domain in R" with compact boundary,
(D.22)
ROBIN-TO-DIRICHLET MAPS AND KREIN-TYPE RESOLVENT FORMULAS 16 1 PROOF. The integral kernel of the operator Kf - Kt/ is given by (D.21 ) . By Lemma A.5, the operator of multiplication by components of E [Cr (8!1)]n belongs to B(H 1 12(8!1)). Hence, it suffices to show that the boundary integral operators whose integral kernels are of the form (D.23) 8jEn (z ; x - y) - 8jEn ( O; x - y)) , x,y E 8!1, j E {1, . . . , n}, belong to B(L2(8!1;dn- 1 w),H 1 (8!1)). This, however, is a consequence of ( A.18 ) , (A. 19) (with s = 1 ) , (C.13), and Lemma D . 1 (with '!f; (t) = t). LEMMA D.5. Let 0 < a < (n - 1) and 1 < p < q < be related by ! = ! - (a + !) .!. . (D.24) pn q p Then the the operator Ja defined by 1 Ja f( X ) = { dn - 1 Y !Rn , E LP(JRn- 1 .,dn-1 x) , ( ) lx - Yl n - 1 - a J Y ' X E + J }JRn- 1 is0, bounded from LP(!Rn- 1 ; dn- 1 x) to Lq (!Rf-; dnx), that is, for some constant C(D.25) a > q v
0
oo
,
p,
(D.26)
PROOF. A direct proof appears in [75] . An alternative argument is to ob serve that M(Jaf) (x) ::; CJa ( I JI ) (x), uniformly for X 81Rf- , and then to invoke the general estimate (D. l l ) in concert with the classical Hardy-Littlewood-Sobolev 0 fractional integration theorem (cf., e.g. , [92] , Theorem 1 on p. 119).
E
Next, we record a lifting result for Sobolev spaces in Lipschitz domains in [51] .
Let n !Rn be a Lipschitz domain with compact boundary. Then,THEOREM for everyD.6. a > 0, the following equivalence of norms holds: (D.27) Lipschitz domain. Then for every z E C,THEOREM D.7. Let n !Rn 2be a bounded (D.28) Sz E B(L (8!1;dn- 1 w),H312 (!1)), and (D.29) Sz E B(H - 1 (8!1), H 1 12 (!1)). In particular, Sz E B(H8 -1 (8!1), Hs + ( l / 2l(!1)), 0 s 1 . (D.30) PROOF. Given f E L2(8!1;dn - 1 w), write Szf = Sof + (Sz - So)f. From (D.17) and Lemma D.6 we know that II Sof i i H3/2 (fl) :S C II J I I £2 (80;dn- lw) > for some constant C > 0 independent of f. Using (C. 13) and Lemma D.5 (with a = 1) one concludes that (D.31) V'2 (Sz - So) E B(£2 (8!1 ; dn - l w), £2 (!1; � x) ) , and (D.28) follows from this. The proof of (D.29) , is analogous and has as starting point the fact that S f E B(H - 1 (8!1), H112(!1)), itself a consequence of (D. 1 7) and the following description of H - 1 (8!1): H- 1 (8!1) = {g + :s;L (81J,k /8rj,k ) I g, hk E L 2 (8!1; dn - 1 w) }· (D.32) j ,k$ c
c
�
0
1
n
�
1 62
F.
GESZTESY AND M. MITR.EA
Then (D.31 ) ensures that
(Sz - So) E B(H-1 (80), H1 (D)) ,
(D.33)
0 and (D.29) follows. We recall the adjoint double layer on an introduced in (2. 122) and denote by
( Kzg)(x)
=
p.v. r dn- lw(y) Ovy En ( z; y - x )g(y) , X E an ,
lao
(D.34)
its adjoint. It is well-known (cf., e.g., (101]) that (D.35) Hypothesis 2.1 � K E B(L2 (80.; dn - 1w) ) n B (H1 (8n)) and (cf. [38] and (D. 1 9)) that (D. 36) 0. a bounded C1-domain � Kz E Boo (L2(80.; �-1w)), z E C. It follows from (D.35), (D.36), (4.4), and Theorem 4.2 that (D.37) f2 a bounded C1-domain � Kz E Bx (H8(of2)), s E (0, 1) , z E C . We wish to complement this with the following compactness result. THEOREM D.8. If 0. C lRn is a bounded C1·r -domain with r E (1/2, 1) then (D.38) K'ff E Boo (H 1 12 (8n)) , z E 0, f a Lipschitz function with compact K(x, y) lx_!yi" F (
A(fl:::�(y)),
support in lRn , define the truncated operator =
(T.J) (x ) =
J
ix-yj > E
dny K (x, y)f(y),
x E lRn .
(D.39)
y ( Rn
Then, for each 1 < p < oo, the follow·ing assertions hold: (i) The ma:L"imal operator ( T* f ) (x) = sup {I (T,J ) (x)l l c: > 0} is bounded on LP(IRn; rflx ) . (ii ) If 1 < p < oo and f E £P(lRn; dnx) then the limit limc--+o (Tef ) (x) exists for almost every x E lRn and the operator ·i::;
lim ( Td) (x) (T.f) (x) = E--> 0
(D.40)
bounded on LP(lRn; dnx) .
A proof of this result can be found in [70] .
THEOREM D.lO. Let A : lRn ----> lRm , B = (B1 , , Be ) : lRn --7 IRe be two Lipschitz functions and let F : lRm x lRe ---) IR be a CN {with N = N(n, m, f) a sufficiently large integer) odd function which satisfies the decay conditions (D.41) IF(a, b) l :::; C ( l + lbl) - n, • • •
IVI F(a, b)l ::; C,
(D.42)
ROBIN-TO-DIRICHLET MAPS AND KREIN-TYPE RESOLVENT FORMULAS 163 (D.43) l V'u F(a,b)J ::::; C(1 + J b l )- 1 , uniformly for a in compact subsets of JR.n and arbitrary b E JRR. (Above, and \7 denote the gradients with respect to the first and second sets of variables.) For x, y E !Rn with x =f. y and t > 0 we set B1 (x) - B1 (y) + t . . . ' Be (x) - Be (y) + t ) . Kt ( x, y) = lx -1 y J n F ( A(x)lx -- A(y) Yl lx - Yl Jx - Yl (D.44) In addition, for each t > 0 we introduce \7 I
I1
'
'
(D.45)
and, for some fixed, positive K,
( D.46)
Then, for each 1 < < oo, the following assertions are valid: (1) The nontangential maximal operator T** is bounded on LP(JR.n; dnx). ( 2 ) For each f E LP(!Rn;dnx), the limit ( D.47) (Tf)(x) = lim 0 (Tt f)(z) ----+t exists at almost every x E !Rn and the operatorT is bounded on LP(JR.n; dnx). PROOF. Fix p E ( 1 , oo) . For x, y E !Rn with x =f. y consider the kernel K( x, y) = lx -1 yJ n F ( A(x)l x -- YlA(y) ' B(x)lx -- YlB(y) ) ' (D.48) and let T, T* be the operators canonically associated with this integral kernel as in Theorem D.9. The crux of the matter is establishing the a.e. pointwise estimate p
! x - z l < ,d z�x,
f LP(!Rn; dnx), M x, z !Rn, t M
( D.49 )
uniformly for E is the Hardy-Littlewood maximal operator where in Then the first claim in the statement of the theorem follows from Theorem D.9 and the well-known fact that is bounded on To this end, fix < and let a > 0 be a large E > 0 such that constant, to be specified later. Then
!Rn.
Kt, dnx). J x -z l LP(JR.n;
I }ff{n{ dny Kt(z, y)f(y) - 1 dny K(x, y)f(y) l ::::; 1 dny J Kt (z, y)J i f (y)J + 1 dny J Kt (z, y) - K(x, y)J i f (y)J
(D.50 )
l x - y i > at
lx-y i < a t
= I+ II.
Clearly, it suffices to show that IJI, il
l x - y i > at
(D.51 )
J l ::::; CMJ. To see this, first observe that J Kt (z, y)J ::::; Ccn uniformly for any z, y E !Rn , z =f. y (D.52) (in fact, this also justifies that Tt is well-defined) . Indeed, using the fact that for each j E {1 , ... , £} one has Ct ::::; l Bj (z) - Bj (y) +t l + l z -y l (easily seen by analyzing
164
F. GESZTESY
AND M. MITREA
the c�et> lz - Y l 2: 2uvij llr, oo and lz - y J � 211vii iiL ), we may infer that '"' n J Bj ( z ) - Bj (y) + tl ) - � C ( -t- ) -n + (D.53) /z - y / lz - yl j= l
(1
t
With this at hand, the estimate (D.52) is a direct consequence of (D.41). Returning to I, from (D.52), we deduce that III � CMJ(x) . Thus, we are left with analyzing II in (D.51 ) . To begin with, we shall prove that (D.5 4) IKt(z, y) - K(x, y) l � Ctlx - Y l-n - l for lx - Yl > at. Let Gy(x , t ) = Kt(x , y). Then (D.55) JKt(z, y) - K(x, y) l = IGy(z, t ) - Gy (x, O) I can be estimated using the Mean Value Theorem by (D. 56) Ct( I'VIGy(w, s) l + /'VnGy(w, s)l) , where w = ( 1 - O) z + Ox , s = (1 - B )t for some 0 < f) < 1 . Next, I'V 1Gy (w , s)l C < ( A(w) - A (y) , B 1 (w) - B1 (y) + s , . . . , Bt(w) - Be(Y) + - lw - yJ n+ l /w - Yl Jw - y j / w - yj C 'V 1 F ( A(w) - A(y) ' B1(w) - B1 (y) + s ' ... ' Bt(w) - Be(y) + s ) I + I l + w l w - Yl l w - Yl Yln l w - Yl l A(w) - A(y) B1 (w) - B1 (y) + s . . , Bt (w) - B�.(y) + s ) C II F ( l lw - yj lw - Yi n l w - Y/ ' '. lw - Y l
s)
IF
+
x
'V I C(Jw - Yl
+
t /Bi(w)lw--BYli (2Y) + s J )
I
(D.S?)
i= l Keeping in mind the restriction.ti on the size of the derivatives of the function F stated in (D.41 )-(D.43) , conclude that the above expression is bounded by Clw - Y l - (n+ l) . Similarly, it can be shown that .
we
IV' uG y(w, s)l �
c
. (D.58) lw y J l To continue, one observes that if we choose a > r;, then, in the current context,
lw - xl � lz - xl � r;,t = and l w - xl + lw - yj 2: lx - y/ . Hence,
_
n+
(;) o:t < (;) lx - yj,
( ;) lx - yl,
lw - Yl 2: 1 -
and, therefore,
(D.59)
(D.60)
Next, we split the domain of integration of JJ (appearing in (D.51)) into dyadic annuli of the form o:t � l x - Y l � 2J+ l at, j = 0, 1, 2, .. . . Then (D.61)
2J
1lx-yl>od dny IKt(z, y) - K(x, y)l /f(y) l
ROBIN-TO-DIRICHLET MAPS AND KREIN-TYPE RESOLVENT FORMULAS 165 (D.62) :::; C L rJ (MJ)(x) = C(Mf)(x). This yields the desired inequality, that is, I III :::; CMJ(x). The proof of last second claim in the statement of the theorem utilizes a well 00
j =O
known principle (cf. , e.g. , [38]) to the effect that pointwise convergence for a dense class along with the boundedness of the maximal operator associated with the type of convergence in question always entails a.e. convergence for the entire space Thus, it suffices to identify a dense subspace V of such that for any E V the limit in question exists for almost every E Then the boundedness of the maximal operator associated with the type of convergence under discussion ensures that this limit exists for any E at almost every E In our situation, we may take V and observe that
LP(JRn; dnx).
LP(JRn;JRn.dnx) x f LP(JRn; dnx)
f x lRn.
=
z---+ x , t---+ 0
where
(T f)( ) = lim t
lim
l x - z l e:
(D.64)
l > l x - v l > e:
Consequently, lim
lim
lim
lim
e:-->0 l x - z l < �t z---+ x , t -+0 E:-->0 l x - z l < �t z-+x , t -+0
whereas
I = 1 �y K(x, y)f(y), II = ! dn yK(x,y)[f (y) - f(x)], lx-yl > l
(D.65) (D.66)
l>lx-yl
III = lim J
dn yK(x,y). (D.67) Now, this last limit is known to exists at a.e. x E JRn (see, e.g., [76]). Once the pointwise definition of the operator has been shown to be meaningful, the boundedness of this operator on LP(JRn; dnx), 1 < p < oo, is implied by that of T** · THEOREM D . l l . There exists a positive integer N = N(n) with the following signi that ficance: Let n ]Rn be a Lipschitz domain with compact boundary, and assume k E CN(JRn \{0}) with k( -x) = -k(x) (D.68) and k(>.x) = >.- (n- l) k(x), >. > 0, x E lRn \{0}. lim
lim
e:-->0 l x - z l < � t z---+ x , t---+ 0
e:-->0
l > l x - y l > e:
T
0
C
166
Fix ry E
Then
F. GESZTESY AND M. l'v1I'T'REA
eN (JR n ) and define the singular integral operator
(Tf)(x)
=
p.v.
l{an dn - 1w (y) (ry(x) - ry (y) )k(x - y)f(y) , T
x E oft.
E
!3(L2 (8fl; d"-1w) , H1 (80)). PROOF. Fix an arbitrary f E L2 (8fl; d'' - 1w) and consider u(x) =
{
lan
d''- 1 w(y ) (ry(x) - ry(y))k( .r, - y)f(y) , x E fl .
(D.69) (D.70) (D.71)
Since Tf = ul an , it suffices to show that (D.72) IJM( V'u) II P(afl;d"-lw) S CJJJIIL2(ofl:dn -lw) > for some finite constant C = C (O ) > 0 (where the nontangential maximal operator M is as in (D.9) ) . With this goal in mind, for a given j E {1, . .. , n}, we decompose (oj u) (x) = 'U t (x) + u2 (x), X E 0, (D. 73) where (D.74) dn - l w (y) k(x - y)j(y) Ut(X) = (Oj1'J)(x) an and (D. 75) dn- 1 w ( y) (ry(x) - ry(y) ) ( ojk)(x - y)f(y). u2 (x) = an Theorem D.2 immediately gives that (D.76) II M ut llu(afl;d"-lw) S CIIJIIL2(ofl;d"-lw)• so it remains to prove a similar estimate with u2 in place of u 1 . To this end, we note t.hat the problem localizes, so we may assume that 11 is compactly supported and 0 is the domain above the graph of a Lipschitz function ;,p : IR.n -t -+ R In this scenario, by pasHing to Euclidean coordinates and denoting g( y' ) = f (y' , ;,p (y' )) , y' E IR."- 1 , it suffices to show the following. l 1 . We then give an application to variable coefficient layer potentials for divergence form elliptic operators with bounded measurable non-symmetric coefficients.
1 . Introduction, statement of results, history The Tb Theorems of Mcintosh and Meyer [McM] , and of David, Journe and Semmes [DJS] , are boundedness criteria for singular integrals, by which the boundedness of a singular integral operator T may be deduced from sufficiently good behavior of T on some suitable non-degenerate test function A "local Tb theorem" is a variant of the standard Tb theorem, in which control of the action of the operator T on a single, globally defined accretive test function is replaced by local control, on each dyadic cube Q, of the action of T on a test function which satisfies some uniform, scale invariant LP bound along with the non-degeneracy condition
£2
b. b, b , Q
(1.1) for some uniform constant C0. A collection o f such local test functions, ranging over all dyadic cubes Q (or over all cubes or balls) is called a "pseudo-accretive system" . The first local Tb theorem, in which the local test functions are assumed to belong uniformly to is due to M. Christ [Ch] , and was motivated in part by applications to the theory of analytic capacity; an extension of Christ's result to the non-doubling setting is due to Nazarov, Treil and Volberg [NTV] . A more recent version, in which Christ's L00 control of the test functions is relaxed to Lq control, appears in [AHMTT] , and this sharpened version (see also [AY] , and the unpublished manuscript [H] ) has found application to the theory of layer potentials associated to divergence form, variable coefficient elliptic PDE (see [AAAHK]) . It is also of interest to consider local Tb theorems for square functions (as opposed to singular integrals) . These have found application to the solution of the
£00,
2000 Mathematics Subject Classification. Primary 42B25; Secondary 35J25 . 8. Hofmann was supported by the National Science Foundation. @2008 American Mathematical Society
175
STEVE HOFMANN
1 76
Kato problem [HMc] , [HLMc] , [AHLMcT] (tree also [AT] and [SJ for related results), and to variable coefficient layer potentials [AAAHK] . In this note, we consider the square function estimate ( 1.2)
where Btf(x) :=
r
./R "
'Wt( X , y)J(y)dy
and {?ft(x, y)}tE (O,oo) > satisfies, for some exponent a > 0, t I 'Wt ( X, y) J � C (t + J x - n+ a yJ) and ( a)
( b)
11/•t (X, Y
I 'W
+ h) - 1/Jt (X , y) J
� C
(1 .3) J h J
a J h J t (X + h , y) - 1/lt (X , y) J � C + J (t x y l)n + a (t + J x
_
y J )n +
(1 .4)
_
whenever J hl � t/2. Our main result. in this paper is the following:
THEOREM 1 . 1 . Let Ot f(x) : = J 'l/J1 (x, y)f(y)dy, where 'l/J 1(x , y) satisfies (1.3 ) and ( 1 . 4 ) . Suppose also that there exists a constant C0 < oo, an exponent q > I and a system {bq} of functions indexed by dyadic cubes Q � R", S'uch that for each dyadic cube Q (i) JIR" Jbq Jq : 1 in the "perfect dyadic" case treated in [AHMTT] , and, bal:ied on [AHMTT], for q = 2 (or q = 2 + E) in the case of st.andard singular integrals in [AY] and [H] . It remains an open problem to treat the case q < 2 for singular integrals that are not of perfect. dyadic type, but rather satisfy standard Calderon-Zygmund conditions. The paper is organized as follows: in the next section we prove Theorem 1 . 1 , and in Section 3 we present an application to the theory of layer potentials for vari able coefficient divergence form operators with bounded measurable non-symmetric coefficients.
Tb
A LOCAL THEOREM
177
2. Proof of Theorem 1 . 1 We begin by recalling the following well known fact, due explicitly t o Christ and Journe but also implicit in the work of Coifman and Meyer
[CJ], [CM]. 2 . 1 . [CJ] Let ()t f(x) fJRn 'l/Jt (x , y)f(y)dy, where 'l/Jt(x, y) satis fies PROPOSITION (1.3) and ( 1.4) (a). Suppose that we have the Carleson measure estimate 1IQT Jro£(Q)Jr IBt l(x) l 2 dxdtt C. (2.1) s p � Q Then we have the square function bound ( 1 .2) . Remark. The converse direction (i.e. that (1.2) implies (2.1)) is essentially due to Fefferman and Stein [ F S). Thus, to prove Theorem 1. 1 , it is enough to establish ( 2 . 1 ) . In fact, by covering an arbitrary cube by finitely many dyadic cubes of comparable side length, it is :=
-
"5:.
enough to establish a version of (2. 1 ) in which the supremum runs over dyadic cubes only. To this end, we shall use the following lemma of "John-Nirenberg" type.
LEMMA 2.2. Suppose that there exist E (0, 1) and 01 < oo, such that for every E .!Rn dyadic Q, withcube Q , there is a family { Qj} of non-overlapping dyadic sub-cubes of ( 2.2 ) and r}Q (1TQ£((x)Q) IBt l(x) l 2 dtt ) q/2 dx C1 1 Q I , ( 2.3 ) where TQ (x) := 2::: 1 Q1(x)€(Qj)· Then (2 . 1 ) holds. TJ
"5:.
1. 1 .
1
We shall defer momentarily the proof of Lemma 2.2, and proceed to the proof of Theorem We may suppose without loss of generality that < q < 2 , as the case > 2 may be reduced to the known case 2 by Holder's inequality. We claim that, in the spirit of and (but using also Lemma 2.2 ) , it is enough to prove that for each dyadic cube there is a family of non-overlapping dyadic sub-cubes of satisfying ( 2.2) for which
q
[S]
q
[AQ,T]
=
{ Qj } Q Q£( ) IBt l(x)l2 _!d ) q/2 dx C 1 (Q l(lt l(x) Atb (x) l 2 _!d ) q/2 dx, (2.4) Q t 1Q (1TQ (x) t Q (1R.0 ) where A t denotes the usual dyadic averaging operator, i.e., At f(x) I Q (x, t) l 1 }rQ x t f, ( ) x with side length at least and Q(x, t) denotes the minimal dyadic cube containing t. Indeed, given (2.4) , we may follow [CM] and write Bt lAt = (Bt l) (At - Pt ) (Bt lPt - Bt ) Bt := R?l R�2 Bt . where Pt is a nice approximate identity, of convolution type, with a smooth, com pactly supported kernel. By hypothesis (iii) of Theorem 1. 1 , the contribution of to the right hand side of ( 2.4) , is controlled by CI Q I , as desired. Moreover, "5:.
:=
+
-:'7 bQ .
-
,
+
+
l
+
178
STEVE HOFMANN
R�2) 1 = 0 , and its kernel satisfies (1.3) and ( 1 .4). Thus, by standard Littlewood Paley/vector-valued Calder6n-Zygmund theory, we have that
fa (1
l(Q)
I R (2) bq (x) j 2
) � t
q/ 2
dx
S Cq llbd � S CJ Q J ,
(2.5)
where in the last inequality we have used hypothesis (i) of Theorem 1 . 1 . Further
more, the same Lq bound holds for RF> (even though (1 .4) fails for this term), as may be o;een by following the interpolation arguments of [DRdeF] . We omit the details. Thus, the right hand side of (2.4) is bounded by C J Q J , so that the conclusion of Theorem 1 . 1 then follows by Lenm1a 2.2 and Proposition 2.1 . Therefore , it is enough to establish (2.4), for a family of dyadic sub-cubes of Q satisfying (2.2) . To this end, we follow the stopping time arguments in [HMc] , [HLMc] and [AHLMcT] (but see also [Ch] , where a similar idea had previously appeared ) . Our starting point is hypothesis (ii) of Theorem 1.1. Dividing by an appropriate complex constant , we may suppose that (2.6) We then sub-divide Q dyadically, to select a family of non-overlapping cubes { Q1 } which are maximal with respect to the property that �e
�
l il
kJ
(2. 7)
bq S 1/2.
By the maximality of the cubes in the family { Q1 } , it follow::> that 1 2 S �e Atbq (x), tf t > Tq (x), 0
so that (2.4) holds with C = 2q. It remains only to verify that there exists "' > 0 such that (2.8) l E I > ryJ Q I , where E = Q \(U Qj ) · By (2.6) we have that IQI
=
r bq = �e r
)q
s I E I 1/q'
)q
(h
bq
J bq jq
)
=
l/q
�e +
r bq + �e L r j
JFJ
)qj
bq
� L IQj j ,
when in the last step we have used (2.7). From hypothesis (i) of Theorem 1.1, we then obtain that ' IQI s CJEI 11q IQI 11q + I Q I ,
�
and (2.8) now follows rea(tily. This concludes the proof of Theorem 1 . 1 , modulo Lemma 2.2, whose proof we now give. PROOF OF LEMMA 2.2.
We
begin by stating
LEMMA 2.3. Suppose that there exist N < oo and {J E (0, 1) such that for every dyadic cube Q, (2 .9) i {x E Q : gq (x) > N } l S ( 1 - {J) I QI ,
Tb
1 79
A LOCAL THEOREM where
(1R(Q) IBt1(xW �t ) 1/2
gq(x) : =
Then (2.1) holds.
We take this lemma for granted momentarily, and prove Lemma 2.2. Fix a dyadic cube Q. For a large, but fixed to be chosen momentarily, let
flN
N
{x E Q : gq(x) > N}.
:=
Under the hypotheses of Lemma 2.2, with E := Q \ (uQ1), we have <
N}l ) 1/ (1 - ry) I Q I + l {x E Q : 1 I Bt1(xW dtt 2 > N} l ( 1 - ry) I QI + q I QI , N
L
( R.(Q)
rq (x)
0, as the same argument carries over mutatis mutandi to the case t < 0
1 83
A LOCAL THEOREM Tb
and to s; . By Lemma 5.2 of [AAAHK] , it suffices to prove (3.6) and (3.7) Moreover, the same lemma shows that (3.7) follows from
!1
dx dt It (8t ) 2 St f( x) i 2 1 t 1 :::; C I I ! I I 2L2(JRn ) · IR±+I -
(3.8)
;�
In addition, from (3.2) for k , , along with the results of [DJK] applied to the solution u(x , t ) := 8t St f( x ) , we have that (3.7) implies (3.6) (we use here that the arguments of [DJK] carry over to the non-symmetric case - see the comments in the introduction to [KKPT]) . Thus, it is enough to prove (3.8). To this end, we first note that by [GW] (if n + 1 ? 3), or (if n + 1 = 2) as a consequence of the Gaussian bounds and local Holder continuity of the kernel of the heat semigroup e - rL (see, for example [AMT]) , we have that
'1/Jt (X, y ) : = t (8t ) 2 f( x , t , y , 0),
t
the kernel of Bt := (8t ) 2 St , satisfies (1 .3) and (1.4) . Thus, it is enough to construct a pseudo-accretive system { bQ } satisfying the hypotheses of Theorem 1 . 1 . We now set
bQ
=
I Q I kLA�- - ·
Observe that condition (i) of Theorem 1 . 1 follows immediately from (3.2). Moreover (ii) is an immediate consequence of the following well known estimate of Caffarelli, Fabes, Mortola and Salsa [CFMS] , extended to the case of non-symmetric coeffi cients (as may be done: see the comments in [KKPT] concerning the validity of the results of [CFMS] in the non-symmetric setting) :
A-
JrQ kL � - ( y) dy ? c · 1
It remains to establish condition (iii) of Theorem 1 . 1 . Let ( x , t) E R� = Q (0, f ( Q )) . Then, since for fixed (x, t ) E JR.�+l , we have that 8lf(x, t, ·, ·) is a solution of L*u = 0 in lR.'.:_+ l , we obtain x
IBtbQ ( x ) l
=
IQI tl
j (8t )2 r (x , t , y , o) k;�_ (y) dy i =
I Q I t l (8t ) 2 r(x, t , Aq ) l =
�
�
I Q I + ( 1 '1/Jt +t (Q) (x, x Q ) I :::; c e( ) " t Q)
where in the last two steps we have used (3.3) and then ( 1 .3). Hypothesis (iii) now follows readily. This concludes the proof of Theorem 3 . 1 . 0
184
STEVE HOFMANN
[AAAHK] M. Alfonseca, P. Auscher, A. Axelsson, S. Hofmann and S. Kim, A nalyti ci ty of layer potentials and L2 solvability of boundary value pmblems for divergence form elliptic equations
References
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[AHMTT] P. Auscher, S. H ofma nn , C. M uscalu , T. Tao, C. Thiele, Carleson measures, trees , extrapolation, and T(b) theorems, Pub!. M at . , 46 (2002), no. 2, 257-325. [AMT] P. Auscher, A. Mcintosh and P. Tchamitchian, Heat kernels of second o rder complex el liptic operators and applications, J. Fu ncti onal A nalysis , 152 ( 1 998), 22- 73 . [AT] P. Auscher and Ph. Tcharnitchian, Square root problem for divergence operators and relat ed topics, Asterisque Vol. 219 (1998), Societe MatMmat ique de France . [AY] P. A us cher and Q. X. Yang, On local T(b) Theorems, preprint . [CFMS] L. Caffarelli, E. Fabe s , S. Mortola and S. Salsa, Boundary behavior of nonnegative so lutions of elliptic operators in divergence form, Indiana Univ. Math. J . , 30 ( 198 1 ) , no. 4, (2002) , 633-654.
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[DJK] B. Dah lb erg, D. Jerison and C. Kenig, Area int egral estimates fur elliptic differential operators with no m moo th coeffir.i en t,s, Ark. Mat., 22 { 1 984) , no. 1, 97-108. [DJS] G. David, J.-L. Journe, and S. Sem mes , Operateurs de Calder6n -Zygmund, fonr.tions para-accretives et interpo lat ion, Rev. M at . 1beroamericana, 1 1-56, 1985. [DR.deF] J. Duoandikoetx.ea and J. L. Rubio de Francia, Maximal and singular integral operator·s via Fourier transform estimates, I nvent. Mat h . , 84 (1986), 541-561. [FS] C . Fefferman, and E. M. Stein, HP spa ces of several v ari ables, Ac ta M at h . , 129 ( 1 9 72) , no. Press, 1986.
3-4, 1 37-193 [GW] M. Griiter and K. 0. Widman , The Green functio n for uniformly elliptic equations, M anuscripta Math . , 37 (1982), 303-342. (H] S. Hofmann, A proof of the local Tb Theorem for stan dard Calder6n-Zygrnund operators, un p u blish ed manuscript, http :/ fwww.rnath.missouri.edu/ - hofmann/ (H2] S. Hofmann, Local Tb Theorems and applications in PDE, Proceedings of the ICM Madrid, Vol. II, pp. 1375-1392, European Math.
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(HMc] S. Hofmann and A. M cint os h , The solution of the Kato problem in two dimensions, Pro ceedings of the Conference on Harmon i c Analysis and PDE held in El EscoriaL Spain in July 2000, PubL Mat., VoL extr a, 2002, pp. 143- 160. [ JK ] D. Jer iso n and C. K enig , The Dirichlet pr·oblern in rwnsmooth domains, Ann. of Math. (2) , 113 ( 1 981), no. 2, 367-382. [K] C. Kenig, Harmonic analysis techniques for second order elliptic boundary value problems, CBMS Regional Conference Series in Mathematics, 83. Published for the Conference Board of the Mathematical Sciences, Washington, DC, A merican Mathematical Society, Providence, R.I, 1994 [KKPT) C. Kenig, H. Koch, H. J. Pipher and T. Toro, A new approach to absolute continuity of elliptic measure, with applica tions to non-:;ymmetric equations, Adv. Math., 153 (2000), no. 2, 231-298. 631.
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[KP] C. Kenig and J. Pipher, The Neumann problem for elliptic equations with nonsmooth coef [KR] C. Kenig and
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[NTV]
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par des
integmlf:'s singulier·es,
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spaces, Duke Math. J . ,
[S]
definis
no.
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(1990) , no.
3, 721 726.
address:
and
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2, 259-312.
DEPARTMENT OF MATHEMATICS, UNIVERSITY OF MISSOURI, COLUMBIA, MISSOURI 652 1 1 , E-mail
hofmannillm at h . missouri . edu
USA
Proceedings of Syntpu�ia in Pure :Ma.therna.tics Volume 79, 2008
Partial differential equations, trigonometric series, and the concept of function around 1 800: a brief story about Lagrange and Fourier Jean-Pierre Kahane
Dedicated to Vladimir Maz'ya and Tatyana Shaposhnikova.
ABSTRACT. Functions of real variables, PDE's and trigonometric series have strong relations. A brief history of these relations as they appeared around 1750 and developed around 1800 is given in the first part of the article. The controversy on vibrating strings, involving d'Alemhert , Euler, Daniel Bernoulli and Lagrange, is well know, and also the birth of Fourier series with the an al ytic theory of heat. Among the many references quoted in the article the main source is the thesis of Riemann on trigonometric series. Riemann showed how difficult is was to acce pt the idea that an arbitrary fWlction could be expressed by a trigonometric series, and he mentioned the strong opposition of Lagrange to the statements of Fourier. A sentence in Riemann's thesis is the source of the second part of the paper, where the author describes his search of two handwritten pages in the collection of manuscripts of Lagrange, supposed to express this opposition, and tries to explain what he found.
Vladimir Maz'ya and Tatyana Shaposhnikova made a significant contribution to the history of mathematics with their authoritative biography "Jacques Hadamard : A Universal Mathematician" [11] . Thus it seems appropriate to include the history of mathematics as one of the themes for the present volume. I will be concerned with the period 1750-1850, and will focus on how the notion of "function" was influenced by the study of partial differential equations (PDEs) and trigonometric series. There are two parts in this article. The first relies on previous studies, it is just a way to look at a well known story. The second relies on the first, it contains a new material, and it is a tentative answer to a series of puzzling questions about a historical document. The study of the relations among PDEs, trigonometric series, and the possible notions of function remains active in modern Limes. A classic example is the theory of distributions developed by Laurent Schwartz in the middle last century. This 2000
Ma.thematir.8 Subjer.t Classification.
Primary 01A50; Secondary 01A55. Lagrange, Fourier ,
Key words and phmses. d'Alembert, Euler, Daniel Bernoulli,
187
Riemann
.
188
JEAN-PIERRE KAHANE
theory extented the notions of functions, derivatives, and Fourier transform [13] and it was first applied to the study of PDEs. Todays, classes of functions and their generalizations play an essential role in research on PDEs, while trigonometric series and Fourier transforms enter as important tools. This story is far from over , and I believe that a look at some of the beginning contributes to our appreciation of current research. 1. Part I I begin with a well known episode, the controversy around vibrating strings af ter 1747. It deals with PDEs as well as trigonometric series [16] , and both subjects introduced important ideas and discussions about functions. The principal charac ters of the story are d' Alembert (1717-1783) , Euler ( 1 707-178:�), Daniel Bernoulli (1700-1782) , and Lagrange (1736-1813). The story was told in many ways, first by the actors themselves, then by many other authors. A volume of the collected works of Euler contains the deo;cription of the debate by C. Truesdell [14] , and this is the most current source of information. The earlier, thorough study by H. Burkhardt on series and PDE (1804 pages) was essentially related to the math ematico; in question, and a large part was devoted to the period we consider [2] . A short and illuminating article of A.P. Youschkevitch (10 pages) appeared in 1975, with the principal niferences on the subject of vibrating strings and the use of "dis continuous" functions [15] . A French thesis has been defended recently in Lyons by Guillaume Jouve ; it contains much new material, comments, translations, un published papers of d' Alembert, together with the relevant part of d'Alembert's Opuscules rnathematiques [8] . The languages used by these authors are English, Germm, Russian, and French. I shall just sketch the story, and I recommend Jouve as a source of further information. The second episode is related to the heat equation. It is a fascinating story, involving mainly Fourier and (again) Lagrange with the participation of many of their contemporaries. I will just sketch the story, but I wish to highlight the appear ance of trigonometric series as a tool, as a mathematical object, and as a source of ideas and problems. The main reference here, apart from Fourier's book "Theorie analytique de la chaleur" [5] , is the historical part of Riemann's dissertation on trigonometric series [1 2] . In fact, this historical part is the best exposition of whole subject, from vibrating strings to Fourier and Dirichlet, that I know. Riemmn's dissertation contains many ideas that have been significant for the development of mathematics in general. His appreciation of the people and their work is well in formed and accurate. I will devote a section to comments on this dissertation, and this will lead to a comment on Dirichlet's ideas. A statement by Riemann serves as motivations for Part II. 1 . 1 . As I already said, the first part of the story, the controversy about vibrat ing strings, is well known. It is kind of dramatic play. The first act begins with d' Alembert in 17 4 7 and Euler in 1748. Both considered a string fixed at two points, say 0 and e on the x--axis, and ordinates y(t, x) above x at time t. Both established the PDE (written here in modern notation) (1)
[J2 y [J2 - w2 y2 2 8t - 8x ·
PARTIAL DIFFERENTIAL EQUATIONS
189
Both found a solution (2)
F(wt + x) - F(wt - x) ,
(3)
F(:r) - F( -:z:) = p(x) , w(F'(x) - F'( -x) ) v(x) .
where F is a 2 e-periodic function. F is well defined by the initial conditions (po sition and velot:ity at time 0) :
=
For d 'Alembert the functions of the form (2) were a particular class of functions, and the functions defined in (3) were particular as well. Therefore (2) provided a solution for a special class of initial data. For other initial data, he said that. t.here could be other solutions, and when the data are not regular enough he said that it rnight become a question of physics, not of mathematics. For Euler, on the contrary, the solution was general for any kind of initial data. Euler's motivation is clear : for any initial data one can compute F, therefore it has to be the solution of the problem. (At first , Euler assumed that the initial velocity vanishes, and in this case (3) means that F is an odd function such that '2F(x) = p(x) when 0 < x < £) . No "continuity" is needed for the initial data ; in particular, broken lines could be allowed. Here is a quotation by d'Alembert, discussing Euler's point of view: On ne trouve la solution du pmbleme que po·u,- le.s cas ou les differentes figtLTes de la corde vibrante peuvent etre renfermee dans une seule et meme equation ([8] , II, p. 72). (One obtains the solution only in the cases when the different forms of the vibrating string can be expressed by one specific equation.) Already here there are two conceptions about the functions you can consider in mathematical analysis. For d'Alembcrt, they should have a well defined expression in terms of known functions. For Euler, they can be defined as well by any graphical representation. A new scene appeared with Daniel Bernoulli. Since a sound is a superposition of harmonics, the general solution of the problem of vibrating strings should be a series of the form (again, I use modern notations)
(4)
� "" k
(
ak
brwt
cos -e
+
)
. knwt . k1rx bk sm -- sm e e
-
[ 1] . Now neither d'Alembert nor Euler would agree. For d'Alembert, and for Euler as well, trigonometric series would represent only a very special class of functions. The last character in this first episode is Lagrange. He wru; able to treat the problem in a complete form when the string is replaced by equidistant weighted points distributed on a thread. Then, finer and finer discretisations of the initial data result in discrete solutions tending to the solution proposed by Euler. It appeared as a justification for Euler's point of view, although it was rejected by d'Alembert. And that is the end of this part of the story [9] . 1.2. The second episode involves Joseph Fourier (1 768-1830) as a principal character. Fourier sent a memoir to the French Academy of Sciences (then called the first class of the Institut de France) in 1807, on t.he propagation of heat in solid bodies. The memoir was read by a committee consisting of La.gmnge, Laplace, Lacroix and Monge. It was not published. The Academy then proposed the subject for a competition. Fourier extended his study and sent his contribution with the
19 0
JEA:"-PIERRE KAHANE
beautiful subtitle "et ignem regunt numeri" (heat also is governed by numbers) . Laplace. Lagrange, and Lacroix were again examiners. Fourier won the Prize, but there were severe reservations, and the work again was not published. Only after 1817, when Fourier became a member of the Academy, did a printed version appeared ; an extended version took the form of an important book, la Theorie analytique de la chaleur, the analytic theory of heat, published in 1822 [5] . The book consists of an introduction, Discours preliminaire, and nine chapters. The fhst chapter expounds the physical aspectH of heat propagation. The second gives the differential equations, first as examples, then in a general way : inside a homogeneous body the heat propagation is governed by the equation
(5 )
where K, C, D are physical constants depending on the body. If we forget the constant, it is what we now call the heat equation. Moreover there are boundary and initial conditions , a. 0) '
where we write 1 for the temperature of boiling water. The temperature inside the body is given by the heat equation in a reduced form : (8) Nowadays we call this a Dirichlet problem. The treatment by Fourier consists of looking first for solutions of (8) in the form u(x, y)
=
f(x) g(y) ,
then, taking into account the boundary conditions on the vertical edges and the fact that temperatures are bounded in the body, considering as general candidates
u (x , y)
= a
exp(-y) cos x + b exp (-3y) cos :h; + c exp(-5y) cos 5x + · · · .
It remains to express that u(x, 0) = 1 when -1T /2 < x < 1T/2, that is 1 = a
cos x + b cos 3x + c cos 5x +
· · ·
(- 1T/2 < x < n/2) .
191
PARTIAL DIFFERENTIAL EQUATIONS
Fourier finds the values of the coefficients and he is pleased with the solution : (9)
(10)
x - �3 cos 3x + �5 cos 5x - �7 cos 7x + ( - �2 < x < �2 ) ' 1 1 1 -u(x. y) = e-y cos x - - e-3Y cos 3x + - e-5Y cos 5x - - e-7Y cos 7x + 3 5 7 4 ( - � < x < �,y > o) .
� u (x 0) = cos ' 4 7r
·
·
·
·
· ·
'
Fourier knows what convergence means and explains that these series converges, indicating the sum of the series (9) on different intervals (n° 177) with a full proof using the so-called Dirichlet kernel (n° 1 79) ; "the limit of the series is positive and negative alternatively. By the way, the convergence is not rapid enough in order to provide an easy approximation, but it suffices for the truth of the equation" (n° 1 79) . The Fourier's main interest is the second series because it is "extremely convergent" and gives a good estimate of the temperature inside the body by using only a few terms (n° 191). Then there is a long digression in the book. Before considering the propaga tion of heat in other domains (chapters 4 to 9) , he spends 50 pages playing with particular functions and their expansions into series of cosines or serieH of sines, thereby giving different extensions of functions defined on an interval. His conclu sion is that arbitrary functions can be represented by trigonometric series and that all series converge. (That was a mistake, but a very fruitful mistake.) He observes that his analysis applies to vibrating strings, therefore justifying the approach of Daniel Bernoulli. As far as the notion of function is concerned, it is clear after Fourier that a function is associated with a domain and that there is no canonical way to extend a function. Vve shall see in part II how trigonometric series played a role in this clarification. 1.3. The first historical study of these matters is due to lliemann (1826-1866). It is the first chapter of his dissertation on trigonometric series [12] . The second chapter introduces the Riemann integral, together with a characterization of the Riemann-integrable functionH. The third chapter is a firework of ideas, methods, examples , and general results. It contains a characterization of the functions ob tained as sums of everywhere convergent trigonometric series. Starting form the series and not from the function forbids the use of integration for computing the coefficients (or needs another definition of the integral, as Denjoy made much later [4] ) . ThiH iH now called the Riemann theory of trigonometric HerieH [16] . The Rie mann theory was completed by George Cantor ( 1845-1 918) ; he proved that if the sum of the series is zero everywhere, it is the null series. The he extended the theorem and proved that the conclusion still holds when "everywhere" is replaced by "everywhere except on some particular set" . This extension is the first paper by Cantor on real numbers and set theory [3] . Riemann's third chapter is a jewel mine, but my purpose here is to use and comment the first chapter. The first chapter is divided into three sections, whose subjects are vibrating strings, Fourier, and Dirichlet ( 1805-1859). The first section is a very clear ex position of the controversy about vibrating strings : d' Alembert rej ecting his own solution when arbitrary initial positions and velocities are given ; Euler claiming that no restriction is needed ; Bernoulli assuming that the motion of vibrating
192
JEAN-PIERRE KAHANE
strings is a superposition of harmonic motions ; and Lagrange's approach, from finite to infinite, supporting Euler's claim. D' Alembert did not agree with Euler and Lagrange, and the three of them rejected the claim of Bernoulli. Riemann says at the beginning of the second section that a new area began with Fourier, namely with the couple of formulas
{
J(x) (11)
=
{
sin x + a2 sin 2x + · · · +2 bo + b 1 cos x + b2 cos 2x + · · .
a1
1
an
= -
bn
=
1
j" f(x) sin nx dx ,
� �-.: j (x ) cosnx dx .
1r
- 7r
He then explains that Lagrange strongly opposed Fourier's method. Let me quote Riemann : Als Fourier in einer seiner ersten Arbeiten uber die Warme, welche er
der franzosische Akademie vorlegte (21 Dec. 1807) zuerst den Satz aussprach, dass eine willkurklich (graphisch) gegebene Function sich durch eine trigonometrische Rcihc ausdruckcn lasse, war diese Behauptung dem greisen Lagrange so unerwartet, dass er ihr auf das Entschiedenste entgegentmt. Es soll sich hieruber noch ein Schrijtstuck im A rchiv der Pariser Akademie befinden.
( When Fourier in one of his first works on heat, communicated to the French Academy on Dec 21 1807, stated that an arbitrary function (given in a graphic way) could be expressed by a trigonometric series, this statement was to the old Lagrange ::;o unexpected that he opposed it in the strongest way. There should still be a written document about this in the Archives of the Parisian Academy. ) Let me explain the phrase, "dem greisen Lagrange. " In December 1807, La grange was 71 ; the other members of the committee were much younger : Monge 61, Laplace 58, Lacroix 42, and Fourier was 39. Concerning the "Schriftstiick" , a footnote explains that the information carnes from Dirichlet, who had known Fourier in Paris. I have looked for this document, and the second part of this article describes what I found. Riemann then discusses matters of priority and concludes :
Durch Fourier was nun zwar die Natur der trigonometrischen Reihen vol lkurnmen ·richt·ig erkannt i sie wurden seitdem in der rnathematischen Physik zu Darstellung willkiirlicher Funktionen vielfach angewandt, und in jedem einzelne Falle iiberzeugte man sich leicht, dass die Fourier 'sche Reihe wirklich gegen den Werth der FUnction convergiere i aber es dauerle lange, ehe d·ieser w·ichtige Satz allgernein bewiesen wurde.
( Through Fourier indeed was the nature of trigonometric series fully understood ; since then they were applied many times in mathematical physics for representing arbitrary functions, and in each case one was easily convinced that the Fourier series really converges to the function ; but it lasted long before this important theorem was proved in full generality. ) As a last comment on this section, the term of "Fourier series" was not classical when Riemann wrote his thesis. He was the first to emphasize the importance of the notion. Nowadays a Fourier series is a trigonometric series whose coefficients are given by the Fourier formulas. Therefore it depends on the kind of integral
PARTIAL DIFFERENTIAL EQUATIONS
193
we consider, and there are indeed many to consider Fourier-Riemann, Fourier Stieljes, Fourier-Lebesgue (the most important now) , Fourier-Denjoy, Fourier Wiener, Fourier-Schwartz series, Haar-Fourier series on compact abelian groups, etc. Before going to Dirichlet in the third :->ection, Riemann mentions several com petitors and several mistakes. Cauchy's mistakes were surprisiug a,ud fruitful : one was about convergent series, another on analytic functions, the very domains where Cauchy made such brilliant contributions. Both were pointed out by Dirichlet and raised important observations by Riemann. The work of Dirichlet on R) -+ R�oo and there exists a strictly convex cone r C !Rn and a y0 1
supp u ( - , 0)
then we must have u
=
C
supp u ( ·, 1 )
Yo + f,
0 on !Rn
x
E
C
!Rn such that
Yo + r,
[0, 1 ] .
Clearly, taking V (x t) = J u J 2 (x, t) , we recover Zhang's result mentioned above. This was extended by [IK] who considered more general potentials V and the case when r !Rf- . For instance, if v E L � (R.n X [0, 1] ) or even v E Lf LH!Rn X (0, 1]) with 2/p + njq :::; 2, 1 < p < oo (n = 1 , 1 < p < 2) or V E C([O, 1 ] ; Lnf 2 (R.n)) n � 3, the result holds with r a half-plane. Our extension of Hardy's uncertainty principle, to this context, now is: ,
=
THEOREM
1 .6 ( [EKPV2] ) . Let u be a solution of iEltu + 6u + Vu 0, in lR.n X [0, 1]. =
Assume that V E L00(lR.n
x
[0, 1] ) , 'Vx V
E
LU(O, 1]; L;' (IR., )) and
}l� J J V I I L! L=(i xi> R) = 0.
If there exists a > 2, a > 0, such that u ( ·, 0), u(·, 1) E H1 (ealxl"' dx), then u = 0.
It is conjectured that Theorem 1.6 remains valid assuming only that u, 'Vu at timeH 0 , 1 are in £2 ((Yo + f), ealx l "' dx) , with Yo + r as in Theorem 1 .5. This extension of Theorem 1 . 6 would clearly imply Theorem 1.5. Let me sketch the prof of this result. Our t:>tartiug point is: LEMMA 1 .7 ( [KPV3]) .
3E >
0 s.t. if I J V I IL1 L;;o :::;
iEltU + 6u + Vu = H,
and uo (x) = u (x, 0) , u1 (x)
X
and u solves (0, 1],
u(x, 1) belong to L2 (e2f3x1 dx) n £2(dx) and
II E Li{L2 ( e 2f3x1 dx} =
in lR.n
t"
n
L2(dx)),
then and
S C { 1 J uo J I £2(e2i3zidx) + J l u1 J l£2( e213"' dx) + [ [ H I I L!£ 2 (e2il"I dx) } with C independent of {3.
This is a delicate lemma. If we a priori knew that u E C([O, 1]; L2(e2f:3x1 dx) ) , a variant of the energy method, splitting frequencies into 6 > 0, 6 < 0 , gives the result. But, since we are not free to prescribe both u0, u1 , we cannot use a priori estimates. This is instead accomplished by "truncating" the weight 2{3x1 and introducing an extra parameter. Or next step is to deduce, from Lemma 1. 7, further weighted estimates:
2 11
QUANTITATIVE UNIQUE CONTINCATION . . .
COROLLARY
1. . Assume that we ar-e under the hypothesis of Lemma 'l and a > 0, a >8 1, u0 , u1 E L2(eal x l "' dx) , H E dx) . Then 3Ca > 0, b > 0 s.t. sup e x l " i u {x, tWdx < Idea for the proof of Corollary 1. 8 : Multiply u by ?1R (x) fJ(x/R), 0 for l x l � 1, 17 = 1 for l x l > 2. We apply Lemma 1.7 to un(x, t) (x/R)u {x , t) , with (3 rRcx_-l, for suitable and the corollary follows. 2 1.
for some
Li,L; (ea lxl
O 0 s.t. a;2 l lea:liH(t) 0, (} > 1, then 3Ce > 0, b > 0 s.t. sup f
O Ce
e*l 9 l u(x, tWdx < oo
when the (complex) potential verifies I I V I I q L� � e, E = E1, . We will next re examine this result and precise it, in the case (} = 2. \Ve will first deal with potentials V V (x), V real valued; I I V I Ioo ::; M1 . We will consider u E C([O, 1] ; L 2 (1R11 } ) which verifies 8t u i(l:,.u + Vu) in !Rn [0, 1] . We will assume that there exist positive numbers o: and ;3 such that ! l e l x l 2/�7u(O) I I , l l e lxl2 / a 2 u(1) II are finite. Here and in the sequel I I II denotes the L2 norm in Then =
=
X
l e:xl2 /(at+( l -t)/:l)2 u(t) l
x.
·
at+( l -t) f:l
is "logarithmically convex" in [0, 1] , i.e. THEOREM 2.1 ( (EKPV5] ).
have
There exists N =
N(o:, ;J)
so that for O
0, ¢R(x) { �J2 JJ xx Jl :S RR ' �
=
choose a radial mollifier ()P and set
r/Jp, R (X, t)
= h(t)Bp
-->
oo,
2 18
CARLOS E. KENIG
where I I V I I u "' � M1 , SUP [o , 1J I I e-r l x l 2 F(t) l l / l lu(t) l l = M < oo, and l l e'lxl2 u(O) I I , 2 l l e'lx l 2 u(1) 1 1 are finite, we have a "log convex" estimate, uniformly in a > 0, smalL In fact, we now repeat the formal argument, but replace ¢(x) = l x l 2 by lx l � 1
l x l 2: 1
and then by ¢ , p (x) = eP * if;€ , where eP E C'Q is radiaL We then have: ¢€,p E C1 , 1 , it is convex and grows at infinity slower that l x l 2-' and 0 , E > 0, p > 0, our argument applies rigurously, since u(O)e'lxl2 E L2 => 0 < t < 1 , u(t)e'Yixl z _, E L 2 , and for a t independent if; ,
€
St + [S, A] = -'Y(a2 + b2) (4\7 (D2¢\7) - 41'2 D2¢\7¢ . \7¢ + .t:-2¢) .
One can see that l l .t:. 2 ¢e,p l l oo ::::; C(n, p)t:, which gives the desired log convexity when 0, then p ____, 0, for a > 0. Once the log convexity holds, for a > 0 again, the "local smoothing" argument applies. The conclusion of these considerations is: ·
€ ____,
LEMMA 2.4. Assume that u E L 00 ([0 , 1]; L2(1Rn)) n L2( [0, 1]; H 1 ) verifies Btu = (a + ib) (.t:.u + V(x, t)u + F (x, t)),
'Y > 0 where a > 0, b E IR, I I V I I oo ::::; M1 . Then, 3N-y s.t.
in lRn
x
[0, 1] ,
sup l le1'1xl2 u(t) l l ::::; [0, 1]
::::; e
N"� [(az +bz)[M�+M�]+ �(M, +Mz )]
l l e-y lxl z u(O) I I l -t l l e'Y i xl2 u(1) W ,
l l vt( l - t) e'Y i xl 2 u iiL (Rn x [O , l] ) ::::; N-y (1 + M1 + Mz ) 2 where Mz
=
SUP[o,1] l l e'Yi xl2 F (t) l l/ l l u(t) l l < oo.
{
sup l l e-r l x l 2 u(t) l l [0.. 1]
}
,
Conclusion of the argument when V ( x, t) = V (x), real. We now consider the Schrodinger operator H = .t:. + V, which is self-adjoint. We consider u E C([O, 1]; L2) solving Bt u = i((L. + V)u) in IRn X [0, 1] and assume that l l e'lxl2 u(O) I I < oo, l l e1'1x l2u( l ) l l < oo. From spectral theory, u(t) = eiHt u(O). Moreover, for a > 0, consider the solution of BtUa = (a + i)((6 + V)ua )
in IRn
X
[0, 1], Ua (O)
=
u(O).
We now have Clearly
= l l e'Y i xl 2 u(O) I I · a Also, ua(1) = e Hu(l). Recall, from the "energy method" that if
l l e'Y i xl2 Ua(O) I I
{
Btv = a(L. + V)v v(O) = vo
'
V real,
l l e'Yalxlz /(a+4-yaz) v (1) I I � exp(Ml ) l l e-rlxl z vo l l '
219
QUANTITATIVE UNIQUE CONTINUATION . . .
= u(1), then v(1) = eaHvo
where that
M1
Let Ia obtain
= 1/(1 + 41a) and apply now our log-convexity result for ua, Ia · We then
=
l l aVI I Ll([o,l];L=)·
Now, if vo
l e'Yixl2 /(1+4rya) Ua (1) I
::=;
=
ua (1), so
exp(Ml) j je'Yi x12 u(1) I I ·
l l e'Ya lx l 2 Ua(s) l l :S eNMl l le'Yalx l z Ua(1) 1 1 1 - s l le'Y" Ixl2 Ua (O)W :S
:::; eNMl exp(Ml ) l l ellxl2 u(1) w-s l l e"Yixl 2 u(O) W .
We then let a - 0 and obtain the "log convexity" bound. To obtain the "local smoothing" bound, we again use the ua , let a ! 0. This establishes Theorem 2.1 when a = (3.
REMARK 2.5. Solutions so that e'Yix l 2 u(O), el l x l 2 u(1) E L2 certainly exist for some I· In fact, if h E L2 (ef l x l2 dx) and u0 = e6Cli.+V) h, our "energy method" gives this for u(t) = ei t (li.+V) u0 , (V = V(x)) . (We are indebted to R. Killip for this remark. ) When V = 0, this characterizes such u! (see [EKPV4]) .
A misleading convexity argument : Consider now f = ea(t) J xl 2 u , where u
solves the free Schrodinger equation
OtU = i6u in IR x [-1 , 1]. Then,
f verifies
otf = Sf + Af, S = -4ia(x8x +
In this case we have
St + [S, A]
=
�)
+ a'x2,
a' 2 --;; S - 8ao; +
(
A = i(o; + 4a2x2 ) 32a3
+ a" - 2
-)
.
(a1 ) 2 - x2 • a
If a is positive, even, and a solution of (a') 2 32a3 + a" - 2 = 0 in [-1, 1], a then our formal calculations show that 8t (a 1 8t log Ha (t) ) � 0 in [-1, 1 ] . Hence, for s < t we have
-
--
a(t)8t log Ha (s) :S a(s)8t log Ha (t) . Integrating between [- 1 , 0] and [0, 1] and using the evenness of a, we conclude Ha( O) :S Ha ( - 1 ) 112 Ha (1) 1 12 .
Now, if a solves
{
32a3 + a11
-
2 (a ) 2
�
a (O ) = l, a'(1) = 0
=
0
a is positive, even, and limR-""' Ra(R) = 0. Also, aR(t) equation. If the formal calculation holds for HaR '
= Ra(Rt) also solves the
CARLOS E. KENIG
220 In particular, u = 0. But
u(x, t) = (t - i) -112 eilxl2 f4(t -i) is a non-zero free solution,
which decays as a quadratic exponential at t
=
±1.
3 . The case a -:;6 (3 ; the conformal or Appel transformation
Assume u(y , ) verifies 85u = (a + ib) (6u + V ( y , s) u + F(y, s)) in !Rn [0, 1] , a + ib -:;6 0, a > 0, (3 > 0, IR and set ..j(ifJ X fjt u(- x, t) = a(l Jcif3 u - t) + fjt a(l - t) + fjt ' a (l - t) + f]t (a - f3)1xl2 x exp 4(a + ib) (o(l - t) + {3t) Then u verifies OtU = (a + ib)(6u V ( x , t)u + F( x , t)) in !Rn [0, 1], ( o/3 fjt x V o ( l..j(ifJ V x t) = (o(l - t) + /3t)2 , - t) + ,6't ' o ( l - t) + ,6't ' X fjt F( x t) = ..j(ifJ F o(l..;c;p , (o( l - t) + f3t ) � +2 - t) + ,6't ' o(l - t) + f3t · Moreover, if = ,6'tj (o(l - t) /3t), [ 7"'/3 �a -,:3)4 ] 1 12 u (s ) I e'"Yixl u(t) II e [ a/3 I I eil x l " F(t) I I = (o(l - t) + (3t) 2 e (•+ll(l-s))2 4(a2+b2)(c.s+;J(l ] 1 y12 F( s) . s
LEMMA 3.1.
x
'Y E
) (
(
)
n/2
(
+
+
s
=
2
I
•
(
x
(
(o.s+ll(L- s))2 + 4(a2+0 )(o.a+il(L-s))
I
)
7nil
+
y
(u-{l)a
•))
I
)
.
)
I
The proof is by change of variableH. Conclusion of the proof of Theorem 2.1: \Ve can assume o -:;6 /3. We can also assume o < (3 (change for u( l - t)). (This gives (a - fj)a < 0.) As before ,
u
Ua e(a +i)tHu(O) eatHu(t) ,
= H = (6 + V), By the "energy estimate" we now have
=
a > 0.
ll e l xl2 /a2 ua ( l) l l s; eaiiVIIoo ll el xl,/a, u ( l)l l and I Jelx l 2 /�2 Ua (O) I I s; ll e ! xl 2 N2 u(O) II · We now have also OtUa = (a + i) ( 6·ua + Vua), so when we do the Appel transform, we have, with 'Ya = l foa f3a, a Otiia = (a + i)((6 + v )ua ) , where
ya (
)
aaf3a
(
V (aa (l� -
)
X x , t = ( a a( l - t ) + f3a t) 2 t) + f3a t . Now, fo r a > 0 we have "log convexity" in this last problem. Moreover, by the Appel Lemma and our definitions, we have
ll elalx l \ia(O)II s; llelxl 21!3\t (O) II ' ll e-ralxl2 ua(l)ll s; eai!VIIoo llel xl 2/a2 u( l) ll ·
22 1
QUANTITATIVE UNIQUE CONTINUATION . . .
Thus,
I e"1a lx l 2
Ua (t)
I
:S:: e N( l +M, +Mf) ea i i V I I "'
II
e lxl l /132 u (O )
1 1 -t ll e lx l 2 /a? u ( 1) w
and the corresponding "local smoothing" eHtimate. But now, letting a -+ 0 and changing variables our result follows. Time dependent, complex potentials: We will consider complex potentials V(x, t) , I I V I I oo :S:: 1'\tfo . We will also assume
N!:!!o I I V I I £l ((O, lj, L oc (lxi>R)) = 0.
We first recall a result in [KPV3] . LEMMA
3.2. There exists N
=
N ( n) , t:o
=
t:o (n) > 0 so that, if X E !Rn,
V E L 1 ( [0, 1]; L00), I I V I I £l ((O.l);Loc) :S:: Eo , then if u E 0([0, 1]; L2 ) satisfies OtU = i(6.u + V(x, t)u + F(x, t )) in !Rn x [0, 1], then
THEOREM
3.3. Let V
0([0, 1 ] ; L2) solve
E
L}L�,
limR-u I I V I I u ( [O, lJ,Loc(l x i>R))
Ot U = i{6u + V (x, t)u)
Assume in addition that V E
L
00
(!Rn -.- l ) , and that
ll el x l 2 /i32 u {O ) II < oo ,
Then, 3N = N (a, {3) s. t.
sup ll e lxl2 / (at+(l-t).6)2 u (t ) l l [0,1 )
+
in JR.n
x
[
0. Le t u E
[0, 1].
l l e lx l2/ 2 u (l ) ll
< oo.
ll vt( 1 - t ) elx!2 /(ot+( l -t) i3) 2 V'u(t) l l
C" N ,NI I VII�
=
l l e l • l ' 1•' u(O) II + II e l • l ' I•' u ( l )
PROOF. We start out by using the Appel transform, 1/a{), (a + ib) = ·i . We now have ii E 0([0, 1 ] ; L2 ),
I
£2(JRn x [0,1J )
+
f,�� l l u (t) I l
:::::;
l
·
ii(x, t) an
THEOREM 4. 1 . Let V
I I V I I oc
< oo , limR_,o
solution of
V(x), V
real, [jV[[oo < oo, or V = V (x, t), V complex,
I I V I Iu([lt,l},T�oo( l xi>R)) = 0 . Assume that u E C ( [O, 1] ; L 2 ) is a =
OtU =
i(.6u + V(x, t)u) in IR.n [0 , 1], such that elx l 2 u(O ) E L2 , e l x l 2 /a2 u( 1 ) E L2, and, Ct,B < 2. Then //3 2
x
u =
0.
Preliminaries: Let 'Y = 1 /a.B . Using the Appel transform and our convexity and "smoothing" estimates we can assume, without loss of generality, that the following holds for 'Y > 1 / 2 : (4 . 1 )
sup ll e�' l xl 2 u(t) ll (0, 1 ]
I 0,1]
+ sup yft( 1 - t)e'Yi x '
£2 [
2 V'u(t ) l
l
£ 2 (R" X [0, 1 ])
0, set f = el-'lx+ Re,t ( I-t) l 2 u , where 0 < J1 < ., , and H (t) = (!, f). At the formal level, it is easy to show (for the free evolution) that oz log H (t) ;:::: -R2 /4J1, so that H(t)e-R2 t(l - t)/&J-L is log convex in [0, 1] and so Letting J1 i
'Y
H ( 1/2) S JI {O ) l /2 H ( 1 ) lf2 e R2 /32�-' .
we see that
J
e 2,[x+ .!!p- [2 [u( 1/ 2) 1 2 S ll er-l x l 2 u( O ) ll ll e' l x l 2 u( 1 )
Thus,
r la(• R/ 4)
I eR2 /32�r .
[u(1/2) 1 2 S ll e/JxJ2 u( o) l l l l e-yJxJ2 u(1) 1 1 e [R2 (l - 412 (1 - E) 2
)Jj:32·r ,
0 < f. < 1 , which implies u(1/2) = 0 as R -+ oo, b > 1/2). The path from the formal argument to the rigorous one is not easy. Vl/e will do it instead with the Carleman inequality: [0,
LEMMA 4. 2 . Let ¢>(t) , 1./J( t) be smooth functions on [0, 1 ] , g (x, t) 1]) , e1 = ( 1 , 0, . . . , 0) . Then, fur· J1 > 0, we have (for R » 0},
jj [·tj/'(t) - 3�: [¢"(tWJe2(t )e1 [ 2+w (t) (i8t + 6)g = 51-' f + A�-' f'
x
224
CARLOS E. KENIG
where S1, = s; , AJJ. inner product ) , and
- A� {the adjoints are now with respect to the L2 ( dxdt)
=
41-L !:._ + ¢e 1 A11 = vVe then have:
( R R
Jl e2w{t) e21' i }4: -¢(t)e, l 2 j ( i8t
)
+
·
V' -
21-Ln
R2
- 2iw.t/
( xR1 + ¢e 1 ) - ·i'lj/.
6 )g j2 =
= ( ( SJJ. + AJJ. ) J, ( SJJ. + AJJ. ) f) = {SJJ.j, SJJ. J) + (AJJ.j, AJJ.f) + + (S11 f, AJJ. f) + (A�' f' SJJ. f ) � { [S�' , A�']! , f ) .
We now compute [Sf! , AIL] and obtain: ' '13Jt [Sf! , A IL] = - R'2 !::,
+
32 /.L3
R4
Thus,
1
1 R + 2/.L ( � X
2 + ¢el + +
¢ e 1 ¢11 + 2t.L (¢1 ) 2 - Bi q/ Bx , + 'lj/1•
�
)
and the Lemma follows. Next, choose ¢(t)
=
't//' (t) -
t ( l - t), lj.•(t) = - ( 1 + E)
R4
( /') 2 (t) = 321-L q
(1
1�: t(l - t). Then
+ t: ) 4 - R4 = .!__ 4 R R
8{.l
and so our inequality reads, for g E C0 (JR.n
x
8{.1
[0, 1] ) ,
8!-L
0
22!j
QUANTITATIVE UNIQUE CONTINUATION . . .
+
We next fix R > 0, recall that u solves i8t'u t:::. u = Vu, and that the estimates (4.1) bold. Choose then rJ ( t), 0 :::; rJ :::; 1 , rJ = 1 where t ( 1 - t) � 1 / R, 1) = 0 near t = 1 , 0, so that supp rJ' c {t(1 - t ) � 1/R }, lrJ' I � C R.
g(x , + (
Choose also M » R, e E COC (IRn), and now set t) rJ(t)(}(x/Af)u(x , t), which is compactly supported in IR" x (0, 1 ) , so that our estimate holds.
( t·at -t-· u. /\ )g Finally,
let
J.L =
+
vg
=
I
Zr]' ( t)e
·
+
(x)
II
M
u
=
x)u 1 " () ( TI
TJ'iu.
+
+ 2vO(x/M)·vu)
III.
M
(1 + t) - 3rR2 • Our inequality then gives:
� ( 1 �t) 3 Rz if e2•i>(tJ ez,, J J'i H( tJed2 JgJ2 :::; �
The
contribution
JJ e21i,(t) e2'' 1 -TI H(t)c t j2 {I + II + TIT} .
of I to the right hand side is bounded hy I I V I I ::>a
jj ez..!(t)e2., j i;"H(t)P, I ! Igl2, =
so that, if R is very large, we can hide it in the left side, t o see that we only have � to deal with II and III. Recall that '1/J(t) (1 + t) 1�>(1 - t) 0, so e2•b(tJ � 1. On the support of 1)1 , we have t( l - t) � 1/ R, so that 0 � ¢(t) � 1/ R. We 11ow estimate 2 J.L
I � + ¢(t)e112
=
(�:��:J { I� 1 +
� ¢(t ) ¢(t)2 } ( 1 �'Yt) :J l xl 2 + , 2, 2"y 2 z 2 ,/x R t) + ( 1 t 3 R ¢ (t ) :::; 21 1:r l + ) ( 1 + E) 3 1 q;( 2
+
2
=
+
C,
+
on supp rJ', where ¢(t) :::; 1 /R. Thus, because of {4. 1 ) , the contribution of II is bounded by C€R. The contribution of II I is controlled by (recalling that rJ = 0 � when t ( 1 - t) � � )
C4 /r{
M
ji:r. IS2M
x
e, j , + i u ( , t) J 2 e21/:( t) e21• 1 1i: H (t) +
c M2
j"Jlx'r �2M I V'tt(x, t ) i 2 e2.;•(t) e2J.t l "fi +cl>(t)c, J 2 t ( l - t )R .
If we use (4. 1 ) , 7/J � 0, the bound above for 2tt j:I;j R + ¢ (t)e1 1 2 beeome::;
�
2r lxl2
( 1 + !) 3
Thus, letting .1t1 -.
+
2"Yi x t l R ( 1 + t) 3 4
R2
21 2 (l + E) 3 16 � 2rl xl + CcR·
we see that, for fixed R, II I -. 0, so that, since
t ( 1 - t) � 1/ R, we obtain: oc ,
+
:_ (1 + t)" 2 R 8 1
/r{
lt(t-t)'?.l/R
e2�·(t) e2�J "ff H( th i, J'tlf � C R .p .
17
=
1
011
22 6
CARLOS E. KENIG
We are now going to restrict to integration over the region where 8 is small, to be chosen. Then,
so that
I � I S o, I t - 1/21 S 8 ,
I.:_ + ¢(t)e1 j 2 - 2._16 - 26 (�2 - 28) . R
>
so that, in our region of integration,
smce J.l = _l_ (l+e)3 R2 . But, if 1 > 1/2, .
"Y
(1 + e) 3
-
(1 + E)4
4�r
>
0'
for some E small, and so, for /j smaller than that we get a lower bound of C,,,sR2 • We thus have But then, since
I
f
ll t - 1/21 � 6 11 7\' l>o
iu1 2 =
e2'Yixl2 e - 2')'lxl2 1 u l 2 < f f llt- 1/21�6 Jl i l 9 s
e- 2'Y82 R2
r llt-1/21 9
by (4.1), we see that , for appropriate c, ,,,t5 we have
(lr
l t- 1/2 1 �6
Letting R ---+ oo , we see that u
=
I iui 2 ) ec, ,,,6 R2
J e2'Yixl2 1ul2
s
s
C'Y e -2'Y!j2 R2
C-.,.,e,t5·
0 on { (x, t) : I t - 1/21 S 8 } , therefore u = 0. References
[A] [BK]
P. Anderson, A bsence
1492-1505. J. Bourgain and
of diffusion in certain mndom lattices, Phys. Review .
C . Kenig
dimensions, Invent. Math
[C KM]
,
109 ( 1 958),
On localization in the Anderson-Bernoulli model in higher
161 (2005), 38�426.
R. Carmona, A. Klein, and F. Martinelli, Anderson localization for Bernoulli and other
singular potentials, Commun. Math. Phys. 108
(1987) , 41-66.
QUANTITATIVE UNIQUE CONTINUATION
22 7
[EKPV1] L. Escauriaza, C. Kenig, G. Ponce, and L. Vega , Decay at infinity of calonc func tio ns within characte1·istic hyperplanes, Math . Res. Lett. 13 (2006 ) , 441-453. [EKPV2] , On uniqueness properties of solutions of SchTo ding er equations, Commun. in PDE 31 (2006), 181 1-1823. [EKPV3] --- , On uniqueness properti es of solutions of the k-generalized KdV equation.�, Jour. Ftmct. Anal. 244 (2007}, 504-535. [EKPV4] , Convexity properties of solutions to the free Schroding er equation with Gauss ian decay, to appear, MRL. [EKPV5] , Hardy 's uncertainty prin cip l e, convexity and Schrodinger evolutions, to appear, JEMS. L. Escauriaza, G. Seregin, and V. Sverak, £3·00 solution.� to the Navier-Stokes equations [ESSJ and backward uniqueness, Russ. Math. Surv. 58:2 (2003) , 2 1 1-250. .J. Frolich and T. Spencer , Absence of diffusion with Anderson tight binding model for· [FS] large dis order or low en ergy, Commun. M ath . Phys. 88 (1983), 1 5 1 -1 84 . [G MP] Y. Goldsheid, S. Molchanov, and L. Pastur, Pure point spectrum of st ochasti c one dime11.� ional Schrodinger operators, Funct. Anal. Appl. 11 (1977), 1-10. A. lonescu and C. Kenig, LP Carleman inP.quali tie s and uniqueness of solutions of [IK] non-linear Schrodinger equations, Acta Math. 193 (2004) , 193-239. [I] V. Isakov, Carleman type estimates in anisotropic case and applications, J. Diff. Eqs. 105 (1993), 217-238. C. K enig , Some recent quantitative unique continuation theorems, Rend. Accad. ::-laz. [K 1 ] Sci. XL Mem. Mat. Appl. 29 (2005) , 231-242. , Some recent applir.ations of unique continuation, Contemp. Math. 439 (2007), [K2] 25-56. [ K PV1] C. Kenig, G. Ponce, and L. Vega, On the support of solutions to generalized Kdll equatio n, Annates de l'Institut H. Poincare 19 (2002), 1 9 1-208 . [K PV2] , On th e unique continuation of s o lu ti o ns to the ge neralized Kd V equati on , Math. Res. Lett 10 (2003), 833-846. [KPV3] , On uniquP. continuation for nonlinear Schro dinge r equatwns, Commun. Pure Appl. Math. 60 (2002), 1247-1262. E. M. Landis and 0. A. Oleinik, Ge ne 1·alized analyticity and some related properties of [LO] solutions of elliptic and parabolic equations, Russ. Math. Surv. 29 (1974), 195-212. V. z. Meshkov, On the poss ib le rate of decay at infinity of s o lut i o ns of second ord fr [M] partial differen ti al equations, Math. USSR S obornik 72 (1992) , 343-360. L. Robbiano, Unicite forte a l 'infini pour KdV, ESAIM Control Optim. Calc. Var. 8 [R] (2002), 933-939. [SVW) C. Shubin, R. Vakilian, and T. Wolff, Some ham�onic analysis questions suggested by Anderson-Bernoulli models, GAFA 8 (1998), 932-964. [SS] E. M. Stein and R. Shakarchi, Complex an alysis, Princeton Lectures in Analysis, II, Princeton University Press, Princeton, NJ, 2003. B . Y. Zhang, Uniqu e continuation for the Korteweg -de Vries equation, SIAM J. Math. (Z1] Anal. 23 (1992), 55-7 1 . [Z2] , Unique continuation for the nonlinear Schrodinger equation, Proc. Roy. Soc. Edinburgh Sect. A 127 (1997), 191-205. ___
___
___
___
___
___
___
DEPATMENT
OF
MATHEMATICS, UNIVF.RSITY OF CHICAGO, CHICAGO, IL 6063 7 , USA
E-mail address: ce k�ath . uchicago . edu
Proceedings of Symposia in Pure Mathematics Volume 79, 2008
Boundary Harnack Inequalities for Operators of p-Laplace Type in Reifenberg Flat Domains John L . Lewis , Niklas Lundstrom, and Kaj Nystrom In this paper we highlight a set of techniques that recently have been used to e stablish bou nd ary Harnack inequalities for p-harmonic functions vani shing on a portion of the boundary of a domain whi ch is 'fiA.t' in the sense that its boundary is well-approximated by hyperplane!;. Moreover, we use these techniques to establis h new results concerning boundary Harnack inequalities and the Martin boundary problem for operators of p-Laplace type with variable coefficients in Reifenberg flat domains. ABSTRACT.
1. Introduction and statement of main results In [LN] , [LN1], [LN2], see also [LN3] for a survey of these results, a number of results concerning the boundary behaviour of positive p-harmonic functions, 1 < p < oo, in a bounded Lipschitz domain D c R,. were proved. In particular, the boundary Harnack inequality, as well as Holder continuity for ratios of positive p-harmonic functions, 1 < p < oo, vani�:;hing on a portion of 8D were established. FUrthermore, the p-Martin boundary problem at w E 8D was resolved under the assumption that D is either convex, C 1 -regular or a Lipschitz domain with small constant. Also, in [LN4] these questions were resolved for p-harmonic functions vanishing on a portion of certain Reifenberg flat and Ahlfors regular NTA-do.tnains. From a technological perspective the toolbox developed in [LN, LN1-L�4] can be divided into (i) techniques which can be used to establish boundary Harnack inequalities for p-harmonic functions vanishing on a portion of the boundary of a domain which is 'flat' in the sense that its boundary is well-approximated by hyperplanes and ( ii) techniques which can be used to establish boundary Harnack inequalities for p-harmonic functions vanishing on a portion of the boundary of a Lipschitz domain or on a portion of the boundary of a domain which ean be well approximated by Lipschitz graph domains. Domains in category (i) are called Reifenberg flat domains with small constant or just Reifenberg fiat domains. They indudc domains with small Lipschitz constant, C1-domains and certain quasi-balls. Domains in category (ii) include Lipschitz domains with large Lipschitz constant and certain Ahlfors regular NTA-domains, which can be well approximated by Lipschitz graph domains in the Hausdorff distance sense. The purpose of this paper is to highlight the techniques labeled as category (i) in the above discussion and to use these techniques to establish boundary Harnack inequalitie!:i as well as to 2000
Primary 35J25, 35J70 . Keywords and phrases: boundary Harnack inequality, p-harmonic function, A-harmonic function, variable coefficients, Reifenberg fiat domain, Martin boundary. Lewis was partially support ed by NSF DMS-0139748. Nystrom was partially supported by grant 70768001 from the Swedish Research Council. Mathematics 81;.bject Classification.
Key words and phrases.
229
©2008 American Mathematical Society
230
JOHN L.
LEWIS . NIKLAS LUNDSTROM, AND KAJ NYSTROM
resolve the Martin boum.lary problem for operators of p-Laplace type with variable coefficients in Reifenberg fiat domains. To state our results we need to introduce some notation. Points in Euclidean n sp ace Rn a.re denoted by x = (xb . . . , Xn ) or (x', Xn ) where x' = (x1 , . . . , x,__I ) E R"- 1 . Let E, 8E, diam E, be the closure, boundary, diameter, of the set E C R" and let d(y, E) equal the distance from y E R" to E. ( , ·) denotes the standard inner product on Rn and lxl = (x, :r:)112 is the Euclidean norm of x. Put B ( x r ) = {y E R" : l x - Y l < r} whenever x E Rn , r > 0, and let dx be Lebesgue n-measurc on Rn . We let -
-
,
h(E , F)
=
max(sup{d(y, E) : y E F} , sup{d(y, F) : y E E} )
be the Hausdorff distance between the sets E, F C R". If 0 c R" is open and 1 :::; q :::; oo, then by w·l,q (O) we denote the space of equivalence clas::;e:; of functions f with distributional gradient 9 f Ux , , . , fxJ , bo th of which are q th power integrable on 0. Let. IIJII1.q = llfllq + ll l 9f l ll q be the norm in Wl. q (O) where ll · ll q denotes the usual Lebesgue q-norm in 0. Next let C0 (0) be the t:et o f infinitely q differentiable functions with compact support in 0 and let Wci'' ( 0) be the closure of C0 (0) in the norm of W 1 ·" (0). By 9· we denote the divergence operator. We first introduce the operators of p-Laplace type which we consider in this paper. =
.
.
Definition 1.1. Let p, ;3, a E (1, x ) and 1 E (0, 1). Let A = (A1 , . . . , A,. ) : Rn X an __, Rn , assume that A = A (x, 17) is continuous in R" X (R" \ {0 } ) and that A(x, 1J), for fixed x E Rn, is continuously differentiable in 17k , for every k E { 1 , . . . , n}, whenever 17 E Rn \ { 0}. We say that the function A belongs to the class A1p(a, 8, -y) if the following conditions are satisfied whP:n.ever x, y, e E Rn and 17 E Rn \ {0} :
(i) (i1: ) (iii) (iv)
a - 1 11JIP - 2 1�1 2 :::;
n 8A
L
� (x , 17)�i�j , i,j=l 17J
(x , 17 ) :::; a i1J I P- 2 , 1 :::; i , j :::; n,
I �� 1
I A ( x 17) - A ( y , 17) j :::; tJi x - Yl7lrJJ P - I , A (x , 77) = l11lp-l A ( x , 11/ l771 ) . ,
For short, we write l'vfp (a) for the class Mp (a , 0, 1) .
Definition 1.2. Let p E (l, oo) and let A E Mp (c., f3, �r) for some (a, (3, -y) . Given a bounded domain G we say that u is A-harmonic in G provided u E W1 ·P(G) and (1 .3)
j (A(x, 9u(x) ) , 98(x)) dx
=
0
whenever () E Wci- ·P(G) . If A(x, 17) = I77I P -2 (171 , . . . , 17 ), then u is said to be p n harmonic in G. As a short notation for (1 . 3) we write 9 · (A(x, 9u)) = 0 in G.
The relevance and importance of the conditions impo�ed through the assump tion A E Mp (a, (3, -y) will be discussed below. Initially we just note that the class Mp (a, (3, -y) is, �ee Lemma 2. 15, closed under translations, rotations and under di lations x __, rx, r E (0, 1] . Moreover, we note that an important class of equations
BOUNDARY HARNACK I�EQUALITIES FOR OPERATORS OF p-LAPLACE 'l'YPE
which is covered by Definition 1 . 1 and \7
( 1 .4)
·
[
2 31
1.2 is the class of equations of the type
]
(A(x)\7u, \7u)PI2-1 A(x)\7u
=
0 in G
where A = A(x) = {a;,1 (x)} is such that the conditions in Definition 1.1 (i) - (iv) are fulfilled. Next we introduce the geometric notions used in this paper. \Ve define, Definition 1 .5. A bounded domain n is called non-tangentially accessible {NTA) if there ex·ist M 2:: 2 and ro > 0 such that the following are fulfilled: (i)
(ii) (iii)
corkscrew condition: for any w E an, 0 < T < To, there exi�-;ts ar(w) E n n B(w, r/2) , satisfying M- 1 r < d(ar (w) , an), R11 \ f2 satisfies the corkscrew condition, uniform condition: if W E an, 0 < r < ro , and Wt , W2 E B(w, r) (1 n, then there exists a rectifiable curve 1 : (0, 1 ]-}n with ! (0 ) = w , �r(1) = w2 , 1 and such that (a) H1 (!) :::; M l w 1 - w2 l , (b) min{H 1 ( 1 ( (0, t] ) ) , H 1 (! ( [t , 1 ] ) ) } :::; M d (! (t) , 00) .
In Definition 1 .5, H1 denotes length or the one-dimensional Hausdorff measure. We note that (iii) is different but equivalent t.o the usual Harnack chain condition given in (JK] (see (BL], Lemma 2.5). M will be called the NTA-constant of n. Definition 1.6. Let n c R11 be a bo unded domain, w E an, and 0 < r < To . Then an is said to be uniformly (J, r0) -approximable by hyperplanes, provided there exists, whenever w E an and 0 < r < ro , a hyperpla.ne A containing w such that h(aO n B(w, r), A n B(w, r)) :::; 8r.
We let :F(8, r0 ) denote the class of all domains n which satisfy Definition 1.6. Let n E :F(J, r0), w E &0., 0 < r < ro , and let A be as in Definition 1.6. We say that &0. separates B(w, r), if ( 1 . 7)
{ x E n n B(w, r) : d(x, an) 2:: 28r} c one component of Rn \ A.
Definition 1.8. Let n C R11 be a bounded domain. Then 0 and &0. a1·e said to be (8, ro) -Reifenberg fiat provided n E F(J, ro ) and {1. 7) hold whenever 0 < r
0. We note that an equivalent definition of a
Reifenberg flat domain is given in [KT] . As in [KT] one can show that a 8- Reifenberg flat domain is an NTA-domain with constant M = M ( n), provided 0 < 8 < J and J is small enough. In this paper we first prove the following theorem.
232
JOHN
L.
LEWIS,
NIKLAS LUNDSTRO M , AND KAJ NYSTR O M
Theorem 1 . Let n c Rn be a (o, r0) -Reifenberg fiat domain. Let p, 1 < p < oo, be given and assume that A E Mp (a., f3, /) for some (a, f3, /) . Let w E an, o < r < r0, and suppose that u, v are positive A -harmonic functions in n n B(w, 4r), continuous in O n B(w, 4r), and u = O = v on an n B(w, 4r) . There exists J < J, a > O, and c1 � 1 , all depending only on p, n, a, (3, /, such that ·if 0 < a < J, then (T u(y2 ) 1 < 11 u(yl) ( 1 YI Y2 i) og -- - l og c1 r v (y2 ) v(y i ) --
whenever Yl , Y2
E n n B(w, r/cl ) ·
We note that in [LN] we obtained for p-harmonic functions u, v , in a bounded Lipschitz domain n, whenever w E an, and Y 1 , Y2 E n n B(w, rfc) . Here c depends only on p, n, and the Lipschitz constant for n. Moreover, using this result, we showed, in [LNl], that the conclusion of Theorem 1 holds whenever u, v, are p-harmonic, and n is Lipschitz. Constants again depend only on p, n, and the Lipschitz constant for n. In this paper we also prove the following theorem.
Theorem 2. Let n c Rn, /), To, p, a, (3, , and A be as in the statement of Theorem 1 . Then there exists o* = /j* (p, n, a, f3, /) < J, such that the following is ,
true. Let w E an and suppose that u, v are positive A-harmonic functions in n with u = 0 = v continuously on an \ { w}. If 0 < a < o* , then u(y) = Av(y) for all y E n and for some constant >..
We remark, using terminology of the Martin boundary problem, that if u is W> in Theorem 2, then u is called a minimal positive A-harmonic function in n, relative to w E on . Moreover, the A-Martin boundary of n is the set of equivalence classes of positive minimal A-harmonic functions relative to all boundary points of n. Two minimal positive A-harmonic functions are in the same equivalence class if they correspond to the same boundary point and one is a constant multiple of the other. Note that the conclusion of Theorem 2 implies that u is unique up to constant multiples. Thus, llince w E an is arbitrary, one can say that the A-Martin boundary of n is identifiable with an. We remark that in (LNl] the Martin boundary problem for p-harmonic func tions was resolved in domains which are either convex, C1-regular or Lipschitz with sufficiently small constant. Also, in [LN4] the Martin boundary problem was resolved, again for p-harmonic functions, in Reifenberg flat domains and certain Ahlfors regular NTA-domains. Theorem 2 is new in the cW>e of operators of p Laplace type with variable coefficients. Recall that n is said to be a bounded Lipschitz domain if there exists a finite set of balls { B(xi , ri ) } , with X i E an and Ti > 0, such that { B ( xi ri ) } constitutes a covering of an open neighbourhood of an and such that, for each i, ,
(1 .9)
n n B( x ) an n B(x; , r;) i , r·i
{x {x
=
=
( x' , x.,. ) E Rn : Xn > ¢; (x')} n B(x; , ri ) , ( x' , Xn ) E Rn : Xn = ¢;(x')} n B(xi , ri ) ,
BOUNDARY HARNACK INEQUALITIES FOR OPERATORS OF p-LAPLACE TYPE
233
in an appropriate coordinate system and for a Lipschitz function c/Ji. The Lipschitz constant of n is defined to be M = maxi IIIV'¢i l lloo · If n is Lipschitz then n is NTA with ro = min rife, where c = c(p, n , M) 2:: 1. Moreover, if each ¢i : Rn - 1 -tR can be chosen to be either C1- or C1·a -regular, then n is a bounded C1- or C1•a -domain. We say that n is a quasi-ball provided n = f(B(O, 1 ) ) , where f = (/I , h , . , fn) : Rn -t Rn is a }( > 1 quasi-conformal mapping of Rn onto R n. That is, fi E W 1 •n(B(O, p) ), 0 < p < oo , 1 ::; i ::; n, and for almost every x E Rn with respect to Lebesgue n-measure the following hold,
.
(i) (ii)
lDf( x) l n
=
sup
l h l=l
I DJ( x )hln :S K lJt (x) l ,
J, (x) � 0 or J, (x) ::; 0.
for the Jacobian matrix of f and In this display we have written D f( x) = (�) 1 J, (x) for the Jacobian determinant of f at x .
Remark 1.10. Let n c Rn be a bounded Lipschitz domain with constant M. If M is small enough then S1 is (6, ro) -Reifenberg fiat for some 6 = o(M), r·o > 0 with 6(M) -t 0 as M -t 0. Hence, Theorems 1-2 apply to any bounded Lipschitz domain with sufficiently small Dipschitz constant. A lso, if n = f(B(O, 1)) where f is a K quasi-conformal mapping of Rn onto Rn, then one can show that an is 6-Reifenberg fiat, with r0 = L where 8-tO as K-t 1 (see [R, Theorems 12.5 -12. 7}). Thus Theorems 1, 2, apply when n is a quasi-ball and if K = K (p , n) ·is close enough to 1 . To state corollaries t o Theorems 1-2 we next introduce the notion of Reifenberg flat domains with vanishing constant. Definition 1 . 1 1 . Let S1 C Rn be a (o, ro) -Reifenberg fiat domain for some 0 < 6 < J, ro > 0, and let w E an, 0 < r < ro. We say that an n B(w , r) is Reifenberg fiat with vanishing constant, if for each E > 0, there exists r = r( E) > 0 with the following property. If X E an n B (w, r) and 0 < p < r, then there ·is a plane P' = P'(x, p) containing x such that h(Gn n B (x , p), P' n B ( x , p)) ::; f.p .
The following corollaries are immediate consequences of Theorems 1-2. Corollary 1. Let n c R" be a domain which is Reifenberg fiat with vanishing constant. Let p, 1 < p < oc , be given and assume that A E Mp(a, {3, ry ) for some (a, {3, ry) . Let w E an, 0 < r < To . Assume that u, v are positive A -harmonic functions in n n B (w, 4r) , u , v are continuous in D n B (w, 4r) and u = 0 = v on an n B(w, 4r) . There exist ri = ri (p, n, o:, {3, ry) < r and C2 = C2 (p , n, o: , ,B, ry) :::: 1 such that if w' E an n B(w, r) and 0 < r' < ri , then < C2
_
whenever
YI , Y2 E n n B( w', r') .
( IYI - Y2 l ) a r1
Corollary 2. Let [! c Rn , p, a, 13, ry and A be as in the statement of Corollary 1 . Let w E an and suppose that u, v are positi ve A-harmonic functions in n with
234
JOHN L. LEWIS, NIKLAS LUNDSTROM, AND KAJ NYSTROM
u = 0 = v continuously on on \ {w}. Then u (y ) = >.v ( y) for all y E n and for some constant >.. Remark 1 .12. We note that if n is a bmmded C1 -domain in the sense of {1.9) then n is also Reifenberg fiat with vanishing constant. Hence Corollaries 1-2 apply
to any bounded C 1 -domain.
Concerning proofs, we here outline the proof of Theorem 1.
Step 0. As a starting point we establish the conclusion of Theorem 1 , see Lemma 2.8, when A E .l'vfp(a), n is equal to a truncated cylinder and w is the center on the bottom of n .
Step A . (Uniform non-degeneracy of IV'ul - the 'fundamental inequality') . There exist u(y, T) , r E [0, 1], for fixed y E n n B(w, 4r* ) , is Lipschitz continuous with Lipschitz norm :S c. Thus uT(y, · ) exists, for fixed y E nnE( w, 4r* ) , almost everywhere in (0,1]. Let {Yv } b e a dense sequence o f 0 n B(w, 4r*) and let W be the set of all r E [0, 1] for which uT ( Ym, ) exists, in the sense of difference quotients, whenever Ym E {Yv } · We note that H 1 ( [0 , 1] \ W) = 0 where H 1 is one-dimensional Hausdorff measure. Next, applying the 'fundamental inequality', established in Step A, to u(-, r) , r E [0, 1], we see that there exist constants c and :X, which depend on p, n, a, /3 , 'Y, but are independent of r, r E [0, I], such that if y E n n B(w, 16r'), r' = r* /c and r E (0, lj, then ·
A
-_1
( 1 . 1 7)
u(y , r) < I ( I < - u(y, r) d(y , an) - Vu y , T) - A d(y , an) . _
One can then deduce, using the fundamental theorem of calculus and arguing as in [LN4, displays ( 1 . 15)-(1 .23)] , that
(1.18)
log
( ) v (ym. ) u (ym )
=
log
(
u(ym , l ) 1i (ym , O)
)
=
f (ym , r ) d !1 u(ym r ,r )
0
whenever Ym E {Yv }, Ym E OnB(w, r' ) , and for a function f which has the following important properties,
( 1 . 19)
=
f 2: 0 is continuous in B(w, r') wit h f f (ym , T) UT (yrn, T)
(i) (ii)
=
0 on B(w, T') \ n,
whenever Ym E {yv } , Ym E n n B(w, r'), r E W. Moreover, f is locally a weak solution in 0 n B (w, r') to the equation
( 1 .20) where
(1.21) whenever y E n n B(w, r') and 1 :S i, j :S we see that
a- 1 � ( y, r)l�l 2
n.
Also, using Definition 1.1 (i ) and (ii)
:S
L ;;ij (y , r)�iej ::; a� cv. r)lel 2 i ,j whenever y E n n B(w, r') and where � (y, r) = I Vu(y , r) I P- 2 • Finally, a key obser vation in this step is that ( = u(-, r ) is also a weak solution to L in n n B(w, r'). Indeed, using the homogeneity in Definition 1 . 1 (iv) we see that (1 .22)
(1.23)
L bii (y , r)u11j (y , r) j
We conclude from ( 1 . 23) that (
=
�A'T]J (y, Vu(y, r) ) u113 (y, r)
=
L
=
(p - l)Ai (y, V u(y, r) ) .
j
u( · , r ) is also a weak solution to L.
236
JOHN L. LEWIS, NIKLAS LUNDSTROM. AND KA.J NYSTROM
Step D . (Boundary Harnack inequalities for degenerate elliptic equations) . Using the deformations introduced in Step C the proof of Theorem 1 therefore boils down to proving boundary Harnack inequalities for the operator L. The idea here is to make use of Step B to conclude that 5,(-, 7), 7 E [0, 1] , can be extended to A2 -weights in B(w , 4r"), r" = r'/(4c2) · Then the operator L can be considered as a degenerate elliptic operator in the sense of [FKS] , [FJK] , [FJK 1 ] , and we can apply results of these authors. In particular, to do this we first observe that the sequence {y,_. }, introduced below (1 . 16), is a dense sequence in n n B (w, r' ) , and V } 0 ! (· , 7) , v2 (·) = u(·, 7), are positive solutions to L, see ( 1 .20)-( 1.22) , vanishing continuously on n n B(w , r'). Second, we observe from Step B that 5, (y , 7 ) = I 'V ii.(y, 7) 1P- 2 can be extended to an A2 ( B(w , 4r") )-weight. Hence, from [FKSJ, [FJK] and [FJK1]: we can conclude that there exist a constant c = c(p, n , a, {3, 1) , 1 :s; c < oo, and (j = (j(p, n, a, ,8, 1 ) , (j E (0, 1), such that if r"' = r" jc, then =
u VI (a,.m (w)) Vt (Yl ) IY1 vl (Y2) Y21 c < ( 1 . 24 ) v2 (yi ) v2 (y2) - v2 (ar"' (w) ) ( r " ) whenever y1 , Y2 E 0 n B(w, ." ) . Hence, assuming ( 1 . 14) we see that Theorem 1 now follows from ( 1 . 18), ( 1 . 24 ) as (1 . 25 ) 0 :s; f(a,.u-('w ), -r) :s; u(ar"' (w) , -r) � c- 1 , whenever 7 E (0, 1]. ( 1 . 25 ) is a consequence of (1. 16) and ( 1 . 14) (b) .
I
1
,
' ,
c,
The proof of Theorem 2 can also be decomposed into steps similar to steps A-D stated above. Still in this case details are more involved and we refer to section 5 for details. The rest of the paper is organized as follows. In section 2 we �>tate a number of basic estimates for A-harmonic functions in NTA-domains and we obtain the conclusion of Theorem 1 when A E Mp (a) , f2 is equal to a truncated cylinder (see (2.7) and Lemma 2.8), and w is the center of the bottom of fl (Step 0) . section 3 we establish the 'fundamental inequality' for A-harmonic functions, u, vanishing on a portion of a Reifenberg flat domain (Step A). In section 4 we first state a number of results for degenerate elliptic equations tailored to our situation and we then extend IY'u i P- 2 to an A2-weight (Step B). In this section we also complete the proof of Theorem 1 by showing that the technical assumption in ( 1 . 14) can be removed. In section 5 we prove Theorem 2. Finally in an Appendix to this paper (section 6), we point out an alternative argument to Step C based on an idea in [W] .
In
2. Basic estimates for A-harmonic functions and boundary Harnack inequalities in a prototype case In this section we first state and prove some basic estimates for non-negative A-harmonic functions in a bounded NTA domain n c R11• We then prove the boundary Harnack inequality for non-negative A-harmonic functions, A E Mv (a) , vanishing on a portion of a hyperplane. Throughout this section we will assume that A E Mp(a, {3, 1) or A E Mp (a) for some (o:, ,8 , 'Y) and 1 < p < oo. Also in this paper, unless otherwise stated, c will denote a positive constant ;:::: 1 , not nec essarily the same at each occunence, depending only on p, n, M, a, (3, 1 where l'vf denotes the NTA-constant for 0 c Rn . In general , c(a 1 , . . . , am ) denotes a positive
BOUNDARY HARNACK INEQUALITIES FOR OPERATORS OF p-LAPLACE TYPE
237
constant 2: 1, which may depend only on p, n, A1, a, {3, "( and a1, . . . , am , not neces sarily the same at each occmrence. If A � B then A/ B is bounded from above and below by constants which, unless otherwise stated, only depend on p, n, M, a, /3, ; . Moreover, we let max u , min u be the essential supremum and infimum of u on B(z.s) B(z,s) B(z, s) whenever B(z, s) C an and whenever 7L is defined on B(z, s). We put .D.(w, r) = an n B(w, r) whenever w E an , 0 < r. Finally, ei, 1 :::; i :::; n, denotes the point in an with one in the i th coordinate position and zeroes elsewhere.
Lemma 2 . 1 . G-iven p, 1 < p < oo, assume that A E Mp (a, /3, ;) for some (a, /3, ;) . Let u be a positive A-harmonic function in B(w, 2r) . Then (i)
there
f B(w,r/ 2)
IV'njP dx :::;
c
( max u)P, B(w,r)
< c min u. - B(w,r) exists iT = iT(p, n, a, ,B ,;) E (0, 1) such that if x , y E B(w, r),
(ii)
Furthermore, then
rp - n
(iii)
B(w,r)
max
u
c(1x;ul)ii
iu(x) - u (y ) l :::;
max u . B(w,2r)
Proof: Lemma 2 . 1 (i), (ii) are standard Caccioppoli and Harnack inequalities while (iii) is a standard Holder estimate (see [S]). 0
Lemma 2.2. Let n c an be a bounded NTA-domain, suppose that p, 1 < p < oo , is given and that A E Mp (a, {3, -y) joT some (o:, /3, ;) . Let w E an, 0 < r < ro , and suppose that u is a non-negative continuous A-harmonic function in n n B (w, 2r) and that u = 0 on .D.(w, 2r) . Then
( i) Furthermore, there exists B(w, r), then ( ii)
i V' iP d f QnB(w,r'/2) u
iT =
x
:::; c ( max u)P. OnB(w,r)
a(p, n, M, a, {3, ;) E (0, 1) such that if x, y E n
( )
iu(x) - u(y) l :::; c lx�yi ii
OnB(w,2r)
max
n
u.
Proof: Lemma 2.2 (i) is a standard subsolution inequality while (ii) follows from a Wiener criteria first proved in [M] and later generalized in [GZ) . 0 Lemma 2.3. Let D. c Rn be a bounded NTA -domain, suppose that p, 1 < p < oo, is given and that A E Mp (a, /3, "() for some (a, /3, ;) . Let w E an, 0 < r < r0, and suppose that u is a non-negative continuous A-harmonic function in fi n B(w, 2r) and that u = 0 on .D. (w, 2r) . There exists c = c(p, n, M, a, /3, "(), 1 :::; c < oo, such that if r = TIc, then max u ::::; c u ( ar (w )) . SlnB(w,•' ) Proof: A proof of Lemma 2.3 for linear elliptic PDE can be found in [ CFMS] . The proof uses only analogues of Lemmas 2 . 1 , 2.2 for linear PDE and Definition 1 .5. In
JOHN L. LEWIS, NlKLAS LUNDSTROM, AND KAJ NYSTRO::vt
238
particular, the proof also applies in our situation.
0
Lemma 2.4. Let n c Rn be a bounded NTA-domain, suppose that p, 1 < p < oo, is given and that A E Mp (a., /3, ''!) for some (a. , /3, ')') . Let w E an, 0 < r < A r0 , and suppose that u is a non-negative continuous -harmonic function in n n B (w , 4r ) and that u = 0 on u(w, 4r ) . Extend u to B(w, 4r) by defining u = 0 on B ( w , 4r) \ n. Then u has a representative in W1·P (B ( w, 4r)) with Holder continuous partial derivatives of first order in n n B(w, 4r) . In particular, there exists & E (0, 1], depending only on p, n, a., /3, 1' such that if x , y E B(w, f/2), B(w, 4r) c n n B(w, 4r), then c - 1 1'\?u(x ) - '\i' u (y ) i :::;
Proof: Given (2.5)
E
(lx - Yl/r)& B(w,r) In§\� l'\?ul
:::; c f-1 (lx - Yi/f)&
rn_ax_ u .
B(w, 2r)
> 0 and small, let
.4 (y, 7J, E )
=
J A (y , 'T) - x)BE(x)dx
whenever
(y, ·r1) E Rn x Rn,
R"
Bdx = 1 and e. (x) = cne(x/e) whenever X E Rn . where e E COO (B(O, 1)) with From Definition 1.1 and standard properties of approximations to the identity, we deduce for some c = c(p, n) 2: 1 that
JRn
(i)
(2 .6)
( ii)
(iii)
(ca:) -l ( t: + I7JI )p-2 1�12 ::;
� ��;
l
Ai
:t � i,j=l
'r/;
(y , .,,, t: ) �i�j ,
(y, 'T), t:) :::; ca:( t + I1J I )P-2, 1 ::; i , j ::; n ,
IA (x , ·q, e) - A (y , 1J, t) l :::; c.Bi x - y i'Y(e + l rJI ) P-l
whenever x, y, 1J E Rn . Moreover, A(y, · , f ) is, for fixed (y, t:) , infinitely different iable . To prove L emma 2.4 we choose u ( · , E), a weak solution to the PDE with struc ture as in (2.6), in SUCh a way that u ( · , E) is continuous in f!nB(w, 3r) and u(·, c) = U on 8[!1nB(w, 3r)]. Existence of u(-, f) follows from the Wiener criteria in [G Z] men tioned in the proof of Lenuua 2.2, the maximum principle for A-harmonic functions, and the fact that the W1·P-Dirichlet problem for these funct ions, in n n B(w, 3r) , always has a unique solution (see [HKM, Appendix I]) . Moreover, from [T] , [Tl] , it follows that u( - , E ) is in C1·& (n n B(w, 2r) ) for some a > 0 with constants inde pendent of t. Letting t---+0 one can show, using Definition 1 . 1 , that subsequences of { u(-, €) } , {Vu(-, t) } , converge pointwise to u, Vu. In view of Lemma 2 . 1 and the result in [T] it follows that this convergence is uniform on compact subsets of n n B(w, 3r ) . Using this fact we get the last display in Lemma 2.4. Finally we note that in [T] a stronger a.o:;sumption, compared t o (2.6) (iii), is imposed. However, other authors later obtained the results in [T] under assumption (2.6) (see [Li] for references) . D Next we show that the conclusion of Theorem 1 holds in the case of a truncated cylinder with w the center on the bottom of the cylinder (Step 0). To this end we
BOUNDARY HARNACK INEQUALITIES FOR OPERATOILO , Q r(y) c Q {1 2 (0) } Q . ( y)
sup
r
,
(3.14)
where XF is the indicator function for the set F. Then using weak ( 1 , 1)-estimates for the Hardy-Littlewood maximal function, (3 .11) and (3.12) we see that 1Qi;2 (0 ) \ Gl :5 CE- 17 1 F I :5 CE - ?JE-pryEa = CE?J (3.15)
BOUNDARY HARNACK INEQUALITIES FOR OPERATORS OF p-LAPLACE TYPE
by our choice for ry. Also, using continuity of
lu2 (y) - u l ( Y) I =
(3.16)
If
�, r) l J
�� I B (
B(y,r)
243
u2 (y) - u1 (y) we find for y E G that lu2 (z) - u1 (z) l dz :::; ce1lu2(en/2) .
y E Q{14(0) \ G, then from (3 . 15) we see there exists fj E G such that I Y - t/ 1 :::;
c(n)e7Jin.
Using Lemmas
�
!u2 (y) - u1 (y) j
2 . 1 , 2.2,
!u2 (fj ) - u1 ( Y) I + ! u2 (y) - u2 (fj) l + l u l ( Y) - u 1 (Y ) I c(e1J + ei11J/n ) u2 (en /2) .
�
( 3.17)
we hence get that
This completes the proof of the first inequality stated in Lerruna 3.1. Finally, using the Harnack inequality we see that there exists T = T(p, n, a, {3, �f ) 2: 1 such that
u2 (en/2) :::; cp-ru2 ( Y)
whenever
y E Qi14(0) \ Q{14,p (O).
D
We continue by proving the following important technical lemma.
R" be an open set, suppose 1 < p < oo, and that A1, A2 E Mv(a, {3, "f). Also, suppose that u 1 , u2 are non-negative functions in 0, that ·ih ·is A1 -harmonic in 0 and that u2 is A2 -harmonic in 0. Let ii 2: 1 , y E 0 and assume that 1 u 1 (y) u1 (y ) � a � d( y , 80) � I V u l (Y)i d(y, 80) ' Let £-1 = (cii)( l +iY) /& , where &- i.e; as in Lemma 2.4. If
Lemma 3.18. Let 0
C
•
_
u2 1 ( 1 - f:) L � -;-- � ( 1 + f:)L in B(y, 100 d(y , 80) ) •
for some L, 0 < L
then deduce that
0 to be chosen.
From (3. 19) with
z
=
z1 , y
=
z2 and (3.20) we
(3.21 ) whenever
IY - :0 1
(3.22)
=
z E B(y, td(y, 80) ) . Integrating, it follows that if fj td(y, 80) , t = ( 1 /&-, then iu 2(y) - u2 (Y ) i � c'( 1 +1 /& u2 (y).
The constants in
(3.21 ), (3.22)
depend only o n p , n , a , {3, "f ·
E
8B(y, td(y, 80) ),
244
JOHN L. LEWIS,
NIKLAS LUNDSTR OM , AND KAJ NYSTRO M
������j1 .
(3.19)
Next we note that also holds with u2 replaced by u1 . Let A = Then from (:U9) for 11 1 and the non-degeneracy assumption on l \7iL 1 1 in Lemma we find that
3. 18,
1 d(y, 80 ) ), { \i'u 1 ( z ) , A} � (1 - c ii() l\i'u1 (y) l whenever z E for some c = c(p, n, o., fJ, I)· If � ( 2cii) - 1 , where c is the constant in the last
display,
B(y, ( /&
(
then we get from integration that
1& (3 .23) c• (u1 (Y) - iLt (Y) ) � ii- 1 ( / u l (Y) with y = y + ( 1/& d(y, 80)A and where the constant c• depends only on (3, From (3. 2 3), (3. 22), that if f. is in Lemma 3.18, then u2 (y) c'(1+1/" ) u (y ) ( (1 f.)t 1"l1 (fj) � 1 + (1 /&f (iic• ) ul (Y ) - ( 1++ c'(lj(ac• +1/& ) A (3.24) (1 + ) 1 (1 /& ) L < ( 1 - c)L provided 1/ (iic) 1/& ( 1 /& � iic € for some large c = c(p, o., (3, 1). This inequal 1 ity and (3. 2 3) are satisfied if €- = ( cii ) Cl+&)/& and (- 1 Moreover, if the hypotheses of Lemma 3.18 hold for this then in order to avoid the contradiction in ( 3.24) it must be true that (3.20) is false for this choice of (. Hence Lemma 3 . 18 is true. Armed with Lemma 3.1 and Lemma 3 . 18 we prove the 'fundamental inequal ity' for A-harmonic functions, A E Mp ( o., f3, /') for some (o., ,B , 1), vanishing on a portion of {y : Yn = 0}. we see
a.'l
0} .
(3.38)
From (3.38)
find that
if we define
A' = {(y' , 0) + 20ose , y' E Rn - l }, 0.' {y E Rn : Yn > 20os }, we
n
=
then
0.' n B(z , 2 s) c 0. n B(z, 2s). Let v be a A-harmonic function in 0.' n B(z, 2s) with continuous boundary values on 8(0.' n B(z, 2s)) and such that v ::::; u on 8(0.' n B(z , 2s)). Moreover, we choose v so that v(y) = u(y) whenever y E 8(0.' n B(z, 2s)] and Yn > 408s, v(y) = 0 whenever y E 8[0.' n B(z, 2s) ] and Yn < 305s. (3.39)
of v follows once again from the Wiener criteria of [GZ] , the maximum principle for A-harmonic functions, and the fact that the W 1 ·P-Dirichlet problem for these functions in 0' n B(z, 2s) always has a solution. By construction and the maximum principle for A-harmonic functions we have v ::; u in 0' n B(z, 2s). Also, since each point of 8 [0.' n B(z, 2s)] where u 1- v lies within 808s of a point where u is zero, it follows from (3.36) and Lemmas 2.2, 2.3 that u ::; v + c8"' u(y) on 8[0' n B(z, 2s)] . In particular, again using the maximum principle for p-harmonic functions we conclude that Existence
v ::::; u S
v
+ c85u(y) in n' n B(z, 2s) .
BOUNDARY HARNACK INEQUALITIES FOR OPERATORS OF p-LAPLACE TYPE
Thus, using the last inequality and
(3.36)
1 ::; u (y) ::; ( 1 - co,_)_1 v (y
(3.40)
we see that
whenever
provided J is small enough. Using Lemma
( ) A• - 1 d vf) yaD) ( ,
(3.41)
::;
247
3.25
,
IV'v (y) l
y E D' n B(y, � d( y, Gn' ))
and the construction we also have
::; A v ( y ) d( y ,l )D) " •
for some j, = j, (p, n ) . In particular, from (3.40) , (3.41) we see for 0 < o < J, and J = J (p, n, a, /3, "! ) suitably small, that the hypotheses of Lemma 3.18 are satisfied with 0 = D' n B(z, 2s) and We now fix J and from Lemma 3.18 we conclude
-
that
for some
Lemma
5. 1
3.35
a = �. (y)
A t d (uY, aD)
=
-
1
:S I V'u
5. 1 (p, n, a, /3, 1 ) . Since f)
is complete.
D
( ') I
-
u (y )
y :S At d( f) , 00 )
E D n B(w, rjc')
is arbitrary, the proof of
4. Degenerate elliptic equations and extension of IV'uiP- 2 to an A2-weight Let w E Rn , 0 < r and let A(x) be a real valued Lebesgue measurable func tion defined almost everywhere on B(w, 2r). A(x) is said to belong to the class
A2 (B(w, r)) (4. 1)
if there exists a con::;taut r such that r:- 2n
J
B(w,f)
A dx .
J
B(w,f)
A-1dx ::; r
w E B(w, r) and 0 < f ::; r. If A(x) belongs to the class A2(B(w, ·r )) then A is referred to as an A2 (B(w, r) )-weight. The smallest r such that (4.1 ) holds is referred to as the A2-constant of A.
whenever
In the following we let D C Rn be a bounded (t5, r0)-Reifenberg flat domain with NTA-constant 1'vf. We let w E aD, 0 < r < ro , and we consider the operator
,
(4.2 )
L
n
=
a
(·
a
L - bij (x) OX · i,j =l ox •
J
)
in D n B (w, 2r). We assume t h at the coefficients {bij ( x)} are bounded, Lebesgue measurable functions defined almost everywhere on B(w, 2r). Moreover, n
c-1 A(x) l�l2 :S L bij (x)�i�j :S cl�l2 A(x) i,j =l for almost every x E B(w , 2r) , where A E A2 (B(w, )) By definition L is a degen erate elliptic operator (in divergence form) in B( w, 2r) with ellipticity measured by the function ,\. If 0 c B(w, 2r) is open then we let W1•2(0) be the weighted Sobolev space of equivalence classes of functions with distributional gradient \i' v (4.3)
r
v
and norm
ll v flt2
=
.
j v2 Adx + j IV'vi2 Adx < oo.
0
0
248
JOHN L. LEWIS, NIKLAS LUNDSTROM, AND KAJ NYSTROM
2 Let W� ' 2 (0) be the closure of C0 (0) in the norm of W1• (0). We say that v is a weak solution to Lv = 0 in 0 provided v E W1•2 (0) and
1 �::)ij Vxi .(x) = >.(wj) when x E Q1 . This defines >. almost everywhere on B (w, 50r) with respect to Lebesgue n measure, since it follows from ( 4.27) that for 8 small enough, an n B(w, r) has Lebesgue n measure zero. From the definition of >., Lemma 3.35, and the Harnack inequality for A-harmonic functions we see that rv
(4.23)
>.(x) = >.(wi)
>. (z) whenever x
E
rv
Qj and z
E
B(wj , d(wj , DD) /2) .
Let .\ = >. if p � 2 and .\ 1/ >. if 1 < p � 2. If w E B (w, r) and d( w, an) /2 < f � r, then from Lemmas 2 . 1 - 2.3, (4.23), and Holder's inequality it follows that �
=
j
(4.24)
Adx :::::; cu ( ar (w ) ) IP - 21 fn- I P - 2 1 .
B(w,r)
Here w E an with Jw - wJ = d(w, an) . Also, from Lemma 4.8 we get for J small enough and y E n n B(w, cf) , that cu(y) � u(ar (w))
(4.25)
(
d(
y�an)
)
l+£"
Here €* > 0 is a small positive number which will be fixed after the display following (4.27). From (4.25) and Lemma 3.35, we see that if d(w, an)/2 < f � r, then (4.26)
j
B(>v,r)
>.- 1dx :::::; cf (l+£") 1P- 21 u(ar (w ) ) - IP- 21
j
nnB(w,ci'J
d(y, OO) _,. I P- 21 dy.
To complete the estimate in (4.26) we need to estimate the integral involving the distance function. To do this we define I ( z, s) =
f
nnB(z, .. )
whenever z E an n B(w, r) , 0
. = 5.(p, n, a ) � 1 , in O(w, ii) n B(w, r 1 ) where fj = ij(p, n, a ) is as i n (5. 2 ) . To this end we show there exists c = c(p, n, a) � 1 such that if c2r' < r < r0/n, and p = rfc, then (5.1) holds for u on O(w, fj) n 8B(w, p) . Here u > 0 is A-harmonic in n \ B(w, r') with continuous boundary values and u = o on an \ fJ(w, r'). It then follows from arbitrariness of r, r', the above discussion, and Lemma 5.4 that Theorem 2 is valid whenever A E Mp (a) and u is a minimal positive A-harmonic function relative to w E 80. With this game plan in mind, observe from Lemma 2.15 and (1.7), that we may assmne r = 1, w = 0, and a,
{3, "f, be
n, H( O , 4n) n {y : Yn :S -J.£} c Rn \ 0 , 1 where J.£ = 500nc5* , 0 < p. < 10- 00 and r' < ( o* )2. Here 8* is temporarily allowed to vary but will be fixed after the proof of Lemma 5.19. Extend fi. to be continuous on Rn \ B(O, r'), by putting u = 0 on Rn \ (0 u B(O, r')) . Using the notation in (2.7), let Q = Qt, 1 _J.I(I1en ) \ B( O , ,fii) and let v 1 be the A-harmonic function in Q with the following continuous boundary values, (5.16) B (O, 4n) n {y : Yn � 1'·}
Vt (Y)
=
vt (y)
c
u(y) , y E 8Q n {y : 2p. :S Yn} , (Yn - l1) u(y) , y E 8Q n {y : p. :S Yn < 211}· j.£
Comparing boundary values and ul:lirtp; the maximum principle for A-harmonic func tions, it follows that ( 5 . 1 7)
We now set
1-£
v1 :::; u in Q .
=
!-£(«'-) = exp(-1/«'-). We ::ihall prove,
Lemma 5. 18. Let 0 < E S €,
J.£ = p.(«'-) be as above and let ij be as in (5.2}. If E is small enough, then there exists B = O(p, n, a) , 0 < {J :::; 1/2, such that if p = J.£1 12-e, then
whenever
y
E 0(0, fi/4) n
1 :S u( y) fvt (Y) :S 1 + t:
[B(O, p) \ B(O, 2fo)].
2G8
JOHN
L. LEWIS,
NIKLAS LUNDSTRO M , AND KAJ NYSTRO M
Lemma 5. 19. Let v1 , f, f., 0 , J.L be as in Lemma 5. 18 and let ij be as in (5.2). If € is small enough, there exist 8 e(p, n, a), O < () < 0/4) . = 5-(p, n, a) > 1, such that
if P
=
J.L l / 2 - 20' a
= J.L - e ' then
=
< I vi (Y) V'v1 (y) l � A d(y, an) d(y, an)
- _ 1 v1 (y)
A
whenever y
E
0 (0, �/2)
n
-
-
[B(O, ap) \ B(O , pja)] .
Assuming Lemmas 5. 18, 5. 19, are true we complete the proof of Theorem 2 when A E lv!p (a) as follows. From these lemmas and Lemma 3.18 we de duce, for sufficiently small f. = f.(p, n, a: ) > 0, that (5.1) is valid for u and for some ). = 5. (p, n, a:) 2 1 in D. (w, ij) n aB ( O, p). With € now fixed we put 8* = p,(E)/ (500n) and conclude from (5.2), Lemma 2.15, arbitrariness of r, that (5. 1) holds in n n [B (w , ri ) \ B(w, r' )] with r1 = r0 /c, r' � r0jc', provided c, c' are large enough, depending only on p, n, a:. Thus we can apply Lemma 5.4 and proceed as in the discussion after that lemma to get Theorem 2 under the assumption A E Mp ( a: ) . Proof of Lemma 5. 18. To begin the proof of Lemma 5.18 observe from (5.17) that it suffices to prove the righthand inequality in this display. We note that if y E aQ and u (y) =!= v1 (y), then y lies within 4J.L of a point in 8Q. Also maxaB (O ,t ) u is non-increasing as a function of t 2 r', as we see from the maximum principle for A-harmonic functions. Using these facts and Lemmas 2 . 1- 2.3 we find that (5.20)
on fJQ. By the maximum principle this inequality also holds in Q. Here iJ is the exponent of HOlder continuity in Lemma 2.2. Using Harnack's inequality, we also find that there exist T = T (p, n, a: ) 2 1 and c = c(p, n, a: ) > 1 such that (5.21)
max{ V; ( z ) , lb (y) } $ c(d( z , fJQ) /d(y , 8Q)t min{ 'f/' ( z ), 1/'(y) }
whenever z E Q , y E Q n B ( z , 4d( z , 8Q)) and 1/J = u or v1 . Also from Lemmas 2.12.3 applied to v1 , we get (5.22)
Let p, {) be as in Lemma 5.18. Using (5.20) - (5.22), we see that if y E 0 (0, ij/4) n [B(O, p) \ B ( O, 2 fo)] , then (5.23)
u(y) $ vi (y ) + CJ.L&/2 u(foen ) �
( 1 + C2f.J,& /2-e-,- ) VI (Y)
$ (1 +
f)VI (Y)
provided f. is small enough and ih = G-/4. The proof of Lemma 5.18 is complete
.
D
Proof of Lemma 5.19. To prove Lemma 5.19 we let VJ , t:, f., e, f-1, be as in Lemma 5.18. Using Lemmas 2.2 - 2.3 and Harnack's inequality we see that there exists ¢> = ¢> (p , n, a:) > 0, 0 < ¢> $ 1/2 , and c = c (p, n, a:) > 1 with
u (y) $ c(sjt) ri>u(sen ) provided y E Rn \ B(O, t) , t � s � 2r'. Using (5.24) with t Lemmas 2.1 - 2.3 we see that (5.24)
(5.25)
=
1 , s = 2fo, and
BOUNDARY HARNACK INEQUALITIES FOR OPERATORS OF p-LAPLACE TYPE
2ri9
where c depends only on p, n, a. Let ii be the A-harmonic function in Q with continuous boundary values ii = 0 on 8Q \ B(O, JJi) , and v = v 1 on 8B(O, fo). Then from (5.25) and the maximum principle, we see that ii S v 1 S v +
( 5 .26)
Let p = J.-L 1 12 - 28 , e small, and a = to '¢ = v we find v
(5.27)
CJ.-L12u( ..fiien ) in Q.
J.-L-8
be as in Lemma 5 . 19. Using (5.21) applied
2 c- 1 (J.-L112jap fu(.Jiie n) = c- 1 !138" u( Jiie n)
on 0(0, ij/8)n[B(O, 2ap)\B(O, pj (2a))], where T is aB in (5.21) and the nontangential approach region 0 was defined above (5.2) relative to w, ij. Abo, since ij depends only on p, n, a, it follows that c = c(p, n, a) in (5.27) . If we define () by () = min{¢/(12T) , B/4}, then from (5 .26) , (5.27) we get
V1 < 1 + € 0 be A-harmonic in 0. \ B(w, r') , continuous in Rn \ B (w, r'), with u := v := 0 on Rn \ [0. u B(w, r') ] . Then there exists 8* , CJ > 0 , c+ � 1, depending o n p, n , a , /3, 'Y , such that 4 0 < 8 < 6. < J (J as in Theorem 1} and r1 = ro fc+ , then
I
log (
wheneveT z, y
E
' r u(z ) u (y) ) - log ( )I < c+ ( min ( rl , lz - w l , v( y ) v(z )
0. \ B (w, c+ r') .
I Y - wl )
)
a
Proof: Once again we assume that r' fr1 < < 1, since otherwise there is nothing to prove. As in (5.5) we may assmne for some c = c(p, n, a, /3, "'!) that u :S: v/2 :S: cu in 0. \ B(w, 2r') .
(5.36)
Let u(·, t ) , t E [0, 1], b e A-harmonic in 0. \ B(w , 2r'), with continuous boundary values, (5.37)
·u (·, t)
=
( 1 - t ) u(·) + t v(·) on 8 [0. \ B(w, 2r') ] .
261
BOUNDARY HARNACK INEQUALITIES FOR OPERATORS OF p-LAPLACE TYPE
We claim there exists c, ). 2: 1 depending only on p, n, a, /3, 'Y such that if t E [0, 1], and y E n n [B(w, fo/c) \ B(w, cr')], then - _1
.A
(5.38 )
u(y , t )
d (y, an) < I Vu ( y , t) l _
b-if>)v(·, t)
on O (w, �/2) n (B(w, 2p) \ B (w, p/2)). Choosing b = b(p, n, a, (3, 'YHa.rge enough in (5.40), using (5.39), Lemma 3.18, it follows that (5.41)
X+ 1 h (y, t)/d( y , an) :::; IVh ( y, t ) l :::; A+h ( y , t)jd (y , an )
whenever y E 0( w, fJ) n 8B( w , p) for some )\+ = >.+ (p, n, a, (3, 'Y) 2: 1 . From (5.2) we see that (5.41) holds on n n 8B(w, p) provided -A-r (p, n, a, p, 'Y) is large enough. With a , b, now fixed, depending only on p, n, a, /3, 'Y, we can use Lemma 2 . 15 and argue as in Lemma 3.1 to conclude for given f. > 0, the existence < of r·1 = ·r1 (p, n, a, (3, 'Y, t) so small that if bp :::; r1 r0 , then 1 - E :::; u (- , t)jh(· , t) :::; 1 + t
on O(w, fJ/2) n (B(w, 2p) \ B(w, p/2)) . In view of this inequality, (5.41 ) , and Lemma 3 . 18, we see that if t = t(p, n, a, (3, 'Y) is small enough, then (5.4 2)
I Vu( ·, t ) l
�
u( ·, t) /d ( - , an)
on fl(w, fj) n 8B(w, p) , where proportionality constants depend only on p, n, a, (3, 'Y In view of (5.2), this inequality holds on n n 8B(w, p) . With r 1 , a, b fixed we see from arbitrariness of p that (5.38) is true. We can now argue as in Lemma 5.4 or just repeat the argument in ( 1 . 1 8) - ( 1 .25) to conclude Lemma 5.35. 0 As pointed out earlier in this section, if u, v are minimal A-harmonic functions relative to w E an, then we can apply Lemma 5.35 and Let r' -tO to get Theorem 2. The proof of Theorem 2 is now complete. D
2
62
JOHN L. LEWIS, NIKLAS LUNDSTROM, AND KAJ NYSTROM
6. Appendix : an alternative approach to deformations
In this section we show that Step C in Theorem 1 can be replaced by a some what different argument based on ideas in [WJ. The first author would like to thank Mikhail Feldman for making him aware of the ideas in [WJ . In the following all con stants will depend only on p, n, a, (3, --y and we suppose that u, v are A-harmonic in n n B(w, 4r) and continuous in B(w, 4r) with u = v = 0 on B(w, 4r) \ n. From Lemma 3.35 we see that if o is small enough, f = rjc, and c is large enough, then for some p, ;::: 1 , (6. 1)
j],
-1
h(y) h(y) 'Vh(y) l d(y, an) :::: I :::: P- d(y, an )
whenever y E 0 n B(w, 4f), h E {u, v }. Also from Lemma 4.8 exists J1,. ;::: 1 , for t:* > 0 fixed, such that ( 6.2)
p, ;
1
() s
f
1 +. E Rn , � E Rn \ {0}, that
j:
we
-.) - Ai(x, 0
t
0
(6.3)
A, (x , t>. + ( 1 - t)�)dt
n
L ( Aj - �j )
·-t J-
1
'
aA ( JOTJJ 0
x,
t). + ( 1 - t )� )dt
for i E { 1 , ... , n}. In view of (6.3), (6 .1), and A-harmonicity of u, v, we deduce that u - v is a weak solution to L( = o in n n B(w, f) , where
-
L((x)
(6.4)
and aij (x)
=
n
a · (a;j (x)(,j ) i,j=l ax t
L
1
=
J �:: (tV'u(x) + (1 - t)'Vv(x) )dt, 0
for 1 � i, j ::; n. Moreover, from the strueture assumptions on A, see Definition 1 . 1 , we find that (6.5) whenever
c:;:1 ( I V'u (x) l + I'Vv(x) I )P-2 1 � 12
X
n
L
a;j (x ) �i�j i,j=l < c+ ( I'V u(x) l + I'Vv(x) l)p-2 1 �12
0 , such that if r* = r· jc, and 0 < o < oo, then (I'V u l + I 'Vv i ) P- 2 extends to an A2 -weight ·in B(w, r*) with Az -constant depending only on p, n, a, (3, --y.
BOUNDARY HARNACK INEQUALITIES FOR OPERATORS OF p-LAPLACE TYPE
263
Proof: The proof is essentially the same as the proof of Lemma 4.9. That is , we use a Whitney cube decomposition of Rn \ D to extend ( I'V'ul + I 'V'v i ) P- 2 to a function >.. on B(w, 4r* ) . Let w E B (w, r*) and 0 < f < r* . Let w E 80 with lw - w l = d(w, 80) and suppose that l w - wl/2 < f < r•. We assume, as we may, that
(6.7)
Let 5. = >.. when p �
max{ u (ar(w )) , v (ar(w ))} = u (ar ( w )) . 2 and 5. = 1/>.. for 1 < p < 2. As in (4. 25) - (4 . 28), it follows ,
for c* > 0, small enough, that
j
(6 .8)
B('w ,r)
and
5.dx
:::;
r.n (ar (7i1 )) 1P-2 1 r:n- IP-2 1
J
onB(w,50f') B(w,r ) 2 < cu(ar ( w )) -IP -21 ;-:n+lp- 1 . (6.9) These inequalities em ain true if r :::; lw - wl/2, follows e ily from (6. 1). Com bining (6.8), (6.9) , and using arbitrariness of w, f, we get Lemma 6.6. D r
as
as
Using the ideas in [W] we continue by proving the following.
oo, w E 80, 0 < r < rn, suppose that u and v are non-negative A-harmonic Junctions in n n B (w , 2r) with v :::; ft. Assume also that u, v, are continuous in B (w , 2r ) with u = 0 = v on B(w, 2r) \ D. Lpt r* be as in Lemma 6. 6. There exists c � 1 such that if f = r* /c, then r( c- 1 u(af'(w)) - v(af'(w)) < u (y) - v(y) < c u(af'(w)) - v(a w)) v(y ) v (a;;(w)) v(ar(w)) whenever y E D () B (w , f) .
Lemma 6.10. Given p, 1 < p
lR. as
x -
Variational Structure of (2 .4) . At this point we make the formal observation
that the transformed system (2.4) has variational structure. This is no surprise since the original free-boundary problem ( 1 . 1) has the variational structure discussed in [2] (see also [10]) and our change of variables (2.3) leads from there to the functional J below. (An analogous variational formulation [7, § 4.1] of the Dubreil-Jacotin equation (1.3) follows similarly. ) For functions � which are periodic in on the semi-infinite strip S = lR. x in the )-plane, with = let
x [-d, 0] f) (x, -d) -d, 2 1 J ( �) � J Ish U2 { ;zf); + f)z } dxdz - � 1 7!' f)(x, 0)2dx, (2.5) where S2 = ( ) ( -d, 0). Then critical points of (2.5) satisfy the system (2.4). Moreover, f)(z) = z is a critical point of J. So let f)(x, z) = z+��: ( x, z) in the formula for J. Then the first term has the form � J /' U 2 { 2 + /\.; + K; } dxdz = C + � J r U2 { II:� + ��:; } dxdz , 2 Js2-. 2 Js2.. 1 + ;;,z 1 + ;;,z (x, z
=
rr
- 1r , 1r
x
where C is independent of ;;,. Therefore where
we
are interested in critical points of J,
= -2l/ 1 ( "'12 ++ll:z"'2z ) U(z)2dxdz - -2g 11r ��:(x, oY.! dx. Critical points of J satisfy the system (2. 7a) ( 1U2 ;;,ll:z ) ( 1u+2 "' ) 2 ( U2(1(;;,+2 ll:+z)11:22 ) ) 0, (2.7b) ��:(x, -d) = 0, (2.7c) U(0):-! ( 1 + K�(x, O)) + (2g��:(x, O) - U(0) 2 )(1 + ��:z (x,0)) 2 = 0, (2.7d) ��:(·, z) is 21r-periodic in x. (2.6)
J(;;,)
S2-.r
+
:r:
x
+
X
Z
ll:z
z
1
_,.
X
Z
z
•
270
M. LILLI AND J.
(2.8a)
(U21ix)x + (U2�z)z = 0,
F.
TOLAND
Linearization of (2.4) . The functional J has a critical point K = 0 , irrespective of U and the linearization of the Euler-Lagrange system (2.7) , with respect to K, about this zero solution is li(x, -d) = 0, 2 g'K(x, 0 ) - U(0) 'Kz (x, 0 ) = 0 , 'K( · , z) is 27r-periodic in x.
(2.8b)
(2.8c) (2 .8d)
We will see that this linear problem is easy to analyze using separation of variables.
3. Parametrized Families of Laminar Streams
Now we consider a parametrized family of laminar running streams, U(y; c) , y E [-d, 0),
cEI
C JR, where, for c E I (an open interval) , U(· ; c) E C2(-d, O) , U(y; c) i= 0 on [-d, O] and c �---+ U(· ; c) E C2 (I; C1•19 [-d, Ol) . Here no physical meaning is assigned to the parameter c, the dependence of U on c being quite general. Let the corresponding
stream function be denoted by
\II (y; c) =
jy
-d
U(z; c) dz,
and the dependence of vorticity on the stream function by 1(\II (y; c) ) = -U' (y; c) ,
With
c1 (c) = \II (O; c) and c (c)
y E [- d, 0] . 1
2 u(O; c) 2, U(· ; c) and 1 = 1 ( · ; c) . The corresponding 2
=
lit ( · ; r:) is a solution of ( 1 . 1 ) when U solution of (2.7) is "' = 0 for all c E I. The question is whether there are other (non-laminar) solutions of ( 1 . 1 ) for the same vorticity function 1(· ; c) for certain values of c. This is a global question, but here we regard it as a question of finding bifurcation points on the line of trivial solutions {K 0 , c E I} of system (2.7) . =
=
4. Bifurcation Theory
We now consider basic bifurcation theory [8] for the nonlinear problem ( U(z; c)2 Kx ) + ( U (z; c ) 2Kz ) 1 + Kz x 1 + Kz z
(4_ 1a)
( 4.1b) (4. 1 c)
-
� U (z; c) 2 (K; + K�) 2
r;, (x, -d) = 0 ;
(
( 1 + Kz) 2
)
z
=
O,
U(O, c) 2( 1 + l'i:; (x , 0)) + (2gl'i: ( X, 0 ) - U( O , c) 2) ( 1 + l'i:z (x, 0) ) 2 = 0 , K ( - , z ) is 27r-periodic in x ,
(4.1d)
regarding c as the bifurcation parameter. To simplify matters we will seek solutions K that are even in x. To this end let X=
{"' E C2·1J (S) : K is even and 21r-periodic in x and K( x , -d) = 0} , 0 y = {,. E C ·1J (S) : /'i, is even and 2tr-periodic in X} , Z = { w E C1•19(JR) : w is even and 27r-pcriodic},
2 71
WAVES ON A STEADY STREAM WITH VORTICITY
which are Banach spaces when endowed with the usual Holder-space norms. Let B denote the open ball of radius 1 about the origin in X x Y and define F B x JR y
X
:
Z by
F (c, K)
-+
=
(
( U(z; c)2i'i:z ) ( U (z; c)2 Kx ) + 1 + "-z z x 1 + Kz
_
� ( U(x; c)2 (K; + K; ) ) z 2 (1 + Kz) 2
)
U(O; c)2(1 + l'i:�(x, 0)) + (2gl'i: (X, 0) - U(c; 0)2) (1 + Kz (x , 0) )2
It is clear that F is twice continuously differentiable from B x I into Y x Z and that F (O, c) = 0 E Y x Z for all c E I. In order to show that a particular c* is a bifurcation point for the problem F ( K, c) = 0 , it will suffice to show that, for some � E X \ {0} , ker d�.1 (c)
is
a
{sup Q(v, c} : v E E C W 1 • 2 [-d, O], v =f:- 0 , v(-d) = O } .
simple eigenvalue .
PROOF. This minimax characterization of Aj (c) is part of the classical theory, see [11, §4.5) , for example. Moreover, solutions of the eigenvalue problem (5. 1} attain these minimax values. In particular, .At (c} is attained at a certain function v. Since Q(lvl; c) � Q(v; c), we may assume that ..\1 (c) is also attained at lvl . Now suppose that >.1 (c) is attained at v1 and v2 , and therefore that lv 1 l and lv2 l are eigenfunctions of (5.1) for the eigenvalue >.1 (c). If lv1 l and lv2 l are not linearly dependent, it follows that lv1 l l v2 l (u + c)2dz = 0. Since this is false, lvd is a
jo
-d
scalar multiple of lv2 l· Since both satisfy (5.1), it follows that v1 is a multiple of v2 , as required for ..\ 1 ( c) to be a simple eigenvalue. D
o j Pc =
Let
-d
dy >0 (u (y ) + c )2
and con�:>ider the eigenvalue problem (5.3a} !" = J.Lf, f( O ) = 0, gf(Pc ) = j' ( Pc), (5.3b)
f ¢ 0.
It is easy to see that there exists a solution with JL = v2 > 0 if and only if gPc > 1 , in which case f(z) = a sinh vz for some a #- 0, where v is uniquely determined by tanh vPc 1 vPc
g Pc .
When gPc > 1 all the other eigenvalues J.L of (5.3) (there are infinitely many) are negative and determined by tan vPc 1 . J.L = -v 2 and f ( z) = sm vz where P. . v
P.
c
=
g
c
By a similar calculation, every eigenvalue of (5.3) is negative when gPc < 1 , and when gPc = 1 all its eigenvalues are non-positive, exactly one (counting multiplicity) being zero with eigenfunction f(z) = z. As with Q and (5.1), these eigenvalues correspond to minimax values of q (f; c) =
J:c f' (z)2 dz - gf(Pc ) 2 J:c f(z)2 dz
over the class of non-zero functions f E W1· 2 (0, Pc) with f(O) = 0. From the above observations we infer that when gPc ::::; 1, inf { q( f, c) : f E W 1 • 2 [0 , Pc) , f #- 0, f (O ) ) = 0 } 2: 0. (5.4a) However, when gPc > 1, ( 5.4b)
inf { q(f, c) : f E W1. 2 [0, PcJ , f #- 0, f(O)) = 0 }
0.
We return now to our study of (5 .1). In addition our basic assumption that u E C2 ( - d, 0) n C1•'9 [ - d, 0] with u(O) = 0, we now as;,ume that u (y) < 0 , y E [- d, 0) .
(5.5)
When f : [0, P,] --) IR is smooth and f(O)
v (y ) =
jy
-d
=
0, let
dt (u(t) + c)2
Then v ( - d) = 0 and, when substituted in (5.2) , we infer from (5.4a) thaL >.1 (c) > 0 when gPc < 1 .
(5.6)
Because (4.8) and ( 5 . 1 ) are equivalent, our main results on the eigenvalue problem (5. 1) represent a significant simplification and extension of [15, Lemma 2.5] . LEMMA 5.2. Suppose that (5.5) holds and that c < 0. Then >.1 (c) --) -oo as c / 0 and >-t (e) > 0 for all c < 0 with lei sufficiently large. Hence, for each k E N, there ex-ists c; < 0 such that - k2 = >.1 (c;) . PROOF. Note first that
Pc --)
(5.7) since u(O) substitute
=
0 and lu'(O)I
.1 (c) --) -oo as c / 0, by (5.7) and the dominated convergence theorem. It follows from (5.4) that >.1 (e) ::; 0 if and only if Peg 2:: 1 , which is true for all c < 0 with lei sufficiently small, by (5.7) . Finally note from (5.6) that >.1 (c) > 0 for all lei sufficiently large, since gPc --) 0 as lei --+ oo. Since >.1 (c) obviously depends continuously on c < 0, the result 0 follows.
M. LILLI AND J.
276
F. TOLAND
To consider the behaviour of >. 1 (c) for positive c suppose that (5.9) u( -d) < u(y) for all y E ( -d, 0] and let y -u( -d) > 0. Note that (5. 10) Pc ---> oo as c \, y. We now restrict attention to c E (�, oo) . =
LEMMA 5.3. Suppose that (5.9) holds. (a)
liminf >. 1 (c) :::;
(5 . 1 1 )
c "..:g
-g
2 iiu + � II P( -d, o)
{b) >.1 (c) > 0 for all c > 0 sufficiently large. (c) Suppose that -k2 > lim infc� >.1 (c) , k E
-k2 . PROOF. Since P, ---> oo as c \, �,
>.r (ct)
.
N. There exists ct > y, su ch that
=
1 1 2: p c
jy
-d
dt
·
( 'IJ, (t ) + c) 2
--->
1 as c \, y for all y E ( - d, 0] ,
and ( 5 . 1 1 ) follow.s from (5.8) . As in the preceding proof, >.1 (c) > 0 for all c suffi 0 ciently large.
REMARK. An example of Lemma 5.3 (a) arises in the problem of bifurcation of waves on flows of constant vorticity with d = 1, in which case u(y) Wo Y , Wo E R When w0 > 0 the hypotheses of Lemma 5.2 are satisfied. Moreover, in Lemma 5.3, y, wo and il u + Y i ll2(-r,o) = w5/3. Hence -k2 = >.r ( c) for some c > y, > 0 is an eigenvalue if w5 k 2 < 3g. In fact, for this example we have seen from an explicit calculation that -k2 = >.1 (c) for some c > y, if and only if w5 (1 - k- 1 tanh k) < g k - 1 tanh k. (In particular, -1 >.1 (c) for some c > w0 if and only if w5 < 3. 194g (which may be compared 0 with w5 < 3g , the criterion in the Lemma) . =
=
=
THEOREM 5.4. The parameter values ct in the preceding two lemmas are furcation points for problem (2.7) when U(z ; c) = u(z) + c.
bi
PROOF. We have shown that (4.2a) is satisfied when c ct , and (4.2c) follows by the self-adjointness of (4 . 3 ) in an L2 setting. Since 8Uj8c(O; ct ) 1, it follows from ( 4 .5 ) that K, 0 if ( 4.2b) is false. Hence hypothesis (4.2) is satisfied, and it follows that the eigenvalues ct are bifurcation points for the problem of waves on 0 a running stream. =
=
=
References [1]
A.
J.
Abdullah, Wave motion
at the
surface of a current which has an exponential distribution
of vorticity, Annals New York Acad. Sci. 51
( 1 949), 425 - 441. Alt and L. A. Caffarelli, Existence and regularity for a minimum problem with free boundary, J. Reine Angew. Math. , 325 (1981), 105-144. [3] T. B. Benjamin, The solitary wave on a stream with an arbitrary distribution of vorticity, J. Fluid Mech. 12 (1962), 97 -1Hi. [4] B. Buffoni and J. F. Toland, Analytic Theory of Global Bifurcation - An Introduction. Prince ton University P ress , Princeton, N. J., 2003.
[2]
H. W.
WAVES ON A STEADY STREAM WITH VORTICITY
277
[5] A. Constantin and W. Strauss, Exact steady periodic water waves with vorticity, Comm. Pure Appl. Math., Vol. LVII (2004), 481-527. (6] A. Constantin and W. Strauss, Exact periodic traveling water waves with vorticity, C. R. Math. Acad. Sci. Paris 335 (10) (2002), 797-800. (7] A. Constantin and W. Strauss, Stability properties of steady water waves with vorticity, Comm. Pure Appl. Math. , Vol. LX (2007) , 911-950. [8] M. G. Crandall and P. H. Rabinowitz, B ifurcati on from a simple eigenvalue, J. Funct. Anal. 8 (1971), 321-340 . (9] R. A. Dalrymple and J. C. Cox, Symmetric finite-amplitude r ot ational water waves, J. Physical Oceanography, 6 (1976), 847-852. [10] I. I. Danil iuk, On i ntegral functionals with a variable domain of integration, Proc. Steklov Inst. Math. 118 (1972). In English, Am er. Math. Soc. (1976). [11] E. B. Davies, Sp�ctral Theory and Differential Operators. Cambridge University Press, Cam bridge, 1995. (12] M.-L. Dubreil-Jacotin, Sur Ia determination r:igoureuse des ondes permanentes periodiques dampleur finie. J. Math. Pures Appl. 13 (1934), 217291. [ 13] D. Gilbarg and N. S. Trudinger , Elliptic Partial Differential Equations of Second Order. 2nd Edition. Springer, New York, 1983. (14] J. N. Hunt, Gravity waves in flowing water, Proc. R. Soc. London A, 231 (1955) , 496-504. [15] V. M. Hur and M. Lin, Unstable surface waves in running water, ArXiv 0708:0541VI [Math. AP] 3rd Aug 2007, to appear. UNIVER.SITii.T AucsBURG, lNSTITUT FUR
MATHF:MATIK,
U!-�IVERSITTSSTR.ASSE 14,
86159 Aves
BURG, GERMANY
Current address : Department of Mathematical Sciences, U niversity of Bath, Claverton Down, Bath BA2 7AY, UK E-mail address: lilli/Dmath .uni-augsburg .de DEPARTMENT OF MATHEMATICAL SCIENCES, UNlVERSlTY OF
BA2 7AY, UK
E-mail address:
jftiDmaths . bath . ac . uk
BATH,
CLAVRR1'0N DOWN, BATH
Proceedings of Symposia in Pure Mathematics Volume 79, 2008
On analytic capacity of portions of continuum and a question of T. Murai Fedor Nazarov and Alexander Volberg This paper is dedicated to Vladimir Maz'ya
We give an answer to an old question of T. Murai concerning the characterization of the boundednesss of the Cauchy integral operator on arbitrary set� of finite Hausdorff length. If the set is a continuum, we got a new proof to a theorem of Guy David characterizing the rectifiable curves on the plane for which the Cauchy integral operator is bounded on L2(ds). In doing that we use also a nonhomogeneous version of a certain Tb theorem first proved by M . Christ in homogeneous spaces. We are going to "compute" in metric terms the analytic capacity of the intersection of an arbitrary continuum and a half-plane ( or a disc, or any domain with piecewise smooth boundary) . ABSTRACT.
1. Introduction Takafumi Murai asked in [Mu] the following question: given a compact set E C C su ch that its Hausdorff !-dimensional measure sat isfies 0 < H1 (E) < oo, is that true that Cauchy integral operator is bounded in L2(E, H1 I E) if and only if H1 (E n Q) S C ')'(E n Q) .
Here
/'
stands for analytic capacity defined in the next paragraph.
We give a positive answer to this question here.
THEOREM 1 . 1 . Let E be a compact on the complex plane such that 0 < H1 (E) < oo. Then the Cauchy integral operator T (and also r•) is hounded on L2 (E, H1) if and only if there exists a constant C such that for every square Q on
the plane
(1. 1 ) 2000
Mathematics Subject Classification.
Key words and phrases.
Primary 47B36; Secondary 42C05. Analytic capacity, Hausdorff content, nonhomogeneous harmonic
analysis, accretive functions. The first author was supported in part by NSF Grant 0501067. The second author was supported in part by NSF Grant 0501067.
279
F. Nazarov, A. Volberg
280
Definition. Let K be a compact set in C.
"f( K) : = sup{ lim l z f (z) l : J E Hol(C \ K) , lf( z) i :::; 1 Vz E C \ K, J(oo) = 0} , Z-+00
"f+ (K) : = sup{}�.� lz f(z)l : f(z) = By definition,
I �f.L�(I , f.L E M+ (K) , lf( z) l :::; 1 Vz E C \ K} .
.
(1.2)
In [T4] Tolsa proved that the opposite inequality also holds with absolute constant. It is a very tough theorem. We discuss its relations with results of this paper in the last section. A natural question arises: how verifiable is this criterion? Strangely enough it is sometimes verifiable, and this is the second main topic of this article. In Theorem 1.3 we compute (up to the absolute constant) the analytic capacity of certain class of sets. This allows us to observe in Section 3 that a famous theorem of Guy David is a one-line consequence of the above criterion ( 1 . 1). David 's theorem we are referring to is the one that characterizes all rectifiable curves on the plane on which the Cauchy integral operator is bounded. Let nH recall that there is another criterion of the boundedness of Cauchy integral. It is obtained in [NTV2] , [Tl] and we want to formulate it now. To do that we need to recall the reader the notion of Menger's curvature of a measure. Given three pointH z1 , z2 , z3 E C we call R(z 1 , z2 , za ) the radius of the circle ( may be oo) passing through those points. Then Menger's curvature of a positive measure f.L is by definition c2(/L) :=
)
1 2
R2 (z1 , z2 , z::�) H 1 1E for a certain compact E, O < H 1 (E) < oo, we will use the following
If f.L notations: =
(.I J .I
df.L ( zl ) df.L(z2 ) df.L(z3 )
cz(E) := c2(H1 IE) . We are ready to quote the criterion proved by Nazarov-Treil-Volherg in [NTV2] and Tolsa in [Tl] . THEOREM 1.2. 1) Cauchy integral operator is bounded in £2 (/L) if and only if there exists a finite constant C such that for every square Q (1.3) f.L(Q ) :::; c f( Q) ' where £(Q) is the length of the side of Q, and ( 1 .4)
C2 (f.LIQ ) 2 :::; c J-L( Q) .
2) In particular, if J-L = H 1 IE, for a certain compact E, O < H 1 (E) < oo, then the boundedness of the Cauchy integral operator in L2 ( E, H1 ) is equivalent to (1.5) and (1.6)
On analytic capacity of portions of continuum
28
1
Remark. Actually one can sometimes Rlcip assumption (1.3) as Tolsa has shown in Lemma 5.2 of [T5] . This is the case for measures J.t such that their upper density lim sup J.t(B(x, r) ) /r r--+0
is uniformly bounded. We are grateful to the referee for this remark. We will ::;how below that Theorem 1.1 implies easily part 2) of Theorem 1 .2. On the other hand, one can deduce Theorem 1 . 1 from Theorem 1 .2, but this requires a much more efl"orts. This deduction is based on already mentioned very tough result of Tolsa [T4] . This deduction is briefly discussed in the last section of this article. Another interesting feature of our criterion (1.1) is that one can prove its multi dimensional analogs, however, the multi-dimensional analogs of criterion (1.6) from [NTV2] , [Tl] are not known now (because they involve the notion of Menger's curvature that did not yet get multi-dimensional understanding) .
1 . 1 . Theorem 1 . 1 implies easily the second part of Theorem 1.2. The difficult part is to prove that ( 1 .5), (1 .6) imply the boundedness in L2 (E, H11E) of the Cauchy integral operator. \Ve want to use Theorem 1 . 1 . So for our goals it is sufficient to prove the following implication ( Q is always a square ) (1.7) VQ c2 (E n Q)2 :S C1 H1 (E n Q) :S C2f(Q) * H1 ( E n Q) :S C21(E n Q) . To prove (1 .7) we need the trivial inequality 1 � I+ and the following charac terization of I+ due to Melnikov (see e.g. [Tl] , it can be found in [Vo] also ) :
(1.8)
I+ ( K)
�
sup
p,: p,(B(x,r ) ) � r Vr 'Vx
11�112
c2 (J.t) +
II J.tl l
Obviously the second inequality in the left hand !:>ide of (1. 7) implie!:l that measure H 1 I E satisfies the growth condition: H 1 (E n B(x, r)) :::; C1 r .
Hence, m>ing (1 .8) for test measure
p, : =
C}1 H1 I E n Q we obtain
Hl (E n Q)2 1 I ( E n Q) � I+ ( E n Q) � a c2(E n Q) 2 + H l (E n Q) � a H (E n Q) .
We got the right hand side of (1.7) , which is the reduction we wanted.
1.2. David's characterization of bounded Cauchy integral operator on rectifiable curves: Ahlfors-David curves. If the set E is a rectifiable curve
r, then a theorem of Guy David describe::; all !:>uch curvet-> on the plane for which the Cauchy integral operator is bounded on L 2 (r, d H1) . This is the class of curves I' (called Ahlfors-David curves) satisfying
( 1 .9)
H1 (D(x, R)
n
r)) s C R
for a.ny disc D(x, R). Lipschitz curves and Lavrentiev (chord-arc) curves give us the examples satisfying (1 .9) . It is slightly strange that for Lipschitz curves (and even for chord-arc curves ) there exists a purely analytic proof of the boundedness of the Cauchy operator on £2 ( r, d H1) (see [CJS] , [Chr] ) , but all the proofs of the theorem of David are the mixtures of analytic and geometric arguments, [Dal] ,
282
F.
Nazarov,
A.
Volberg
[DaJ] . In the present paper we are going to show, in particular, that Guy David's characterization follows from two ingredients: a) Theorem 1.1, h) a simple computation in geometric terms of 1(f n Q) , where Q is a square and r is an arbitrary continuum, Theorem 1.3 belows. In the present paper our main idea is to use a local Tb theorem of M. Christ [Chr] (and not the usual Tb theorem or Tl theorem). The difference with [Chr] is that we have to use a nonhomogeneous version of a local Tb theorem. This nonhomogeneous version of Christ 's local Tb theorem leads us naturally to the "computation" of the analytic capacity of the intersection of our E with a square (or a disc) . It is quite well understood now that the analytic capacity of an arbitrary compact cannot he measured in terms of simple geometric characteristics of the compact. The result of Tolsa [T4] (see also the exposition of this result in [Vo] together with its multi-dimensional analogs) only confirms this because the analytic capacity is proved to he computable in metric terms, but only in quite complicated ones. See also, for example, [JM] , where it is shown that the Buffon needle probability cannot serve as a metric equivalent of analytic capacity in general. Surely, one can derive that on compact subsets of an Ahlfors-David curve the analytic capacity is equivalent to just H 1 measure. We will discuss this later, but now let us notice that the equivalence constants are not absolute. They depend on the geometry of the ambient curve. Secondly, to establish this equivalence one needs heavy tools: either the theorem of David or geometric arguments of Jones [J] and Melnikov [Me2] . However, to our surprise, there exists a class of compacts for which one can get the simple geometric measurement equivalent to the analytic capacity up to absolute constants. And this class is large enough to enable us to use our version of Tb theorem resulting in a new proof of the theorem of David. This class consists of intersections of any continuum with any closed half-plane {or any closed disc, or any closed square, ... ) . Let us introduce some notations. In what follows, a, a' , a" , A denote various positive finite absolute constants. Letters II, Q and D stand for various half planes, squares and discs respectively. \Ve will use the symbol h1 (E) to denote !-dimensional Hausdorff content of E, namely ,
h1 (E)
:=
inf{
L
r1
:
E C Uj D(xj , rj ) } .
It is clear that H1 is larger than h 1 , and they vanish simultaneously. For any continuum r the Hausdorff content is equivalent with the diameter. But the same is true for the analytic capacity 1(r). Thus, for a continuum r
(1. 10) We are going to prove {1.10) for sets ourselves to the the case of half-planes. THEOREM
Then (1.11)
r n II, r n Q, and r n D. We restrict
1 3 . Let II be a closed half-plane, and let r be a continuum. .
a
h 1 (r n ll)
:::; 1(r n II) :::; A h 1 (r
n II) .
283
On analytic capacity of portions of continuum
We prove this result below. Our proof was superseded by the proof by John Garnett [JG] that is considerably easier than ours. But we still decided to present our proof for the possible future generalizations to higher dimension. The simpler Garnett's proof relies on complex analysis observations.
2. The Cauchy integral operator on sets E, 0 < H 1 (E) < oo , and nonhomogeneous accretive system Tb theorem Let us remind that the function K(x, y) , x, y E IR.n is called a Calder6n Zygmund kernel of dimension m if there exist finite and positive constants C, c: such that
\ K(x, y) \ :::; Cdist (x , y ) -m ,
' ( ) I K x, y) - K ( x, y ) \ + \ K ( y, X
Cdist(y,) y'+)
-
dA(w)
dH 1 (() ::; c::_ H1(E n D( z, p)) P
n Q(z, p) ) � 9:.H1 (E n Q(z, p)) ::; !!_'"'f(E cp P
a
::; -"((Q(z, p)) ::; A C a , cp where c denotes the constant. from (3.1), C = c-1 , and Q(z, p) denotes the smallest square with the same center as the disc D(z, p) and containing thiti disc. We used here (3.1) and an obvious estimate that the analytic capacity is bounded by the diameter of a set. Lemma 2.2 is proved. 0
p=
Now one can apply this Theorem 2.1 to the kernel K((, z) H1 IE. Of course m = 1 now, and the assumption
=
O
:=
�
c,
Ii .
T
E :F
The segment Ii is well inside D(yi, lO ri) and v is outside of it, therefore sup
xEI,.
r
r>3r;
v*(x) ::; inf (M1 v)(x) . xEl;
Consider :Fo d� {i E :F : E h v* (x ) � 1�0 } . On Io := UiEFo Ji we have ( this is just
:lx
(4.2)) 100 M1 v L (5.2) On the other hand, 100 } I S: 5L II v i L . l {x : (M1 v) (x) � L 100 20 This is a usual weak type result. Consider \ :Fo {i E :F : Vx Ji , v* (x) 1�0 }. We conclude from :F1
:=
:F
�
=
E
=
::;
(5.2) and the last inequality that ( J1 �f uiEFJi) :
(5.3)
l ·h l
�
20 £ , and , \lx 19
E J1 ,
100 . v (x) ::; L •
Without the loss of generality we can think that 1 . v�
E F : l . 'Yi ::;
2oooo L . In fact, if we have the opposite inequality for a certain io , then V1 =f because we choose just ci 1 if i = io ci = 0 otherwise, and see that f = 2: cdi = fio belongs to V1 n V2. But we assumed V1 V2 Let us call a subset :F' of :F1 admissible if the following holds 1 1 L. L kj( q ) mq.e = Uj£
(2.6)
q
(2.3).
D
Here is a second way to see this result in a more general context : \Vrite j
so we can define an (n + 1)
x
Xj (z ) = L aj k Zk k =O (n + 1) triangular matrix a l n) by (n) = aj ajk k
Then (the Cholesky factorization of k)
(2. 7) (2.8)
k ( n) = a Cnl ( a( n) ) *
(2.9)
(xi , xc) = �J!.
(2 . 1 0 )
with * Hermitean adjoint. The condition
says that
(2 . 1 1) (a( n) ) * m(nl (a( n ) ) = 1 n 1 ( l ) the identity matrix. Multiplyi ng by (a ) * on the right and [ (a Cn )*] on the left yields (2.3). This has a clear extension to a general Gram-Schmidt setting. 3. The Christoffel-Darboux Formula The Christoffel-Darboux formula for OPRL says that
Kn ( Z, -,�') - an+ 1 _
(Pn+l (z) p,.((z) -- Pn(Z} p,.+ I (( ) ) ., ;
(3.1)
and for OPUC that
l (() - c,on + I (z ) c,on + I (( ) l ) Kn ( z, �"., ) = 'P�+ (z cp�+
1 - z(
(3.2)
The conventional wisdom is that there is no CD formula for general OPs, but we will see, in a sense, that is only half true. The usual proofs are inductive. Our proofs here will be direct operator theoretic calculations.
300
B. SIMON
We focus first on (3.1 ) . From the operator point of view, the key is to note that, by (1 . 1 7) , (g , [Mz , Trn]f> =
J g(() (( - z)Kn ( z, ()f(z) dJ.L(()dfl (Z )
(3.3)
where [A, B] = AB - BA. For OPRL, in (3.3) , ( and z are real, so ( 3.1) for z, ( E a(dJ.L) is equivalent to (3.4)
While (3.4 ) only proves (3.1) for such z, ( by the fact that both sides are polynomials in z and (, it is actually equivalent. Here is the general result:
THEOREM 3.1 (General Half CD Formula) . Let J1. be a measure on C with finite moments. Then:
( 1 - 1rn) [Mz , 1rn] (1 - 1rn) = 0 1fn [Mz , 1fn]1fn = 0
(3.5)
(3.6)
Xn (3.7) ( 1 - 1rn ) [Mz , 1fn l 1fn = II +lll 1\Xn , · ) Xn+ I JJ Xn ll REMARK. If J1. has compact support, these are formulae involving bounded operators on L2 (C, dJ.L). If not, regard 7rn and Mz as maps of polynomials to polynomial�:>. PROOF. (3.5)
follows from expanding [Mz , 7rn] and using
If we note that [Mz, 1fn] (3.8) again,
1fn (1 - 1fn) = (1 - 1fn)1fn = 0
=
(3.8)
- [Mz , (1 - 1fn)], (3.6) similarly follows from (3.8). By
(3. 9) On ran(1fn- d , Trn is the identity, and multiplication by z leaves one in Trn , that is, (3. 10) (1 - 1rn)Mz 1rn I ran(1rn-d = 0
On the other hand, for the monic OPs, since Mz1fn Xn
Xn+l · Since
=
(3. 11) ( 1 - 1fn) Mz1fnXn = Xn+l n+ l + lower order and (1 - 7rn) takes any such polynomial to z
JI Xn+l ll (Xn 1 Xn)Xn+I = Xn+l II Xn ll we see (3.4) holds on ran(l - 7rn ) + ran(7rn _1) + [XnJ , and so on all of £ 2 .
0
From this point of view, we can understand what is missing for a CD formula for general OP. The missing piece is ( 3. 1 2)
The operator on the left of (3. 7) is proven to be rank one, but (1 - 1fn)M;7rn is, in general, rank n. For Jn) (b)h l2
so that if el is the vector (1 0 . . . oy ' then
(6.3)
is the spectral measure for ln;F (b) and e 1 , that is, {el , Jn;F (b) eel) =
'II
2.': 5-)n)(b)iJn)(b/
j =l
{6.4)
for all f.. We are going to begin by proving an intermediate quadrature formula: THEOREM 6 . 1 . Let p, be a probability measure.
0, 1, . . . , 2n - 2,
For any b and any e =
t >-)nl (b) x)n) (b) e j =l If b = 0, this holds also for e = 2 n - 1 .
j xe dp,
PROOF.
=
(6.5)
For any measure, { aj , bj }j�1 determine {Pj }j��, and moreover, (6.6)
If a measure has finite support with at least n points, one can still define {Pj } j��, Jacobi parameters { aj , bj }j;:f , and bn by (6.6). n dp, and the measure, call it ap,i ) , of (6.3) have the same Jacobi parameters {aj , bj }j;:f , so the same {PJ } j�J , and thus by k = 0, 1 , . . . , j - 1 ; j = 1, . . . , n - 1
(6.7)
THE CHRISTOFFEL-DAR.BOUX
305
KERNEL
we inductively get (6.5) for f = 0, 1 , 2, . . . , 2n - 3. Moreover, determines inductively (fi.fi) £ = 2n - 1.
for
/ Pn-1 (X) 2 df..l = 1
f. = 2n - 2. Finally,
(6.8 ) if b
= 0, (6.6) yields (6.5) for
0
As the second step, we want to determine the x)nl (b) and ).)n) (b) .
THEOREM 6.2. Let Kn ; F = 1r11 - t A1z 1fn -1 r ran( rrn- t) for a general finite mo ment measure, ft, on tC . Then (6.9)
PROOF. Suppose Xn. (z) has a zero of order £ at z0. Let r.p = X11 (z)/(z - z0 ) e. Then, in ran(n11 ) , (6. 10) (Kn ; F - zo) j r.p =J 0 j = o, 1, . . . , e - 1 \ (6. 1 1 ) (Kn ; F - zo) ? = 0
=
since (J1,1z - zo) �' cp X (z) and 1fn- 1 Xn = 0. Thus, zo is an eigenvalue of Kn :F of n algebraic multiplicity at least €. Since Xn ( z) has n zeros counting multiplicity, this accounts for all the roots, so (6.9) holds because both sides of monic polynomials 0 of degree n with the same roots. COROLLARY
6.3. We have for OPRL det (z - .ln;F (b))
=
Pn (z) - bPn- l (z)
The eigenvalues x)n) (b) are all simple and obey for 0 < b < ( with X11.q (O) = oo) , n n n x(J ) (0) < a;(J ) (b) < x(J +)l (0) and for
-oo
< b < 0 and j
=
1, . . . , n ( with Xn - t (O) =
x)�\ (0) < xjn) (b) < xjnl (O)
- oo ) ,
(6. 12) oo
and j
=
1, . . . , n (6. 1 3) (6. 14)
PROOF. (6.12) for b = 0 is just (6.9). Expanding in minors shows the determi nant of ( z - .ln;F(I!)) is ju::;t the value at b = 0 minus b times the (n - 1) x (n - 1) determinant, proving (6. 12) in general. The inequalities in (6 . 13)/(6. 1 4) follow either by eigenvalue perturbation theory 0 or by using the arguments in Section 4. In fact, our analysis below proves that for 0 < b < oo , 1 iJn) (O) < XJ11) (b) < x;n- ) (0)
(6.15)
The recursion formula for monic OPs proves that p1 ( xj (b) ) is the unnormalized eigenvector for l ; F (b). Kn_ 1 (xj (b) , Xj (b)) 112 is the normalization constant, so n since Po = 1 (if f..L (lR) = 1 ) : P ROPOSITION
6.4. lf f..L (R.) = 1 , then
>-.)n) (b)
= (Kn-t (X�n) (b) , XJ11) (b) )) - 1
(6. 16)
B.
306
SIMON
Now fix n and xu E R. Define
Pn (xo ) Pn - l (xo ) with the convention b = oo if P,,_l (xo) = 0. Define for b -=f. oo , ) j = 1, . . . , n x)n ( xo ) = x)n) ( b (x o ) )
(6.1 7)
b (xo ) =
and if b(xo ) = :xJ,
- (n - l) ( O) xj(n ) (xo ) - x1
j = 1, . . . , n - 1
(6.19)
AJn) (xo) = (Kn-1 (xJn) (xo) , xjnl (xo))) - 1
(6.20)
_
and
(6.18)
Then Theorem 6.1 becomes THEOREM
6.5 (Gaussian Quadrature) . Fix n, x0 . Then
j Q(x ) d�-t = j=l :t >-Jn) (xo)Q(x,;n) (xo)) for all polynomials Q of degr·ee up to: ( 1) 2n - 1 if Pn ( xo ) = 0 (2 ) 2n - 2 if Pn (xo) =/- 0 =/- Pn-l (xo ) (3) 2n - 3 if Pn -1 (xo) = 0. REMARKS. 1. The sum goes to n - 1 if Pn -l ( xo ) 2 . We can define x)n ) to be the solutions of
=
(6.21)
0.
Pn-l (xo )Pn (x) - Pn (xo)Pn-l (x) = 0 which has degree n if Pn -l (xo) -=f. 0 and n - 1 if Pn- l (xo) = 0.
(6.22)
3. (6.20) makes sense even if �-t(IR) =f- 1 and dividing by p,(IR) changes J Q(x) dp, and >.)nl by the same amount, so (6.21) holds for all positive J-t (with finite mo ments) , not just the normalized ones. n 4. The weights, AJ ) (x0), in Gaussian quadrature are called Cotes numbers. 7. Markov-Stieltjes Inequalities
The ideas of this section go back to Markov [6 3] and Stieltjes [92] based on conjectures of Chebyshev [21] (see Freud [34] ) . LEMMA 7. 1 . Fix X l < . . . < Xn in IR distin ct and 1 :::; e < n. Then there is a polynomial, Q, of degree 2n - 2 so that (i) 1 j = l, . . . , £ Q(xj ) = ( 7.1) 0 1 = £ + 1, . . . , n (ii) For all x E IR, (7.2 ) Q(x) � X( - oo,xeJ (x)
{
REMARK.
Figure 1 has a graph of Q and X(- oo,xe ] for n = 5, C = 3,
Xj
=
j - 1.
THE CHRISTOFFEL-DARBOUX KERNEL
-1
0
3
FIGURE
4
307
5
1 . An interpolation polynomial
PROOF . By standard interpolation theory, there exists a unique polynomial of degree k with k + 1 conditions of the form Q(yj ) = Q'(yj ) = . . . = Q(nj)(yj) = 0
2:j nj = k + 1 . Let Q be the polynomial of degree 2n - 2 with the n conditions in (7 .1) and the n - 1 conditionti ( 7.3) Q'(xj ) = O j = l , . . . , £ - 1, £ + 1, . . . , n Clearly, Q' has at most 2n- 3 zeros. n - 1 are given by (7.3) and, by Snell's the orem, each of the n - 2 intervals (xr , x2 ) , . . . , (xt- 1 , x.e ), (xH l > XH 2 ) , . . . , (xn -1 , Xn ) must have a zero. Since Q' is nonvanishing on (xe, xe+d and Q (xe) = 1 > Q (xHI ) = 0, Q'(y) < 0 on (xe, X£+1 ) . Tracking where Q' changes sign, one sees
that (7.2) holds.
THEOREM 7.2. Suppose df.l is a measure on JR. with finite moments . Then 1 � f.l(( -oo , xo] ) L (n) ( n)
,X {jlx)nl (xo)�xo } Kn- 1 (xj (xo) j (xo)) �
f.l((-oo, xo )) �
L
{ J.1 x j(n) ( xo ) <xo }
1
0
(7.4)
( ) ( ) Kn -l(xjn (x o) , xjn (xo ))
REMARKS. 1. The two bounds differ by Kn -1 (xo , xo)-1 . 2 . These imply
f.l ({xo }) ::::; Kn- 1 ( xo , xo ) - 1 In fact, one knows (see (9.2 1) below)
f.l ( {x o } ) = lim Kn- l (xo , xo) - 1
If f.k( {xo}) = 0, then the bounds are exact as n ........, oo. n - oo
(7.5) (7.6)
308
B.
PROOF. Suppose
Pn - 1 (xo)
SIMON
=/:. 0. Let ( be such that
the polynomial of Lemma 7.1. By (7.2),
t-t( ( -oo, xo]) $
x�n) (xo) = xo. Let Q be
j Q(x) dt-t
and, by (7.1) and Theorem 6.5, the integral is the sum on the left of (7.4). Clearly, this implies 1
t-t((xo, oo)) :::::
(x 0 )> xo } {). 1 x (nJ j
2:::
Kn - 1 (xj(n) (xo) , xj( n) (xo))
which, by x ---> -x symmetry, implies the last inequality in (7.4). COROLLARY 7. 3 �
k- 1 L.
j=H 1
.
If ( $ k - 1, then
< 11.([x(£n) (x 0 ) - ,_., ' (n) (n ) K (xj ( xo ) , xj (x 0 )) 1
PROOF. Note if X1 =
'
x (kn) (x 0 )] )
D
(7. 7)
x�n\ xo) for some e, then x}n) (xo) = x;n) (x1), so we get D
(7.7) by subtracting values of (7.4).
Notice that this corollary gives effective lower bounds only if k - 1 2: £+ 1, that is, only on at least three consecutive zeros. The following theorem of Last-Simon [57] , based on ideas of Golinskii [41], can be used on successive zeros (see [57] for the proof) . THEOREM 7 4 . If E, E'
� IE - E' l ,
then
.
IE - E' l
2:
are distinct zeros of Pn (x) ,
o2 - ( 2.! I E - E' l 2)2 3n
[
E
=
� (E + E')
Kn (E ' E) sup lv-EI:$ (7.8)
8. Mixed CD Kernels
Recall that given a measure JL on lR with finite moments and Jacobi parameters
{an , bn }�= 1 , the second kind polynomials are defined by the recursion relations ( 1 .5)
but with initial conditions
qo(x) = 0 so qn (x) is a polynomial of degree n - 1.
(8.1) In fact, if [l is the measure with Jacobi
parameters given by then
qn (x; dt-t) = a1 1Pn - 1 (x; d[l)
It is sometimes useful to consider
KAq) ( x, y)
n
=
L qj (x) qj (y)
j=O
(8.2) (8.3)
309
THE CHRISTOFFEL-DARBOUX KERNEL
and the mixed CD kernel
Pj (Y)
(8.4)
- - 2K y,
(8.5)
"
K,V'ql (x, y) = L qj (x) j=O
Since (8.2) implies
K(q)( n
X, y ,· d/-1)
n - 1 (X,
al
· d-) /-1
K.
there is a CD formula for K(q) which follows immediately from the one for There is also a mixed CD formula for K!fql. OPUC also have second kind polynomials, mixed CD kernels, and mixed CD formulae. These are discussed in Section 3.2 of [80] . Mixed CD kernels will enter in Section 21.
9. Variational Principle: Basics If one thing marks the OP approach to the CD kernel that has been missing from the spectral theorists' approach, it is a remarkable variational principle for the diagonal kernel. We begin with:
Fix (a1 , . . . , a,.) E em . Then l min (f 1zj l2 1 f, ajZj 1 ) = (fl aj 1 2 )J =l J=l J=l with the minimizer given uniquely by LEMMA
9.1.
(9. 1)
=
zj(0)
=
· 12
c'ij j L...- =l I Q:J "\"m
(9. 2)
REMARK. One can use Lagrange multipliers to a priori compute z.� o) and prove this result. P ROO F .
then
If
m L: aj Zj j=l
t,
!zj - z) 0l l2
from which the result is obvious.
=
=
(9.3)
1
t, l z1 1 2 - ( t, )
If Q has deg(Q) :S n and Qn(zo)
la1 l2
(9.4)
-l
0
= 1, then n
Qn (z ) = L ajXj ( z)
j= O with Xj the orthonormal polynomials for a measure dJL, then 2: O:jXj ( z0) II Qn ii 1,2(C,dJ. 0, sup I Qn (x, xo) l -+ 0 (11.1) lx-xol>o X E (]"(dp)
THE CHRlSTOFFEL-DARBOUX KERNEL
31 3
While this happens in many cases, it is too much to hope for. If x1 E a(dJ.L) but J.L has very small weight near x 1 , then it may be a better strategy for Q n not to be small very near x 1 . Indeed, we will see (Example 1 1.3) that the sup in ( 1 1 . 1 ) can go t o infinity. What is more likely is to expect that I Qn(x , x0)1Z df.l will be concentrated ne ar x0 . We normalize this to define
dry( xo ) (x) = I Qn (X , xoW2df.L(x) JI Qn (x , xo) l dJ.L(x)
( 1 1 .2)
I Kn (X , xo W df.L (X )
(1 1.3)
n
so, by
(9.6)/ (9.7) , in the OPRL case,
dTfn(xo ) (X ) =
Kn(x, xo)
We say f.L obeys the Nevai 8-convergence criterion if and only if, in the sense of weak (aka vague ) convergence of measures,
( 1 1.4)
dry�xo) (x) --+ xo
the point mass at x0. In this section, we will explore when this holds. Clearly, if xo � a(df.l), (1 1 .4) cannot hold. We saw, for OPUC with df.L = d0/21r and z � alDl, the limit was a Poisson measure, and similar results should hold for suitable OPRL. But we will see below (Example 1 1 . 2) that even on a(df.l-), (11.4) can fail. The major result below is that for Nevai class on eint, it does hold. We begin with an equivalent criterion: DEFINITION. We say Nevai's lemma holds if
1Pn (Xo ) l 2 n-x· Kn (Xo , xo ) lim
=
( 1 1 .5)
O
THEOREM 1 1 . 1 . If df.l is a measure on IR with bounded support and then for any fixed xu
E
( 1 1 .6)
inf n an > 0
IR,
REMARK. That (11.5) Breuer-Last-Simon [14] .
(11.4)
=}
{:}
(11. 5 )
( 1 1 .4) is in Nevai [67] . The equivalence is a result of
PROOF. Since
1
Kn-l(xo , xo) = 1Pn(xo ) l 2 Kn (xo , xo) Kn (xo , xo) x K (11. 5 ) n-t(Xo, o) 1
_
Kn (xo, xo)
so We thus conclude
(U.S) 1 n -> oo
( 1 1 . 1 1)
lim sup
( 1 1 . 12)
IPn (xoW 0 > n-+= Kn (xo , xo) for if ( 1 1. 1 2 ) fails, then (11.5) holciH and, by (1 1.7), for any for n 2: No ,
We claim that
so
t: ,
we ean find N0 so ( 1 1 . 1 3)
lim Kn (Xo , Xo) l/n :S 1 So, by (1 1.5), ( 1 1 . 1 1) fails. Thus, (11.11) implies that (11.5) fails, and so (11.4) D fails. REMARK.
lim inf > 1.
As the proof shows, rather than a limit in (11.12), we can have a
The first example of this type was found by Szwarc [94] . He has a dp, with pure points at 2 - n- 1 but not at 2, and so that the Lyapunov exponent at. 2 was positive but 2 was not an eigenvalue, so ( 1 1 . 1 1) holds. The Anderson model (see [20)) provides a more dramatic example. The spectrum is an interval [a, b] and ( 1 1 . 1 1) holds for a.e. x E [a, b] . The spectral measure in this case is supported at. eigenvalues and at eigenvalues (11 .8), and so (11.4) holds. Thus (11.4) holds on a dense set in [a, b] but fails for Lebesgue a. e. x0 ! ExAMPLE 1 1 .3. A Jacobi weight has the form
with a , b > -1. In general, one can show [93] has
Pn (l) "' cna+l/ 2
(11.14) ( 1 1 . 15)
so if xo E ( - 1, 1) where IPn(xoW + IPn - l (xoW is bounded above and below, one I Kn (Xo , 1) 1 Kn(xo , xo)
rv
na+ l/2 na - 1 2 = / n
so if a > ! , 1Qn (x0, 1)1 --+ oo. Since dp,(x) is small for x near 1, one can (and, as 0 we will see, does) have (11.4) even though (11.1) fails. 'With various counterexamples in place (and more later!), we turn to the positive results:
315
THE CHRISTOFFEL-DARBOUX KERNEL
THEOREM 1 1 .4 (Nevai [67] , Nevai-Totik-Zhang [69] ) . If dp, is a measure in the classical Nevai class (i. e., for a single interval, e = [b - 2a, b + 2a] ) , then ( 1 1 .5) and so ( 1 1.4) holds uniformly on e .
THEOREM 1 1 . 5 (Zhang [108] , Breuer-Last-Simon [ 14 ] ) . Let e be a periodic finite gap set and let p, l·ie in the Nevai class for· e. 'l'hen ( 1 1 . 5) and so ( 1 1 .4) holds uniformly on e .
1 1 .6 (Breuer-Last-Simon [14] ) . Let e be a general finite gap set and let f-L lie in the Nevai class for e. Then ( 1 1. 5) and so (11 .4) holds uniformly on compact subsets of eint . THEOREM
REMARKS. 1 . Nevai [67] proved ( 10.4)/(10.5) for the classical Nevai class for i every energy in e but only uniformly on compacts of e nt . Uniformity on all of e using a beautiful lemma is from [69] . 2. Zhang [108] proved Theorem 1 1 .5 for any 11· whose Jacobi parameters ap proached a fixed periodic Jacobi matrix. Breuer-Last-Simon [14] noted that with out change, Zhang's result holds for the Nevai class. 3. It is hoped that the final version of [14] will prove the result in Theorem 11.6 on all of e, maybe even uniformly in e .
EXAMPLE 1 1 . 7 ( [14] ) . In the next section, we will discuss regular measures. They have zero Lyapunov exponent on O'ess ( M) , so one might expect Nevai's lemma could hold-and it will in many regular cases. However , [14] prove that if bn = 0 and an is alternately 1 and � on successive very long blocks (1 on blocks of size 3n2 and ! on blocks of Hize 2"\ then dfl, is regular for r:r(dp,) = [ - 2 , 2] . But for a.e. x E [-2 , 2] \ [-1 , 1], (10.4) and (10.3) fail. 0
1 1 . 8 ( [14] ) . The following is extensively discussed in [14] : For of compact support and a.e. x with respect to f-L, (10.4) and so (10.3)
CONJECTL"RE
general holds.
OPRL
12. Regularity: An Aside There is another class besides the Nevai class that enters in variational problems because it allows exponential bounds on trial polynomials. It relies on notions from potential theory; see [42, 52, 73, 102] for the general theory and [9 1, 85] for the theory in the context of orthogonal polynomials.
Let 11 be a measure with compact support and let is regular for e if and only if lim (a1 . . . an )lfn = C(e) n-oc the capacity of e. DEFINITION.
We say
f-L
e
=
O'ess (p,) .
(12.1)
For e = [ -1, 1 ] , C ( e ) = � and the class of regular measures was singled out initially by Erdos-Turan [32] and extensively studied by Ullman [103] . The general theory wa::; developed by StahJ-Totil< [91]. Recall that any set of positive capacity has an equilibrium measure, p. , and Green's function, c., defined by requiring c. is harmonic on C \ e, G. (z) = log izl + 0(1) near infinity, and for quasi-every x E e , ( 12.2) lim c. (zn) = 0 Z n --t X
3 16
B SIMON
(quasi-every means except for a set of capacity 0) . e is called regular for the Dirichlet problem if and only if { 12.2) holds for every x E e. Finite gap sets are regular for the Dirichlet problem. One major reason regularity will concern us is:
Let e C lR be compact and regular for the Dirichlet problem. Let p, be a measure regular for e. Then for any �::, there is o > 0 and Ce so that sup IPn (z, dp,) l ::; G_, ec- l n l ( 12.3) THEOREM 1 2 . 1 .
dist(z,•) 0}. Then N \ �ac has Lebesgue measure zero.
{x
PROOF. If Xo E lR \ �ac and is a Lebesgue point of f.J., then Theorem 14.1 , E lR \ N. Thus,
xo
(IR \ �ac) \ (IR \ N) has Lebesgue measure zero.
=
w(xo) = 0 and, by
N \ �ac
0
REMARK. This is a direct but not explicit consequence of the Mat8-Nevai ideas [64] . Without knowing of this work, Theorem 15. 1 was rediscovered with a very different proof by Last-Simon [55] .
On the other hand, following Last-Simon [55] , we note that Fatou's lemma and
J � Kn (x , x) df.J.( X)
implies
so
THEOREM
=
1
/ lim inf � Kn(x,x) df.J.(x) :::; 1 15.2 ( [55]) . �ac \ N has Debesgue measure zero.
(15.3) (15.4)
320
B. SIMON
=
Thus, up to sets of measure zero, 2:ac N . What is interesting is that this holds, for example, when e is a positive measure Cantor set as occurs for the almost Mathieu operator (an = 1, bn = >. cos(1ro:n + B), 1>.1 < 2, >. =J- 0, a: irrational). This operator has been heavily studied; see Last [54]. 16. Variational Principle: Nevai Trial Polynomial
A basic idea is that if dp,1 and df.l2 look alike near xo , there is a good chance that Kn (xu, xu; df.l t) and Kn (x0 , xu; dp,2) are similar for n large. The expectation (1 3.8) ::;ays they better have the same support (and be regular for that support), but this is a reasonable guess. It is natural to try trial polynomials minimizing >.n (x0 , df.l1) in the Christoffel variational principle for An (x0, dJ.L2 ), but Example 1 1.3 shows this will not work in general. If df.ll has a strong zero near some other x 1 , the trial polynomial for df.l l may be large near x1 and be problematical for df.l2 if it does not have a zero there. Nevai [67] had the idea of using a locallzing factor to overcome this. Suppose e C JR., a compact set which, for now, we suppose contains a(dp, 1 ) and O'(df.l�). Pick A = diam(e) and consider (with [ · ] = integral part) ( x - xo ) 2 [en] 1(16.1) = N2 [cn j (x ) AZ
(
Then for any 0,
) -
jx-x0 j > cl xEe
sup N2[enj (X) ::;; e - c(.S ,e)n
(16.2)
so if Qn- 2 [wj (x) is the minimizer for J.ll and e is regular for the Dirichlet problem and J.ll is regular for e, then the Nevai trial function
N2 [cn] (x) Qn- 2 [cn] (x) will be exponentially small away from xo. For this to work to compare >.n (x0, df.l 1) and >.(x0 , dp,2) , we need two additional properties of >.n ( x 0 , dp1): (a) An(x0 , df.l 1) 2: C�e-m for each c < 0 . This is needed for the exponential contributions away from x0 not to matter. (b)
1. 1. 1m 1m sup e:!O n-oo
An (xo , dJ.lJ ) =1 An -2 [e :n] (x, df.li) so that the change from Qn to Qn- 2[en] does not matter. Notice that both (a) and (b) hold if
.....
nlim n>. n (Xo , df.l) = c > 0 oo
(16.3)
If one only has e = aess(df.l2 ), one can use explicit zeros in the trial polynomials to mask the eigenvalues outside e. For details of using Nevai trial functions, see [87, 89]. Below we will just refer to using Nevai trial functions.
321
THE CHRISTOFFEL-DARDOUX KERNEL
17. Variational Principle: Mate-Nevai-Totik Lower Bound In [66] , Mate-Nevai-Totik proved: THEOREM 17. 1 . Let dJ-L be a measure on 8j[)) w(O) dfJ = � dO + dJ-Ls
which obeys the Szegfi condition Then for a. e. 0=
E
g
J lo (
8ID,
w
( 17.1)
dO (B) ) 27!' > -oo
(17.2)
( 1 7. 3 ) This remains true if >-n((}oo) is replaced by >-n (Bn ) with On _. Boc obeying sup niBn Boo l < 00 . REMARKS. 1 . The proof in [66] is clever but involved ( [89] has an exposition); it would be good to find a simpler proof. 2 . [66] only has the result (),. 000• The general (}n result is due to Findley [33] . 3 . The 000 for which this is proven have to be Lebesgue points for dJ-L as well as Lebesgue points for log( w) and for its conjugate function. 4. As usual, if I is an interval with w continuous and nonvanishing, and J-Ls(I) = 0, ( 17.3) holds uniformly if 000 E I. =
By combining this lower bound with the Mate-Nevai upper bound, we get the result of Mate-Nevai-Totik [66] : THEOREM 17.2. Under the hypothesis of Theorem 1 7. 1 , for a.e. 000 E (Jj[]),
lim n >.n (Boc )
n ---+ =
=
w(()co)
( 17.4)
This remains true if >-n (B00 ) is replaced by An(Bn ) with ()n _. 000 obeying sup niBn Boc l < oo. If I is an interval with w continuous on I and J-Ls (I) = 0, then these re.mlts hold uniformly in I.
REMARK. It is possible (see remarks in Section 4.6 of [68]) that (17.4) holds if a Szcgo condition is replaced by w(B) > 0 for a.e. 0. Indeed, under that hypothesis, Simon [88] proved that 12rr dO _. 0 l w (O) (n.An (0) ) - 1 - 1 1 2 o
rr
There have been significant extensions of Theorem 1 7.2 to OPRL on fairly general sets: 1 . [66] used the idea of Nevai trial functions (Section 16) to prove the Szego condition could be replaced by regularity plus a local Szego condition. 2. [66] used the Szego mapping to get a result for [- 1, 1]. 3. l.Jsing polynomial mappings (see Section 1 8) plus approximation, Totik [96] proved a general result (see below); one can replace polynomial mappings by Floquet-Jost solutions (see Section 19) in the case of continuous weights on an interval (see [87]). Here is Totik's general result (extended from cr(dfJ) C e to aess(dtt) C e):
B. SIMON
322
THEOREM 1 7 .3 (Totik [96, 99] ) . Let c be a compact subset of !R. Let I C e be an interval. Let dfL have Uess( df..L } = e be regular for e with
Then for a. e. X00 E I,
fz log(w) dx > -oo . hm - Kn ( Xoo, Xoo ) 1
n->oo n
W
(17. 5 }
(
= ---
Pe (x oo )
X00 )
The same limit holds for �Kn(Xn , Xn) if sup n nlxn - Xoo l
(17.6}
< oo .
If fls (I)
=0
and
w is continuous and nonvanishing on I, then those limits are uniform on x00 E I and on all Xn 's with sup n n lxn - X00 I ::; A ( uniform for each fixed A) .
REMARKS. 1 . Totik [98] recently proved asymptotic results for suitable CD kernels for OPs which are neither OPUC nor OPRL. 2. The extension to general compact c without an assumption of regularity for the Dirichlet problem is in [99] .
18. Variational Principle: Polynomial Maps
Iu passing from [- 1, 1 ] to fairly general sets, one uses a three-step process. A finite gap set is an e of the form
(18.1)
where (18.2)
£1 will denote the family of finite gap sets. We write e = e 1 U · · · U el+ 1 in thiH case with the Cj closed disjoint intervalH. £p will denote the set of what we called periodic finite gap sets in Section 10-ones where each Cj has rational harmonic measure. Here are the three steps: (1) Extend to e E £p using the methods discussed briefly below. (2) Prove that given any e E £1, there is eCn) E Ep , each with the same number of bands so Cj C e;n) C e)n-l) and nne)n) = e.i · This is a result proven independently by Bogatyrev [12] , Peherstorfer [71] , and Totik [97] ; see [89] for a presentation of Totik's method. (3) Note that for any compact e, if eCm) = { x I iliHt( x , e) ::; 7k }, then e( m) is a finite gap set and e = nme(m). Step (1) is the subtle step in extending theorems: Given the Bogatyrev Peherstorfer-Totik theorem, the extensions are simple approximation. The key to e E £p is that there is a polynomial A : C -> C, so A - 1 ( [-1, 1] ) = e and so that Cj is a finite union of intervals ek with disjoint interiors so that A is a bijection from each e k to [ - 1 , 1]. That this could be useful was noted initially by Geronimo-Van Assche [36]. Totik showed how to prove Theorem 17.3 for e E £p from the results for [- 1, 1] using this polynomial mapping. For spectral theorists , the polynomial A = �b. where b. is the discriminant for the associated periodic problem (see [43, 53, 104, 9 5 , 89]). There is a direct com;trm:tion of A by Aptekarev [4] and Peherstorfer [70, 71, 72].
323
THE CHRISTOFFBI�DARBOUX KERNI!:L
19. Floquet-Jost Solutions for Periodic Jacobi Matrices
16,
As we saw in Section models with appropriate behavior are useful input for comparioon theorems. Periodic Jacobi matrices have OPs for which one can study the CD kernel and its asymptotics. The two main re�mlts concern diagonal and just off-diagonal behavior:
19.1. Let be the spectral measure associated to a periodic Jacobi matrix with essential spectrum, a finite gap set. Let dJ-1. w(x) dx on e (there can also be up to one eigenvalue in each gap). Then uniformly for x in compact subsets of x ..!:. Kn (x , x) Pe ( ) (1 9. 1 ) w(x ) and uniformly for such x and a, b in with la l A, l bl B, Kn (x + � . x + �) sin(rrp.(x ) (b - a ) ) (1 9.2) Kn (x, x) rrp. (x)(b - a) 1. (19.2) is often called bulk universality. On bounded intervals, it 1-1
THEOREM
e,
=
i nt e ,
n
�
lit
:::;
:::;
--������ � --������
REMARKS.
goes back to random matrix theory. The best results using Riemann-Hilbert meth ods for OPs is due to Kuijlaars-Vanlessen [51]. A different behavior is expected at the edge of the spectrum-we will not discuss this in detail, but see Lubinsky [62] . For [- 1 , ] , Lubinsky [60] used Legendre polynomials as his model. The references for the proofs here are Simon [87, 89].
2.
1
The key to the proof of Theorem solutions of
19.1 is to use Floquet-Jost solutions, that is, (19.3)
n E Z where {an , bn} are extended periodically to all of Z. These solutions obey n +p (19.4) n For x E Un and Un are linearly independent, and so one can write in terms of u . and .. Using (19.5) Pe ( x ) p � I�� I one can prove (19.1) and (19. 2 ). The details are in [87, 89] . for
e
int
U
_
ei8(x) u
P· - l
,
u
=
20. Lubinsky's Inequality and Bulk Universality
Lubinsky [60] found a powerful tool for going from diagonal control of the kernel to slightly off-diagonal control-a simple inequality. z , (,
THEOREM
20.1. Let J-1. :::; 1-1* and let Kn, K� be their CD kernels. Then for any (20.1) I Kn (z, () - K� (z , ( ) 1 2 Kn (z, z)[I
Pn
dv
Pn
d�J-
- APn
X
Since and
n l
Xo ,
M
J! Kn- l (Xo,xo; d�J-W dOx0 K - ( xo xo d ) 2 and Kn -l(xo ,xo ) is bounded (by (24.22)), we see that ff K ( xo · ; dp,) f [ P(dv) is bounded. Thus, by (24.22) and (24.23) , =
"-n ( dM) "-n (dv) - 1
;
,
M
n- l
;
= ( 1 + ..\)-1 1 2 + O (e -C1 n )
which leads to (24.18) . This in turn leads to (ii) , and that to (iii) via (24.22) , and, for example, an (dJ.L) an (dv)
=
J XPn (x ; dJ.L)Pn -l (x; dM) =J (x; d XP n (x; dv)P n - 1
dv)
dJ.l
(24.24)
v
(24.25) 0
This shows what happens if the weight of au isolated eigenvalue changes. 'What happens if an isolated eigenvalue is totally removed is much more subtle-sometimes it is exponentially small, sometimes not. This is studied by Wong [107] .
References [1] S. Agmon, Lectures on Exponential Decay of Solutions of Second- Order Elliptic Equa tions:
Boundt!
on Eigenfunctions of
N- body
Schrodinger Operators,
Mathematical Notes,
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(30]
Asymptotic properties of polynomials orthogonal on a system of contours,
and periodic motions of Toda chains,
THE CHRISTOFF�L-DARBOUX KERNEL
[31] [32] [33] [3·1] [35] [36] [37]
[38] [39] [40]
[41] [42) [13]
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334
B.
[60] D.
SIMON
to Ann. of Math. [61] D. S. Lubinsky, Unive1·sality limits in the bulk for arbitrary measures on compact sets, to appear in J. Anal. Math. [62] D. S. Lubinsky, A new approach to universality limits at the edge of the spectrum, to appear in Contemp. Math. [63] A. A. Markov, Dbnonstration de certaines inegalites de M. Tchebychef, Math. Ann. 24 S. Lubinksy, A new approach to universality limits involving orthogonal polynomials,
appear in
and P. Nevai,
Bernstein's inequality in LP for
(1884), 172-180. [64] A. Mate
orthogonal polynomials, A nn . of Math .
[65]
A. Mate ,
0
.kYcpk (x) , k=l
A S AINT- VENANT PRINCIPLE FOR LIPSCHITZ CYLINDERS
33 9
where { IPk
: k � 1} is an orthonormal basis of L� ( 0) consisting of Dirichlet eigen functions of � x , ipk E HJ (O) , �x iPk = ->.}pk, 0 < At < .\2 S:: >.3 / oo, and ](k) = (!, 'Pk)£2 (0) · In such a case, >.k Ckl/(n-l ) , L; k >.�i](k ) 1 2 < oo, and (1.8) i� both asymptotic and convergent, in HJ (O) . Such a separation of variable� approach would work for solutionH to �2 u 0 if 0 were replaced by a compact manifold without boundary, or if the lateral boundary conditions in ( 1 . 1) were replaced by "'
=
u (y, x) = 0 ,
{1 .9)
but for ( 1 . 1 ) this method fails.
�xu(y , x ) = 0,
x E 80,
y
2: 0,
2. The Dirichlet problem on OR
as
Here we fix R E (0, oc ) and discuss solutions to (1.3), which can be rephrased
(2. 1 )
where aN is the unit normal to anR . Here !R = f on {0} X 0, !R = 0 on the rest of anR, while 9R = g on {0} X 0, 9R = 0 on the rest of anR . The hypothesis (1.2) yields
(2.2 ) Work of [2] yields a unique solution to (2. 1), smooth in the interior of OR, and satisfying
(2 .3 )
l l u * II L2(8flR) + II ( V'u) * II L2(&nn) ::::: C I!/R I I H1 (&flR ) + GII 9R I I L2(8nn) · Here, given a function v, continuous on the interior of On, we denote by v* the nontangential maximal function of v, v• (x) (2.4)
r (x )
= =
sup lv(z) l ,
zEI' (x)
x E 80n,
{z E flR : dist (z, x) ::::: K dist(z, 80 R ) },
for some fixed (large) positive K. The following additional information will be useful. PROPOSITION
2 . 1 . The solution to (2. 1) satisfying {2.2)-{2.3) also satisfies
(2.5) PROOF.
u
E H312 (0R)·
It is shown in Theorem 2.4 of [7] that, for such u ,
(2 .6) where (2.7)
A(V' u )(x)
(
One has
(2.8)
J dist (x, z)2-niV'V'u(z) l2 dz
=
(x)
J Xr(x) (z) dist(x, z)2-n dS(x)
ann
)1�
�
dist(z, 80R ) ,
340
MICHAEL E. TAYLOR
which, by Fubini'::; theorem, implie::;
(2.9)
j dist (z, &nR) j\7\i'u(zW dz :::; C / IA(V'u) (x) l2 dS(x).
OOR
OOR
From here, Proposition S of [1] yields (2.5).
D
We investigate higher regularity of n away from the top and bottom pieces of the boundary. We parametrize 0.R by z = (y, x ) , y E [0, R] , x E 0. Pick tp = r.p(y) E 00 ((0, R)). (2.10) PROPOSITION
2.2.
For u as in Proposition 2. 1, tp as in (2. 10), tpu E H5/ 2 -" (0.R),
(2.11) PROOF.
(2. 12)
'II t: > 0.
First note that �2 (r.pu) = (a� + � x )2 (r.pu) = (a� + 2a;�x + �;) (tpu)
where
Q = [a: , Mcp ] + 2�x [a; , Mcp] = Q 3 (y , ay) + �xQl (y , av ) , where Qj (y, ay) have compact y-support in y E (0, R) and order j. We hence have �2 (tpu) = Qu E H-312 (rtR) , tpu l anR = aN ( tpu) l anR = 0 (2.1 4)
(2. 13)
,
the degree of regularity of Q u following from (2 . 5) . Now Theorem 2.1 of [1J applies to (2.14), to give (2.11), with (2. 15) II'Puii Hs/2-•(r!R ) ::; Cs IIQull H-3/2 (f!R) . D
' In fact, Theorem 2.1 of [1] gives 5 2
3 2
-- < s < --.
(2. 16) We next establish the following result. PROPOSITION
2.3.
r.pa;;'u E H5 / 2-< (OR) ,
(2 . 1 7) PROOF.
have (2. 18)
Given u as in Proposition 2.1, tp as in {2. 10), m E z+ ,
Define ay,h by ay,hv(y, x )
and, since �2 (ay,hu)
=
=
'\/ t: > 0.
h- 1 [v(y +
h, x) - v(y, x)] . For small h
0,
(2 . 19) 6.2 (r.p&y,h u) = Q(&y,hu), bounded in H-3/2 -e(nR) · Since tp(y)8y,hu has vanishing Dirichlet data on &nR, (2.16) yields (2.20) li 'POy,huiiH5/2-c(f!R) :::; Ce ,
A SAINT-VENANT PRJNCIPLE FOR LIPSCHIT:t; CYLINDERS
34 1
with CE: independent of h . Taking h � 0, we have cpay ,h u � cpay u , and the bounds in (2.20) imply this convergence hold!:l weak* in H512 -= (DR) and This gives (2.17) for m
m.
llcpay uii H�/2 -•( nR ) :::;
c"'.
= 1 , and iterating this argument
gives
the
result for larger
D
The following corollary will be useful in §:�. COROLLARY 2.4. 000 ( [0, R])
such that
(2.21)
Then
tf;(y) = 1
Take u as in Proposition 2. 1. Also choose t/J 2R R for 0 :::; y :::; 3 ' 'lj;(y ) 0 for 3 y R. :::;
=
tf;(y) E
.jylpjt(x) + llle (y, x ),
(1. 1) constructed
LL
where
(4.12) holds and
(4.2 0)
j=l e=I
II W"e (Y, · ) llnJ (O) = O (e -KY),
We have
y ---+ oo.
(4.21) Remark. Making one more use of (4.4), we can improve (4.20) to (4.22) l l llle ( Y, · ) II Ho/2 -> (o) = O ( e - KY ) , y ---+ oo ,
for each o > 0. Other bounds, involving £P-Sobolev spaces and Besov spaces, can be deduced from regularity results of [1] and [8] . Of course, if 80 has additional regularity, one has further estimates, in stronger norms. 5. Another semigroup We define T8 for s � 0, acting on functions on 0 by (5.1) T8u (y , x) = u(y + s, x). We have T8 acting on various function spaces, such as £ 2 (0) . Here we investigate the action on Y = {'u E H2 (0) : !::!.. 2 u = 0, u and ON U = 0 on �+ x 80}. (5.2 ) We denote the restriction of rs to Y by ys . The study of T8 : Y ---+ Y is more closely parallel to the general set-up of [3] than the study of SY : X ---+ X . One advantage of T8 is that it is obviously a contraction semigroup on Y, strongly continuous in s E [0, oo) . On the other hand, an advantage of SY is that it incorporates an existence result for the Dirichlet problem. The next result, parallel to (4.5), makes use of result�; of §§2-3 as much as (4.5) does. PROPOSITION 5.1. For s > 0, T" : Y ---+ Y is compact. PROOF. Consider
(5.3)
pu( x) =
( u(O, x) ) 8y u(O, x)
,
p : Y ---+ H312 (0)
n
HJ (O) EB H112 (0) .
Since X = HJ (O) $ L2(0 ) , we see that (5.4)
p
:
Y
---+
X is compact.
Now, given u E Y, s > 0, we have T8U = T8Y:.pu, (5.5) where Y:. is the solution operator to (1.1) constructed in §3. Results of §3 imply T'Y:. : X ____. Y, V s > 0, (5.6)
A SAINT-VENANT PRINCIPLE FOR LIPSCHITZ CYLINDERS
345
so (5.5) represents 78 : Y -+ Y as a composition of a compact operator and a 0 continuous operator, for each s > 0. The next result paiallels Proposition 4.1. LEMMA 5.2. If ( E
Spec T1 , then 1(1 < 1 .
T1 is compact on Y, 0 -1 ( E Spec 71 ::::} 3 nonzero u E Y such that 'T1 u = (u ::::} u(y + k, x) (ku(y, x).
PROOF. Since
( 5 . 7)
Again (4.7) holds, and implies 1(1
. . . , f3n ) E N0 then we put 1 (monomials) .
If cp E S(JR") then (2.1)
iP ( O
=
(F cp) ( O
=
(211' ) - n/2
[ e- ix� cp (x) dx , }[f.n
� E :IR",
denotes the Fourier transform of cp. As usual, p- 1 r.p and cpv stand for the inverse Fourier transform, given by the right-hand side of (2.1) with i in place of -i. Here x� denotes the scalar product in IR". Both F and p - 1 are extended to S' (!Rn) in the standard way. Let cp0 E S(!Rn) with (2.2 )
cp0 (x) = 1 if lx l ::::; 1
and cp0 (y) = 0 if IYI 2: 3/ 2 ,
and let (2.3)
X
E :!Rn ,
k E N.
Then ��0 cpj (x) = 1 in !Rn is a dyadic resolution of unity. The entire analytic functions (cpj j) v (x) make sense pointwise for any f E S'(:!Rn ) .
DEFINITION 2 . 1 . Let cp = {cpj }�0 be the above dyadic resolution of unity. (i) Let 0 < p ::::; oo, 0 < q ::::; oo, s E R Then B �q (!Rn ) is the collection of all f E S' (!Rn) such that
(2.4)
II! I B;_, (R" l ll.
�
(� =
2; " 1 1 ( �Jl v I L,(R") I I '
)
' ''
< 00
(with the usual modification if q oo) . (ii) Let 0 < p < oo, 0 < q ::; oo, s E R Then F;q (!Rn ) is the collection of all f E S' (JRn) such that (2.5) (with the usual modification if q = oo) .
< oo
WAVELETS IN FUNCTION SPACES
349
REMARK 2.2. The theory of these spaces may be found in [27, 29, 32]. In particular these spaces are independent of admitted resolutions of unity
0, x > 0, and all m E zn . The theory of periodic distributions and related periodic spaces B�q ('II'n ) and F;q ('JI'n) has some history which is not the subject of this survey. We rely on [24, Chapter 3] and [27, Chapter 9] where one finds also further references. Let { 'Pj }� 0 be the same dyadic resolution of unity in Rn as in (2.2), (2.3) and in Defi nition 2.1. Let f E D'('JI'n), given by (2.34) , be extended periodically to JRn . Then f E S' (lRn) (using the same letter f) and m ( j f)v (x) = '2: am 'Pj (2rrm) ei2n x
'P
m.Ezn
trigonometrical polynomials. This justifies the following periodic counterpart of Definition 2.1. arc
2.10. Let cp = { cp1}�0 be the above resolution of unity in JRn (i) Let 0 < p ::; oo, 0 < q ::; oo, s E JR. Then B�q ('JI'n) is the collection of all f E D'('JI'n), given by (2.34), such that DEFINITION
(with the usual modification if q = oo ) . (ii) Let 0 < p < oo, 0 < q ::; oo, s E JR. Then F;q ('JI'n) is the collection of all f E D'('JI'n), given by (2.34), such that 'l l/ q am cpj ( 2rrm) e i21rmx 2jsq l l f I F;q ('li'n ) II'P = I Lp ('II'n ) < 00
(f: 1 2:: J =O
mEZ"
l)
WAVELETS IN FUNCTION SPACES
( with the usual modification if q
355
oo ) .
=
REMARK 2.1 1 . One has a periodic version of Remark 2.2, including the special cases mentioned there.
There are natural periodic counterparts of t he related sequence spaces and wavelet expansions. But the rigorous justification requires some care which may be found in [34] and to [35, Section 1]. We restrict ourselves to a description. Let wb ,m he the same wavelets as in (2.14) where we choose (and fix) L E N0 such that supp IJ!�, o
Let
C
{x E lRn : l x l < 1/2} ,
IP'j = { m E zn
G E G0 = {F, M}n.
0 $ m r < 2i+L } , be the 2 (i + L)n lattice points in 2i +LTn. Let (2.35)
w{J�:<x ) =
:
j E No,
2:: w�,m < x - z) = z:.= wb,m+2J+q (x),
with j E N0 and m E IP'j be the periodic extension of the distinguished wavelets wb m with off-points 2 -j - L m E Tn, restricted afterwards to Tn . Then one has the foll;wing counterpart of Proposition 2.3. PROPOSITION
2.12. Let u E N in (2. 12), (2.13) and (2.14), (2.35) . Then
{'I!{]�::
is an oTthonormal basis in REMARK
2.13. Let
:
j
E No ,
£2 (Tn).
G E GJ , m E
IP'j }
(J, g)'lr = r f(x) g (x) dx
}Tn
be
the dual pairing in ( D(Tn) , D' (Tn ) ) , appropriately interpreted. Similar as in (2.22), (2.23) (and with the same justification as there) we abbreviate
( 2.36)
,per = ""' ).._J ,G 2 -jn/2 \I!j,per ""' ""' )..j,G 2 -jn /2 \I!j�m ""' G� � � � � m j=O GEG.i mEl'j j ,G.m 00
m
in what follows. First we remark the obvious counterpart of (2. 16)-{2. 18): Any f E L2 (T") can be represented by f = ""' )...j ,G 2 -jn/2 \I!j ,per � m G,m
j,G,m
with and
(Recall that w{J�::; is real). The periodic counterpart of the sequence spaces in Definition 2.5 can be de scribed as follows.
356
HANS TRIEBEL DEFINITION
2.14. Let s E !R, 0 < p ::; oo, 0 < q ::; oo. Then b;;rr is the
collection of all sequences
(2.37)
,\
such that
=
{,\{_;.0 E C : j E N0,
G E Gi, m E lP'j }
and f;;r is the collection of all sequences (2.37) such that r
< oo with the usual modifications if p = oo and/or q = oo, where Xim is the characteristic function of a cube with the left comer 2-J-Lm and of side-length 2-j -L (a subcube of 'fn). After these preparations one gets now the following counterpart of Theorem 2. 7. Recall the abbreviation (2.:35) . Furthermore, O'p and O'pq have the Harne meaning aH in (2.24) . Let u E N be as in (2.12), (2.13) and (2.14), (2.35). THEOREM
2.15. Let { wb;:;} be the orthonormal basis in L2 ('fn) according to
Proposition 2. 12. (i) Let 0 < p ::; oo, 0 < q ::;
oc , s
E JR. and
u >
rnax(s, O'p - s) .
Let f E D' ('fn). Then f E B�q ('Jl'n) if, and only if, it can be represented as i,G j /2 j,per f= � (2 . 38) � ).. m 2- n w G,.,n. '
j,G,m
unconditional convergence being in D' ('IT'n) and in any space B;q ('IT'n ) with < s. The representation (2 38 ) is unique, 0'
.
(2 . 39 ) and
(2 40) .
is an isomorphic map of B;q('fn) onto b;;J'er . IJ, in addition, p < oo, q < oo, then { wb;;�} is an unconditional basis in B;q ('fn). (ii ) Let 0 < p < oo, 0 < q ::; oo, s E IR and u > max ( s, O'pq - s). Let f E D' ('fn). Then f E F;q ('JI'n) if, and only if, it can be represented as � ,\j,G ,per ' f � (2.41) m 2-i n/2 iJ!jG,m =
j,G,m
WAVELETS
IN
357
FUNCTION SPACES
unconditional convergence being in D'('JI'n) and in any space F;q ('JI'n) with u < s. The representation (2.41) is unique with (2.39) , and I in (2.40) is an isomorphic is an uncondi map of F;q (1I'11) onto f;:rr . If, in addition, q < oo, then tional basis in F;q ('JI'n ) .
{ llr·b�:·}
Discussion 2.16. This is the direct and to some extent expected periodic counterpart of Theorem 2. 7 . Basically one extends functions and distributions f E D'(1'") periodically to IRn. But these extended distributions do not belong to any space A�q(IRn) with p < oo (with exception of f = 0) . One has the same unpleasant effect if one periodises the 1Rn-wavelets as in (2.35) with x E !Rn in place of x E 1'" . Thi�:> obstacle can be circumvented if one deals first with suitable weighted spaces on !Rn. Let 'lL'a ( x
and
) = (1 + j xj 2)"12 ,
X E !Rn ,
A�q (IRn , 'Illa ) = {f E S' (IRn)
:
a E IR,
waf E A�q (IR" )} ,
naturally quasi-normed. There is a complete counterpart of Theorem 2.7 with the . same wavelets iJ!3a , ,m and the same restrictions for u and suitably modified sequence spaces. This goes back to [16] and may be found in [32, Section 6 . 2] . As for a refined version needed in the above context we refer to [34]. If 0 < p � oo and a < -njp then
A;;ier (IRn, wa ) = {f E A�q (IRn , wa ) : /( ·) = f( · - m) , m E zn } is the closed subspace of A� q (IRn, Wa) consisting of the indicated periodic distri butions on !Rn. It is isomorphic to A;q (1'" ) . First one proves a representation of
these distributions on !Rn in terms of the wavelets iJ!&P�: according to (2.35) in IR'n . Reduction to 'JI'n gives the above theorem. We refer t� [34] and [35]. 3. Spaces on arbitrary domains
3.1. Definitions. The remaining sections of thi�:> survey deal with wavelet bm:;es for function spaces on domains. First we fix some notation. Let n be an arbitrary domain in !Rn. Domain meam open set vvithout any further restrictions. Then Lp(!l) with 0 < p s; oo is the standard quasi-Banach space of all complex valued Lebesgue rnem:;urable functions in n such that
I (ln lf(x)IP dx)
II! I L ( D ) = p
l/ p
(with the natural modification if p oo) is finite. As mmal , D(D) = C0 (n) stands for the collection of all complex-valued infinitely differentiable functions in JR!n with compact support in n . Let D' (!1) be the dual space of all distributions in D . Let g E S' (!Rn ) . Then we denote by g j !l its restriction to n, =
g (cp) for
t } is an orthonormal u-wavelet basis in L2 (0) then any f E L 2 (0) can be represented as oo
J = L: L: >.t Tjnf2 4>t
(3. 12) with
NJ
j=O r=l
).,� = >.� ( f) = 2jnf2 (.f, 4>t) = 2jnj 2
and II/ 1£, (!1) I
�
(t, �
k f(x) 4>�(x) dx
2 -'nl!,i l'
-
)
,
, ,
By (3.9)-(3.11) based on (3.5) the functions 2 jn/2 4>� are L00-normalised what is convenient for our
later considerations.
The extension of Proposition 3. 7 and of the representation (3. 12) to other func tion �>paces on domains requires appropriate counterparts of the sequence spaces b�q and f;q in Definition 2.5. Let Xir be the characteristic functions of the balls B(x�, c2 2 -j) in (3.8) . DEFINITION 3.9. Let n be an arbitrary domain in ]Rn with n "1- IPI.n and let Z n be as in (3.2)-(3.4) . Let s E IR, 0 < p :S oo, 0 < q :S oo. Then b�q (ll.n ) is the
collection of all sequences
(3.13)
such that
361
WAVELETS IN FUNCTION SPACES
and f;q (Zn) is the collect·ion of all sequences (3.13) such that ; 2; " I At x,. ( l l ' I L. ( !l ) < oo IIA IJ;, (zo) I
)
(t, �
�
' '
with the usual modification if p = oo and/or q = oo. REMARK 3.10 . The structure of the sequence spaces f;q (Zn} is somewhat com plicated. A relevant discussion may be found in [32, Section 1.5.3] . One has =
b�P (Zn)
As usual
n such that
s
f;p (Zn),
E
IR, 0 < p �
oo.
Li0c(n) collects all complex-valued Lebesgue-measurable functions in
L lf(x) l dx
.(!), >.t (f) = 2jn/2 ( !, ibt.) = 2j nf2
(3.16) and
J
:
f � >.(f)
=
1n f(x) � (x) dx
{ 2jn/2 (!, ibt.)}
is an isomorphic map of Lp(O) onto fg,2 (Zn) (equivalent norms) . Discussion 3.12. Again there is a striking interplay between wavelets and building blocks in function spaces. We give an idea bow to prove Proposition 3. 7 and the above theorem. First one decomposes n in Whitney cubes Qln centred at some points 2-1m E 0 (m E zn, l E No) and of side-length 2-1 , 00
(3.17)
(modification for l = 0). Then
(3. 18) where
(3.19 )
fz,. Xtr
l,1"
is the characteristic function of Qlr and
= Xtr f,
HANS
::Jti2
TRIEBEL
9lr one has a canonical situation which admits a uniform 2. 7. Pulling back one gets flr in a (small) neighbomhood of Qlr · Clipping together
For the dilated functions
application of the �n-expansion according to Theorem wavelet expansions for
ii!-{-;.,m in
these expa,nsions (with some tm:hnical care) and using that the wavelets
(2. 14) are well-ada,pted to dyadic dilations and translations one gets expansions for f which are first steps towards ( 3 . 1 5 ) . But there are some obstacles. In contrast to
wb m in(2.25) with j E N , Lhe dilated starting ii!� , m do not fulfil the moment conditions needed for the
the dilated terms originating from i nterpretation as atoms in
terms originating from
Lp(JRn).
But this applies only to the starting terms
It
and this difficulty can be removed by direct Lp-arguments.
is more serious that
other dilated wavelets) . This is the point where the multiresolution structure of the
just these dilated starting terms spoil the desired orthogonality (in contrast to the wavelets is of great service. Let the origin in the one-dimensional case
'lj;c where
breaking point with the dilation With
GE
at the left and the dilation
(21 +Lx - m)
=
n
= 1 be a
at the right.
L cfm ,t 1/JF (2l+l+L x - t)
consisting of finitely many elements with supp
1/Ja
2-l-l
{F, M} as in ( 2 . 1 2 ) one has ncar the origin the multiresolution
1/Ja
property
2-1
¢F (21 + 1 +t
tEZ
-t )
·
c supp
'lj;p
,Pn
(21+L · -m ) .
VVith some local orthogenalisation at the origin one can remove the disturbing terms
(21+£ -m) ·
a,t the expense of
(21+1+L -t) .
If n :2':
·
In case of
n
=
1
all
breaking points are isolated endpoints of intervals and the above orthogonalisation can be done at each such point separately. structure of to
n
2 then one can rely on the product
wb, m in (2 . 14) which transfers the orthogonalisation from one direction
directions (applying Fubini's theorem) . Some care is necessary, especially in
corner points and along edges. But all this can be done and results finally in the boundary wavelets
if?�
according to (3. 1 1 ) .
In this way one
proves first Proposition
3 . 7 and afterwards (with the help of the indicated atomic arguments) Theorem 3.11.
The first step of this procedure (without the final orthogonalisation) was
done in [33] , where we denoted the outcome as a Proposition 3.7 and Theorem proofs will be given in
3.1 1
para-basis.
The above versions of
arc published here for the first time. Detailed
[35].
Discussion 3 . 1 3 . The above arguments for the spaces Lp (O) in arbitrary do n rely mainly on the localisation (3. 18) , the homogeneity ( 3 . 1 9 ) and the 1Rn-wavelet theory. There is little hope that other spaces A�q (O) and A�q (n) ac mains
cording to Definition 3 . 1 in arbitrary domains fit in this scheme.
But there is a
remarkable modification which even lays the foundation of all what follows. sketch the basic ideas. Let
( 3.2 0 ) Let
CTpq
0 o-pq • O < q � oo, (4.6) O < p � oo, PROPOSITION
( q = oo if p u > s,
=
oo
) be the spaces as introduced in Definition 3. 1 . Let for u E N with
with Nj E N { � : j E No ; r = 1, . . . , Nj } be an orthonormal u-wavelet basis in L2(0) according to Propo:;ition 3. 'l and Defi nition 3. 5. Then n n max(l , p) < v .::; oo , s - - > - - , (4.7) v p
(what means v = oo if p oo). Furthermore, f E Lv (fl) is an element of .F;q (O) if, and only if, it can be represented as =
(4.8)
j
=
NJ
oo
L L A� Tjn/2 ClJ�. ,
j=O r=l
absolute ( and hence uncondil'iona� convergence being in L, (n) . If f E the representation (4.8) is unique with A = A(/),
(4.9)
I
:
f�
then
2jn/2 l f(x) i.(x) dx A(j) = { 2j n 2 (!, t )}
A� (f) = 2jnf2 (!, �)
and
F';q(O)
=
/
is an isomorphic map of .F;q (O) onto f;q (Zo.). lf p < oo, unconditional basis in F';q(Q).
q
< oo then { f. } is an
Discussion 4.4. First we remark that (4. 7) is a continuous Sobolev embedding. It ensures that f E .F;q (O) admits an expansion of type (3.15) with /� 2 (Zo. ) in place , of f�. 2 (Zn) (locally if p oo ) . By Definition 3.1 one gets f E i';q (O) from =
(4. 10)
f = giO
with g E F:q(fi)
C
F;q(lR11).
366
HA�S TRIEBEL
This reduces (4.8 ) to corresponding expansions in �n as considered in Theorem 2.7 and Discussion 2.8. One does not need moment conditions for the atoms in (2.29) with . (!),
(4 . 1 7 ) and ( 4. 18)
J
:
I
f-t
>.(f) =
{ 2jn/2 (!, V } on r similar as in {3.7), (3.8) based on the counterpart Zr of (3.2), (3.3). Afterwards one obtains wavelet bases for all spaces A;q(r) of the same type as in Theorem 2 . 15 with n 1 . =
5 . 2 . Decompositions. Let 0 be a bounded coo domain in IR " and let r = an be its boundary. Let A�q(n) and A�q (O) be as in Definition 3.1 and let A�q (r) be their counterparts on r as introduced in Definition 5. 1 . First we recall some more or less known trace assertions and decompositions for the spaces
with The linear bounded
trace
1 ::; p
ljp .
370
HAKS TRIEBEL
has the usual meaning. Let K = [s - ;1- be the largest integer K K < s - l. Let v be the (outer) normal on r. Then () g 1 K = [s - - ] - , tr�'P : g ,...... trr ovk : 0 � k � K , (5.2) p k maps
p
{
E
N0 with
}
K
II s;; r;-k(r)
s; ( O ) onto q
1
k=O
K
and
II s;; "-k(r).
F;q (O) onto
1
k=O Futhermore there are linear and bounded extension operators extL� p ·. {go , . . . , gK } I-> g (5.3) mapping K
IT s;; �- k (r)
(5.4)
into s;q ( n)
k=O
and
K
II s;; i -k(r)
(5.5)
into F�(n)
k=O
such that
trr's p o extr's p
=
id ,
identlty m •
•
K
Il Bpsq- lP -k ( f ) . k=O
We refer to [27, p. 200] with corresponding assertions in [26] as a forerunner. But the extension operators constructed there are not good enough for our purpose. One needs wavelet-friendly extension operators extending functions 9 on r with supports in an c-neighbourhood in f of some point "'/ E f into functions with supports in a corresponding c-neighbourhood of "'! in n. This can be done but will be shifted to !35] . Its paves the way to clip together wavelet bases of the spaces A�q (O) according to Theorem 4.8 with wavelet base.s for s;q (r) (if exist ). This procedure relies on the following assertion. Recall that A.;q (O) is the completion of D(O) in A;q (O), whereas l;q (O) has the same meaning as in Definition 3.1. PROPOSITION 5.4. Let n be a bounded c= domain 'tn JR.n accm·ding to Defini tion 4, 1(iii) and let r = 80, Let 1 � p < oo ,
1�
q
1 0 < s - - It No ,
< oo,
p
Then
(5.6) with tr�'P as
( 5 . 7)
in
(5.2 ) .
Furthermore, s;q (n) = i3;q (n)
x
K
II s;; r;-k(r)
k =O
1
3 71
WAV ELETS IN FUNCTION SPACES
and F%9 (0)
(5.8)
REMARK 5.5. If 1 :::; p, q
=
i';q(n)
< oo
x
K
k II B;; r; - (r) . k=O 1
then (5.6) can be complemented by
-1
Ti j No, r =f. r'. Let be the characteristic function of B(xt , 2-j ) for some s IR, 0 < q Then b;q (Z11) is the collection of all sequences (5.10 ) .A = { .At E C : j E No; r = 1, . . . , Nj } such that =
c1
E
X.i r
r
p,
:::;
oo .
r' - c1
'
E
C2
n
n
C2
> 0.
Let
372
HANS TRIEBEL
and f;q ('J.P.) is the collection of all sequences (5.10) such that II A l f;, in Theorem 4.8. This may explain the difference of the above part (ii) and Definition 3.5. After these preparations we can now formulate the main result of Section 5. As before we write A;q ( D ) with A E {B, F} and similarly a;q(z n ) with a E {b, f} if the assertion applies equally to the B-spaces and F-spaces. THEOREM 5.10. Let m E No . Let ( 5. 1 1 )
s = m+u
1 1 with 1 � p < oo and - - 1 < (]' < -. p
p
(i) Let D = 1 = (a, b) with -oo < a < b < oo be an open interval in JR. Then for any u E N with u > m there is a common u-wavelet basis accord·ing /,o Definition 5.8(ii) for all spaces A�q(I) with 1 � p, q < 00 and s as in (5. 1 1 ) . Furthermore,
N1 1 = 2::: :�:::t:-� u) 2-j 12 t oc
j=O T=l
and f � >-(!) is an isomorphic map of A�q(I) onto a�q (:l/ ) . (ii) Let n be a bounded coo domain in the plane R.2 . Then for any u E N with
WAVELETS IN FUNCTTO'< SPACES
373
u > m there 1:s a common u-wavelet basis according to Definition 5.8(ii) for all spaces A�q(n) with 1 :S: p, q < oo and s as in (5.11) . HtTthe'Tmorc, oo
Ni
f = L L .Xt( f) Ti .(f) k E !2,2 (Zn ) where >.{ (f) k = 2jn/2 L I (f, D" �{) I
(
)
JC>!J9
(equivalent norms). Sometimes �� are called vaguelettes. It remains to be seen if this observation is of any use. COMMENT 6.5. We excluded in Proposition 5.4 and Theorem 5.10 the case s - � E N0 • But there are some negative results. Let again n be a bounded c= domain in JRn and 1 < p, q < oo. Then D(rl) is dense in A��P (f!) . On the other hand there is no orthonormal u-wavelet basis according to Proposition 3.7 which is also a basis in A��P (n) (in contraHt to A"!�P(n) according to Theorem 4.8). COMMENT 6.6. We mention a second negative result. By Remark 2.2(iv) the spaces B�(l�n ) with p, q, s as in (2.9) can be equivalently quasi-normed by (2.10), (2. 11). This is no longer valid if s < n 1 + . But one can define corresponding spaces as subspaces of Lp (lRn), B�q (JR.n ) {J E Lp(JR.n ) : IIJ IB�q (JRn) ll m < 00 } where 0 < s < m E N, 0 < p, q oo, (6. 1) and
=
(� - )
1 on e
For a nonnegative Borel measure w on n, we set (1 .4) and (1. 5)
c1 (w, n) = sup
{ l_.,. Ju(x)l 2 {
c2 (w , n) = sup
dw
:
}
u E C0( il ) , IIV'u i i P CO)
w(e ) : cap (e, n )
.
S:: 1
}
where the supremum above is over compad. sets e c n of positive capacity. As was shown in [Ml] (see also [M4] , Sec. 2.5) , (1.6)
WEIGHTED NORM INEQUALITIES
3 79
where the constants 1 and 4 are sharp. Measures w obeying conditions of the type c2 (w, D) < oo which characterizes the imbedding L1•2(0 ) c L2(D, ch;.;) will be called Maz'ya measures (or sometimes admissible measures) . The class of Maz'ya measures associated with (1 .2) in the case of Riesz poten t i als T = Iu for 0 < a < n on n Rn was characterized completely in terms of Riesz capacities by Adams, Dahlberg and Maz'ya (see [AH] , [M4] ) in the 1970s. Alternative characterizations were given later by Kerman and Sawyer [KS] who used more precise local energy conditions, and Maz'ya and Verbitsky [MVl] in terms of yet more localized pointwise inequalities. The latter turned out to be especially well adapted for applications to nonlinear PDE (HMV] , [KV] , [PV1] [PV3], [VWI]. These results are discussed in Sec. 2. In Sec. 3, we consider two weight inequalities and nonlinear potentials for the so-called dyadic model [COV2], [ C OV3] , [NTV] where T is the integral operator =
Tf(x) =
(1. 7)
{
Kv (x, y) f(y) da(y)
{
k ( i x - y i ) f(y) da(y) ,
fw,
with the kernel Kv (x, y) = L Q E'D K(Q)xQ (x) xQ (y) . Here v = {Q} is the family of all dyadic cubes on Rn, K(Q) are arbitrary nonnegative constants, and XQ are characteristic functions of Q. Weighted norm inequalities for general convolution operators
(1.8 )
Tf(x)
=
fw,
where k = k(r ) is an arbitrary positive nonincreasing function of r > 0, can be deduced from the dyadic model. This makes it possible to answer a question raised in (AH] , Sec. 7.7, of how to define an analogue of Wolff's potential for general radially decreasing kernels. In Sec. 3 we give an appropriate definition of Wolff 's potential, which is far from being obvious, and prove Wolff's inequality for radially decreasing kernels [COV2], [COV3] . We observe that Woltr's potentials and their modifications have become an important tool in modern theory of quasilinear and fully nonlinear PDE [KiMa] , [L] , [MaZi] , (PVI]--[PV3] , [TW] . In Sec. 4 we will present recent developments [MV2] concerning trace inequal ities with "indefinite" weights of the type:
where (1. 10)
I Ln w l � I L, wl � a l u l2
(1.9) n
C l l \7 u l li2(R" ) '
u E C8" (Rn),
:2: 3, and their inhomogeneous analogues: lu l
2
l l \7 u l l i2 (Rn) + b l l u l l i2(Rn ) •
for some positive constant:-; a , b, where w E D'(Rn), n 2 1 . Here D(Rn) = C0 (Rn ) , and the left-hand sides are understood in the sense of distributions. Both inequalities (1 .9) and (1. 10) are important to the Schrodinger operator theory. The first inequality expresses the domination (in absolute value) of the potential energy associated with w by the kinetic energy, while the second one is equivalent to the classical notion of the relative form boundedness of w with respect to the kinetic energy operator Ho = -�. This concept is used in the so-called KLMN theorem which makes it possible to define a self-adjoint operator H = Ho+w so that the quadratic form domain Q(H) coineides with Q(H0) provided w is real-valued. See, e.g., [EE] , [RS] , [Sch] . It is worth mentioning that the
I. E. VERBITSKY
380
quadratic form inequality ( 1 .10) is equivalent to the boundedness of the operator H : W l,:t (Rn )
__..
w - 1 ,2 (Rn ) .
When the form bound a > 0 i n ( 1 . 10) can be arbitrarily small (with b depending on a) w is said to be infinitesimally form bounded relative to C. This notion is used extensively in mathematical quantum mechanics (see [RS] , [Sch] ) . Another important version of these form boundedness properties occurs when one replaces the nonrelativistic kinetic energy, namely I I V'u l l i2 (Rn ) ' with its relativi:;tic counterpart, I I ( -� + 1 ) 114u l l i2 (Rn ) associated with the Sobolev space W 1f 2 ,2 (Rn ) of order 1 /2 . The form boundedness problem for H = -� + w where w E D'(Rn), along with its infinitesimal version, was recently solved by Maz'ya and Verbitsky [MV2] [MV6] . The main idem; discussed below can be expressed as follows: (1 .9) holds for a distributional potential w if and only if it holds for I V' � - 1 w l 2 in place of w. This nonlinear transformation w --> IV'� -1wl 2 reduces the form boundedness problem for "indefinite weights" to the well-studied case of nonnegative weights. In Sec. 4 we will outline a proof of this form boundedness criterion for H = -� + w based on sharp estimates of equilibrium potentials and related weighted norm inequalities [MV2] .
Similarly, the class of w E D' (Rn) associated with the relativistic form bound edness of 1i = ( -� + 1 ) 112 + w is invariant under the transformation w --> 1 ( 1 �)- 112wl 2 . The relativistic problem can be reduced to a similar nonrelativistic one for the operator H = -� + w with a distributional potential w on a higher dimensional Euclidean space as was shown in [MV4] . In [MV5] necessary and sufficient conditions were obtained for the infinitesimal form boundedness of the potential energy operator associated with w with respect to the kinetic energy operator H0 = -6 on L2 (Rn) . Here w is an arbitrary real or complex-valued potential (possibly a distribution). Furthermore, a related form subordination property of Trudinger type (see [Tru] , and also [Sim] , [RSS] , and the literature cited there) is characterized explicitly in [MV5] . More precisely, we will present in Sec. 5 a characterization of the class of po tentials w E D'(Rn) which are -�-form bounded with relative bound zero, i.e., for every € > 0, there exists C(t: ) > 0 such that (1.11)
For complex-valued w,
D(H)
c
it
follows that II is an m-sectorial operator on L2 (Rn) with
W1 , 2 (Rn) ( [EE], Sec. IV.4).
The characterization of (1.11) obtained in [MV5] uses only the functions I \7 ( 1 -
A ) - 1 w l and 1 ( 1 - A)-1 w l , and is based on the representation :
(1. 12)
w = div f + 1,
In particular, f E Lf0c (Rn)",
f(x) = -\7(1 - �) - 1 w, 1 = (1 - 6)-1 w.
1 E Lfoc (Rn) , and, when
n ;::::
3,
(1.13)
once ( 1 . 1 1 ) holds. Here BJ (x0) is a Euclidean ball of radius b centered at xo.
WEIGHTED
NORM
38 1
INEQUALITIES
In the opposite direction, it follows from the results of [MV5] that ( 1 . 1 1) holds whenever (1 . 14)
sup t52r-n
lim
XoER"
o-+0
(
)
r f !f(xW + h(x) l dx = o, .IBa(xo)
for some r > 1. Such potentials form a natural analogue of the Fefferman-Phong class [F] for the infinitesimal form boundedness problem, where cancellations be tween the positive and negative parts of w come into play. It includes functions with highly oscillatory behavior as well as singular mea.•mres, and contains properly the class of potentials based on the original Fefferman-Phong condition where lwl is used in {1. 14) in place of lfl 2 + h'l- Moreover, one can expand thiH cla...:;s fur ther using a sharp condition due to Chang, Wilson, and Wolff [ChWW] applied to lfl2 + hi· In the proofs given in [MV5] considerable technical difficulties have been overcome using sharp estimates for powers of equilibrium potentials, factorization of functions in Sobolev spaces, and theory of Av-weights, along with appropriate lo calization arguments, and good understanding of trace inequalities for nonnegative potentials w. In [MV5] , we also study quadratic form inequalities of Trudinger type where C(t:) in ( 1 . 1 1 ) has power growth, i.e., there exists Eo > 0 such that ( 1 . 1 5)
for every E E (0, co), where f3 > 0. Such inequalities appear in studies of elliptic PDE with measurable coefficients, and have been used extensively in :spectral theory of the Schrodinger operator [AS] . As it turn�> out, it is still possible to characterize (1. 15) using only lfl and I 'Y I defined by ( 1 . 12 ) , provided /3 > 1 . In this case (1. 15) holds if and only if both of the following conditions hold: sup o2�:;� - n
(1.16)
xoER" 0. = 3n + 2 - 2p(n + 2) if p * < p < 1. The form boundedness problem for the general second order differential operator
=
=
( 1 .21)
C. =
n
L
i,
j=l
aiJ f)J}J + L j=l
bJ Oj + c,
where aij , bi , and c are real- or complex-valued distributions was solved in [MV6] . Here C. is not even assumed to be elliptic. \Ve will discuss in Sec. 6 quite complicated necessary and sufficient conditions for the quadratic form inequality
(1.22) to hold for some a , b > 0. It is easy to see that the symmetric part of the matrix (aij ) must be uniformly bounded, and the skew-symmetric part reduces to the first order terms. The main problem here is to inve::;tigat.e the interaction between the first-order and zero-order terms. The proofs make use of compensated compactness arguments (a vector-valued version of the div curl lemma) , along with the gauge transform involving powers of equilibrium potentials. Applieations to multidimensional Riccati's equations, quasilinear and fully non linear PDE, global estimates of Green's functions, etc., can be found in [FV] , [HMV], [KV] , [ Ma Z i] , [PV1] -{PV3] . -
2. Basic trace inequalities
We start with the following important theorem [M3] .
THEOREM 2 . 1 . (Ma:.-:'ya) Let 1 < p < 00. Let n be an arbitrary open set in Rn ' and let w be a nonnegative locally finite Borel measure on n. Then the inequality
(2 . 1 )
holds if and only if, for any compact set E c n, (2 . 2)
w ( E)
:::;
C cap1,p (E, n),
383
WEIGHTED NORM INEQL"ALITIES
where C is a constant which is 'independent of E. Here cap1 ,p (E, n) is the capacity defined by (2.3)
cap 1 ,p (E, n) = inf
{ fo l'\7u iP
dx :
u ( x ) � 1 on E, u E Cgo(n) .
}
Theorem 2.1 has numerous applications in harmonic analysis, operator theory, function spaces, linear and nonlinear PDE's, etc. (see, e.g., [AH] , (FV] , [M4] ,
(MSh] , [MaZi], [PV2]). For simplicity, we will only consider some analogues of Theorem 2 . 1
n = R"' for Riesz potentials defined by Ia f
=
( - A) - � f = c( n , a: ) ( l l a - n * !),
·
in
the case
0 < a: < n,
where c ( n , a:) is a normalization constant. We also set la (f duJ) ( x) = c( n , a)
1
f(y )
· -a dw ( y ) , R" I x - y I n for potentials with a Borel measure duJ in place of dx, and law = Ia ( ldw) if f = 1 on Rn . The Riesz t:apacity of a measurable set E c Rn is defined by ( 2.4)
(2.5 )
Capa ,p ( E ) = inf
{ fo l giP dx : Iag(x) � 1
on
E,
g E
}
L�{Rn) .
In the case a: = 1 it is known that C ap1 ,p (E) :::::: cap1 p (E, Rn) for compact sets E, , where cap 1 p (-, Rn) is defined by (2.3), and constants of equivalence depend only , on p (see (AH]) . The following theorem for a: = 1 and q = p is equivalent to Theorem 2.1 when n = Rn . THEOREM 2.2. (D. Adams-Dahlberg-Maz 'ya) Let 1 < p < and let 0 < a: < Let w be a nonnegative Borel measure on R"' . Then the following statements are equivalent. (i) The inequality oo
n.
(2.6)
holds where C is a constant which is independent of f . (ii) For every compact set E C Rn , (2 . 7)
w (E) � C Capa,p (E),
where C is a constant which is independent of E. The statement {i)=}(ii) in Theorem 2.3 is obviou:o; the converse follows from the so-called strong capacitary inequality
100 Capa,p({x : llaf(x) l > t}) tP -1 dt
�
C l lfl li,P , f
E
LP(Rn ) ,
discovered by V . Maz'ya i n 1972 for a: = 1. In a series of papers by D . Adams, B. Dahlberg and V. Maz'ya in the late 70-s, the preceding inequality was established for aU a: > 0 and p > 1 . Another proof valid for more general convolution operator::; is due to K. Hansson. (See [AH] , [M4] , [H].)
384
I.
E.
VERBITSKY
REMARK 2.3. Condition (2.7) combined with a standard estimate of the ca pacity from below, C ar ,p E ) ?:: C JEJ1- � , immediately yields the following well known sufficient condition: holds if (E) � C JEI1 - � , for every compact set E. This can be restated in terms of weak spaces: cU.u = p(x) dx, where p E L"•00 (Rn), for r = ; P.
a (
(2.6)
w
Lr
A substantial improvement of the sufficient condition mentioned in Remark 2.3 was found by C. Fefferman and Phong [F]. THEOREM
2.4. If dJ.u = p(x) dx where
l pl+< dx
(2.8)
� C JB J 1 - "P\';+-�)
,
E
> 0,
A sharp version of the Fefferman-Phong condition is clue t.o Chang, Wilson, and Wolff [ChWW] .
for every ball B
=
Br (x), then (2. 6) holds.
2.5.
REMARK An improved version of both the Fefferman-Phong and Chang \Vilson-Wolff conditions where w is not necessarily absolutely continuous with re spect to Lebesgue measure is given in (MVl] (see Corollary below).
2.10
It is easy to see that the capacitary condition (2. 7) is equivalent via duality to the inequality (2. 9)
for every compact set E C Rn, where � + -/;; = It was noticed by Kerman and Sawyer [KS] that in this dual form it is enough to restrict oneself to E = where = is a hall (or cube) in Rn.
1.
B
B Br(r)
THEOREM (Kerman-Sawyer) Let 1 < p < oo and let 0 < a: < n. Let be a nonnegative Borel measure on R" . Then the trace inequality (2. 6) holds if and only if
2.6.
w
(2. 10) for every ball B
=
r x) .
B (
The following theorem [MVl] shows that, in a sense, balls may be replaced with single points x.
B Br(x) in (2. 10) =
THEOREM 2 . 7. (Maz'ya-Verbitsky) Let < p < oo and let 0 < a: < n. Let w be a nonnegative Borel measure on Rn . Then the following statements are equivalent. (i) The trace inequal-ity {2. 6) holds. (ii) For every compact set E C Rn , the capacitary inequality (2. 7) holds. < oo dx-a. e. and {iii)
1
law
(2.11)
Ia[(I,w)P'](x) � C Iaw(x)
dx-a. e.
(iv) The trace inequality (2. 6) holds with w replaced with the absolutely contin uous measure dv = dx, or, equivalently,
(Iaw)P'
(2. 12)
v (E
)
=
l (Iaw )P' dx � C Cap0,p (E) ,
where C is a constant which is independent of a compact set E.
WEIGHTED
NORM
385
INEQUALITIES
REMARK 2.8. A simple direct proof of Theorem 2.7 from which Theorem i::; deduced as a corollary can be found in [V4] (see also [VW2]) .
2.6
Another useful characterization of Maz'ya measures in terms of discrete Car leson measures was given in (V4]. COROLLARY 2.9. (Verbitsky) Let 1 < p < oo and let 0 < a < n . Let w be a nonnegative Borel measure on Rn . Then any one of the conditions {i)-(iv) of Theorem 2. 7 is equivalent to the following pmperty: (v) For every dyadic cube P,
)p' IQI :s; Cw(P), � ( I Qw(Q) il-�
(2. 13)
where the sum is taken over all dyadic cubes Q contained in P, and C does not depend on P. w
The following sufficient conditions which are applicable to broader cla.o.;::;es of than those considered in Remark 2.3 and Theorem 2.4 follow from Theorem 2.7
(iv) .
COROLLARY
2 . 10 . (i) If [Otw
E
L8•00 (Rn ) , where s
a(pn- 1 )
I
then the trace
inequality (2. 6) holds. ' {ii) lf t > 0 and J�(!Otw) Cl+.Q
[ sup la xEQ n
da(x ) , s
(�L ) ( L ) a( )
QED
'
a
Q'CQ
(lQ)
AQ ·
Q ' CQ
>.Q'
-1 '
s
da(x) .
3.3. Let a be a positive locally finite Borel measure on Rn . Let 1 < s < oo. Then there exist constants Ci > 0, i = 1, 2, 3, which depend only on s, such that, for any A = (>.Q ) Q e TJ, >.Q E R+ , A1 (A) :::; C1 Az(A) :::; C2 A3(A) ::; C3 A1 (A) .
Theorem 3.1 is deduced from Lemma 3.3 in the case
s = p'.
>.Q
=
K(Q)w ( Q) a ( Q ) and
We next treat continuous versions of the above theorems for integral operators with radial kernels, Tk [f da] ( x ) =
{
Jan
k ( ix - yl) f(y) da(y).
Here k = k ( r) , r > 0, is an arbitrary lower semicoutinuous nonincreasing positive function. The corresponding nonlinear potential is defined by Wk, u [w](x) =
where
-
{
+oo
Jo
k (r) a (Br (x))
(1
Br(x)
k(r) (y) dw (y )
ds k (·r) (x ) = a 1r ( ) Jor k ( s) a (B s (x) ) � , ) (B x
)p'-l
d
_!_ , r
I. E. VERBITSKY
390
for x E Rn, T > 0.
THEOREM 3.4. Let 0 < q < p < +oo, 1 < p < oo, and lei. w and a be nonnegative Borel measures on Rn . A ssume that a satisfies a doubling condition, and the pair- ( k, a ) has the following logarithmic bounded oscillation property (LBO) : yEBr (x)
sup k(r) (y) � A
(3. 10)
.Ln
holds if and only
inf
k(r)(y) ,
E Rn, r > 0. Then the trace inequality f E LP(da ) , I Tk [fda] iq dw � C ll fii1P(da}'
where A does not depend on x (3. 1 1)
yEBr(x)
if Wk,,- [w] E L q(p-1) v-·� (dw ) .
The (LBO) property is satisfied by all radially nonincreasing kernels in the case a satisfies a reverse doubling condition (see [COV2] ) . In particular, the following theorem holds for convolution operators Tk [/] k*f and da = dx. We define the corresponding \Volff's potential by
da = dx, and also by Riesz kernels k(x) = lxla-n, 0 < a < n, if
=
(3.12)
Wk [w] (x)
=
roo o k(r) k(r·) ,:1 w (Br(x)) v!:l rn-l dr, J
1 k(r)(x) = n
where
_
(3.13 )
r
for x E Rn, r > 0.
ir k( s) sn-l ds , o
THEOREM 3.5. Let 0 < q < p < +oo and 1 < p < oo. Let w be a nonnegative Borel measure on Rn. Suppose k = k( I x - y I), where k( r) is a lower semicontinuous nonincreasing positive function on R+, and Wk [w] is defined by (3. 12). Then the trace inequality (3.14 ) holds
if and only if
(3. 15)
Wk[w] E L � (w). p-o
REMARK 3.6. For Riesz kernels, a proof of Theorem 3.5 was given in (V4] . Some technical details related to passing from a discrete to continuous version using shifts of the dyadic lattice, as well as generalizations, can be found in [COV3] . REMARK 3.7. A characterization of (3. 14) for Riesz or Bessel kernels in terms of capacities was given in (MN] (see Sec. 2). The special case q = 1 of Theorem 3.5 leads by duality to Wolff's inequality for radially nonincreasing kernels [COV2] , [COV3) . THEOREM 3.8. Let 1 < p < oo. Let w be a nonnegative Borel measure on Suppose k = k(lx - y l), where k(r) is a lower semicontinuous nonincreasing positive function on R+ , and Wk[w] is defined by (3. 1 2). Then there exist positive constants cl ' c2 which depend only on k / p and n such that
Rn.
(3 . 16)
Ct l lk * wl l�v' (Rn)
�
ln Wk[w) dw � C2 ll k * wi i�P' (R")'
WEIGHTED NORM TNRQUALITIES
39 1
Theorem 3.8 demonstrates that (3. 12) is an appropriate definition of Wolff's potential for radially nonincreasing kernels. This solves a problem posed in [AH] , p. 214. 4. Indefinite weights and Schrodinger operators
We start with some prerequisites for our main results. Let D(Rn) = CBults become vacumL if n = 1 and n = 2. Analogous results for inhomoge neous Sobolev spaces WL2 (Rn) hold for all n � 1. For V E D'(Rn), consider the multiplier operator on D(Rn) defined by < V u. v > := < V, u v > ,
u, v E D(R") , where < · , · > represents the usual pairing between D(Rn) and D' (Rn). Let us denote by £-L2(Rn) = £1•2(Rn)* the dual Sobolev space. If the scsquilinear form < V · > is bounded on £ 1 • 2 (Rn ) £1•2 (Rn): (4. 1 )
(4.2)
x
·,
where the constant c is independent of u, v E D(Rn), then V u E L - 1 •2(Rn), and the multiplier operator can be extended by continuity to all of the energy space £ 1•2 (Rn ) . (As usual, this extension is also denoted by V . ) We denote the class of multipliers V : £1•2(Rn) --. £ - 1 •2(Rn) by
M ( L t ,2 (Rn ) -t £ -t,2 (Rn)). Note that the least constant in (4.2) is equal to the norm of the multiplier operator: I I V I I M(L' ·,(R")-.L-12(R")) = sup { 11 Vul i£-1.2(R") : llullv.2(R") :::; 1 } . For V E M(£1•2(Rn) --. L -1•2(Rn)), will need to extend the form < V, u v > defined by the right-hand side of (4.1) to the case where both u and v are in c
£ 1•2 (Rn ) . This can be done by letting
we
< V u, v >= Nlim < V uN, VN > , -+oo
where u = limN-oo UN , and v = limN-.oo VN , with uN , VN E D(Rn) . It is known that this extension is independent of the choice of UN and 1'N · We now define the Schrooinger operator H = Ho + V, where Ho = -Ll, on the energy space £1•2 (Rn). Since H0 : £1•2 ( Rn) --. £-1•2 (Rn) is bounded, it follows that H is a bounded operator acting from £1• 2 (Rn) to L - 1 • 2 (Rn ) if and only if V E .M (£ 1 , 2(Rn) --. £ -1 •2 (Rn )) . Clearly, (4.2) is equivalent to the boundedness of the corresponding quadratic form: I < V u , u > I = I < V,
lu l2 > I :::; c i i'Vull i2(R" ) •
I.
392
E. VERBITSKY
where the constant c is independent of u E D(Rn). If V is a (complex-valued) measure on Rn, then this inequality can be recast in the form: (4.3) For positive distributions (measures) V, this inequality is well studied (see Sections 1 and 2). We now state the main result for arbitrary (complex-valued) distributions V. By Lfoc(Rn)n = Lfoc (Rn) ® C" we denote the space of vector-functions :f = (ft . . . . r ) such that ri E L�c (Rn), i = 1, . . . , n. THEOREM 4.1. Let V E D' (Rn ) . Then V E M(Ll.2(Rn) ,
n
the inequality
__.
L-1•2(Rn)), i. e.,
I < V u, v > I S:: c l lul l u . 2 (R"J l lvi1Ll, 2 (R" l
(4.4)
holds for all u, v E L1•2{Rn), if and only if there is a vector-field :f E qoc (Rn) such that v div r, and =
(4.5) for all
f 2 1f(x ) l 2 dx s:: c f I Vu(xW dx, jR" 1 '1L(x) l }Rn
u E D(Rn). The vector-field :f can be chosen in the form :f
=
\7 .6. -l 17.
REMARK 4.2. For :f = \7 .6.-1 V , the least constant C in the inequality (4.5) is equivalent to I IV I I 2M(Ll,2 (R" )-L-1.2 (R")) .
The proof of the "if" part of Theorem 4.1 is easy as long as V is represented in the divergence form. This idea was discovered by mathematical physicists in the 1970s (sec [MV2] , p. 265) . Indeed, suppose that V div f , where f satisfies (4.5). Then using integration by parts and the Cauchy-Schwartz inequality we obtain: =
I < v u , v > I = I < v, u v > I
f, v vu > + < r, ·u. vv > I S:: l l fv i i L2(R")" l l \i' u l l u (R"' ) + l l fu i iL2(R" ) l l\i'vi i P(R") S:: 2 VC [IV'u[ [P(R") [ [V'vl lucR") : =
I
0) and f > 0, (4.7)
}{
BR(xo )
j :f(xW dx :::; C (n, f) Rn-2+< I ! VIi�(£1·2(R")-+L -l,2(R")) •
where R 2 max{ 1 ,
lxo j } .
The following statements are concerned with sharp estimatet> of equilibrium potentials ast>ociated with a set of positive capacity.
PROPOSITION 4.4. Let (j > ! and let P = Pe be the equilibrium Newtonian potential of a compact set e C Rn of positive capac·ity. Then
(4.8)
i ! V'P i !L2( R")
REMARI< 4.5. For 8 :S: PROPOSITION 4.6.
v E L 1•2(Rn ) . Then
(4.9)
I ! 'Vv li£ 2 (R" ) :S:
2
=
� Jcap (e) . 26 - 1
�' it is easy to see that \7p& ¢ L2 (Rn ) .
Let 8 > 0, and let v be a real-valued function such that
r
}R
"
' \i' (v P� ) (x) l 2 P dx 20 :S: (x)
''2
( 8 + 1 ) {48 + 1 ) l i'Vv £2(R")·
The proof of the preceding inequalities is based on multiple integration by parts, along with the following estimates for Newtonian potentiaL'> of positive (not necessarily equilibrium) measures w.
PROPOSITION 4.7. Let w be a positive Borel measure on Rn such that P(x) = l2w (x) ¢. oo . Then the following inequalities hold:
(4.10 ) and
v E D ( R" ) ,
2 j\7 P(xW r 2 Jn n v ( x) P (x) 2 dx :::; 4 I I 'Vv l i £2 (R" ) '
(4. 11) REMARK 4.8. The constants 4 and 1 respectively in (4. 10) and (4. 11) are sharp (see [MV2]) . REMARK 4.9. An inequality more general than (4.11), for Riesz potentials of order a E (O, n) and Lp norms (with nonlinear Wolff's potential in place of P (x)) , but with a different constant was proved in [V4].
Vle will also need the following proposition which is deduced from the facts that P(x)28 is an A2 weight (this was proved earlier in [MVl]) , and that the Riesz transforms are bounded in weighted L2 spaces with such weights [St] . (See details in [MV2] .) PROPOSITION 4.10. Let w = � - 1div ¢' where 1 < 28 < n�2 • Then
(4.1 2)
¢'
E
D
0
C".
2 dx r r - z dx Jn" j\i'w (x) j P (x) 21i :S: C( n, 8) }R " j¢(x) j P ( x ) 2� ·
Suppose that
I . E. VERBITSKY
394
\Ve now sketch the proof of the "only if" part of Theorem 4. 1. S upp ose that I < V, u v > I :S: I IV I I M(J,' ·2(R"J-D;-' C R")) l l"vu i i P CR"") I I Y'vi i P C R"" J ·
($ = ( ¢ 1 , . . . , ¢ n) b e a n arbitrary vector-field in V ® en, and let w = �-l div ($ = - J2 div {$, (4.13)
Let
so that
¢ = V'w + s, div s = 0. Note that w E C00 (Rn ) n L1,2(Rn ) , since w(x) = O( l x l 1 -n) and I Y' w(x) l = 0 ( lx l -n)
Hence,
I < V, W > I = I < f, ¢ > I =
l ln
as
lxl --> oo .
f(x) ;f(x) dx ·
l
where by Lemma 4.3, f E LfocCRn) n . We pick 8 so that 1 < 28 < n:::_2 , and factorize w (x) = u (x) 'u(x), where w (x) v = P(x 6 " ) Consequently, -
-
0 I < r , ¢ > I :::; I I V I I M(Ll·Z (R")-- > L21(R" )) I I V'P I I £2( R" ) I I V'vi iL2{ R" ) · B y Proposition 4.6 { 2 dx 2 dx 2 r 0 I I V'vi i£2 (R") :S: J j V'( vP ) (x) l P(x) 2 5 = J " IY' w(x) J P(x) 26 < oo . R R" From this, applying Proposition 4.4, we obtain:
I .Ln
f(x) · ¢(x) dx :S: 0 (1 - 28) - � I IV I I M(£1·2(R" )--.L2" l (R" ))
l
x
cap (e
Notice that by Proposition
Hence,
1 )2
4.10,
(
{
2 dx JR" I V'w(x) l P(x) 28
.
{ 2 dx 2 dx r }Rn IV''w (x) l P (x) 26 s C( n , o) }Rn l ¢(x) l P(x) 25 .
x
cap (e) �
(J�..
2 l ¢(x) j
From the preceding inequality we deduce
(L,
)�
2 lf(xW P(x ) 5 (x) dx
)
1
2
:s: C(n,
� 5) p 2
1 2 •
o) I I V I I cM(£L2 (R")--> L-1,2 (R" ) ) cap (e) � .
Note that P is the equilibrium potential of e , and hence P(x) � 1 dx-a.e. on e. Thus,
WEIGHTED NORM INEQUALITIES
395
for every compact set e c R11, and by Theorem 2.2 this gives ( 4.5), which completes the proof of Theorem 4 . 1. There is an analogue of Theorem 4.1 in terms of ( -D.)- 1 1 2 V .
THEOREM
4. 1 1 . Under t.h.e a.ss?J.m.ptirYn.
f E L]0c(R11) such that I IXBl(xo) f iiLr(an ) < 00.
For 0 < r < oo, we denote by L�nif(Rn) all
IIJI IL� Dif
sup
=
xoERn
}
1 on e .
By U(R11)11 = U(R11) :29 en we denote the class of vector fields :f = {ri }j= 1 R11 ---+ en, such that rj E Lr(R"), j = 1 , 2, . . . , n, and use similar notation for other vector-valued function spaces. By M + (R" ) we denote the class of nonnegative locally finite Borel measures on R11 • If V E V'(R11) is nonnegative, i.e., coincides with w E M + (R11 ) , we write fan J u (x ) l 2 dJ..V in place of (V, J u J 2) = (Vu , u) for the quadratic form associated with the distribution V, if u E C0 (Rn). Sometimes we will use fan J u(x) l2 V(x) dx in place of (Vu, u) even if V is not in L foc (Rn) . We set 1 r f (x) dx mB ( f ) = for a ball B
C
TBl ./B
Rn, and denote by BMO(R") the class of f E Lioc (R11 ) for which sup
xuER" , ti>O
�
I Bli
Xo ) I
r
.JB6(xo)
lf( x ) - mn6(xo) U)I" dx < +oo,
for any 1 .::; r < +oo. It follows from the John-Nirenberg inequality that this definition does not depend on the choice of r E (1 , +oo) . We will also need an inhomogeneous version of BMO(R11) (the so-called local BMO) which we denote by bmo(R11) . It can be defined in a similar way as the set of f E L�nif(Rn ) such that the preceding condition holds for 0 < o .::; 1 (see (St] , p. 264) . The Morrey space _Lr. .>. (Rn) ( r' > 0, A > 0) consists of f E L!oc (Rn) such that sup
xoERn , 6 >0
1
r lflr dx < +oc . J B.s (xo)J-n JB8 ( x o) >.
396
I. E. VERBITSKY
In the corresponding iuhomogeneom; analogue, we set 0 < /j � 1 in the preceding inequality. It will be clear from the context which version of the Morrey space is used. We now state the main results of [MV5] .
.
5 1 Let V E V'(Rn), n ? 2. Then the following statements hold. (i) Suppose that V is represented in the form:
THEOREM
.
(5.1)
where f E Lfoc (Rn)n and 'Y E Lfoc(Rn) satisfy respectively the conditions: (5.2)
lim
o_.+D xoE R" . 1 1m
(5.3)
f8
sup sup u
sup sup
o�+DxoE R"
6
(xo ) Jf(xW Ju(x)i2 dx
2 J J V'uJ J£2 (B�(x0) )
f86 (xo) lr(x) J J u (x) i 2 dx
2 I J Y'uJI P (Bo�(xo) )
=
=
0,
0,
where u E CQ'" (B0 ( x0)), u =:/= 0. Then V is infinitesimally form bounded with respect to -6.. , i. e., for every E > 0 there ex·ists C(E) > 0 such that (1.11) holds. (ii) Conversely, suppose V is infinitesimally form bounded with respect to 6.. Then V can be represented in the form (5.1) so that both (5.2) and (5.3) hold. Moreover, one can set f = -Y' ( 1 - 6.. ) -1 V and 1 = ( 1 - 6.. ) -1 V in the representation u
-
.
(5. 1 ) .
REMARK 5.2. I n the statement of Theorem 5.1 one can replace conditions (5.2) and (5.3) with the equivalent condition where J(1 - 6.. ) - � V J 2 is used in place of J f J 2 in (5.2). The importance of Theorem 5.1 is in the means it provides for deducing explicit criteria of the infinitesimal form boundedness in terms of the nonnegative locally integrable functions J fl2 and l r l · THEOREM
5.3. Let V E V' (Rn ) , n ? 2. The following statements are equiva
(i) V is infinitesimally form bounded with r-espect to (ii) V has the form (5. 1) where f = -\7(1 - 6.. ) - 1 V, measure w E Af+ (Rn) defined by lent:
(
-6.. . 1 =
(1 - 6._)-1 V, and the
)
dw = l f (x W + !r( x) J dx
(5.4)
has the property that, for ever-y E
>
0, there exists C( f) > 0 such that
J u (x W dw � € J J V'uJ Ji2( R" ) + C(t: ) J J V'uJJl,( R") '
{ Jnn (iii) For w defined by (5.4),
(5.5)
(5.6)
. hm
J-.+U P.o: diam Ra _< {) sup
1 -(R0 ) W
w ( P)2 -0 � I P J 1 - l. Pt:;.Po
Vu E C0 (Rn) .
"'
n
'
where P, Po are dyadic cubes in Rn, i.e., sets of the form 2i(k + [0, l) n ), where i E Z, k E Z".
397
WEIGHTED NORM INEQUALITIES
{iv} For w defined by (5.4), o-++O " ' di am e�o
(5.7)
lim
sup
w(e) -- =
cap (e)
0,
where e denotes a compact set of positive capacity in Rn . {v) For w defined by (5.4),
ii wB6 (xo ) ���- 1 2 (Rn) = 0, w(Bt� (xo)) o-++ 0 xo ERn restriction of w to the ball B,s(xo). .
(5.8)
hm
sup
where wB6 (xo) is the (vi} For w defined by (5.4) , (5.9)
where G1 * w = (1 - t!,) - 1 w is the Bessel potential of order 1.
It is worth noting that although Theorem 5.3 holds in the two-dimensional case, its proof requires certain modifications in comparison to n ;::: 3. In the one dimensional case, the infinitesimal form boundedness of the Sturm-Liouville oper + V on L2 ( R1 ) is actually a consequence of the form boundedness. ator H =
-l;.
THEOREM 5.4. Let V E 1)' ( R1 ) . Then the following statements are equivalent. (i) V is infinitesimally form bounded with respect to i. e., (ii) V is form bounded with 1espect to I( V u, u ) [ :::; canst l lull�fl.2( ' ' 'r/u E cgo (R 1 ) .
�.
d�2 ,
R)
{iii} V can be represented in the form V
(5. 10)
sup
xE R 1
=
�;; + "'f , whe1e
1x+l (jr(x) 2 + i"'f(x) ) dx < l
l
x
+ oo .
(iv} Condition (5. 10) holds where
r(x)
=
{
jR '
sign (x - t) e - l x-tl V (t) dt,
'Y(x)
=
{
jR'
e - ! x - t l V(t) dt
are understood in the distributional sense. The statement (iii)=> (i) in Theorem 5.4 is known ( [Sch], Theorem 1 1 .2.1), whereas (ii)=>(iv) follows from [MV2] . We now state a characterization of the form subordination property (1 .15). It was formulated originally in [Tru] , in the form of the inequaliLy:
(5. 1 1)
i ( Vu, u) l
:S
€
i [ Vu[ [i2(Rn) + C€- v l lu l l i•(Rn)'
'r/u
E Cgo (Rn) ,
for V ;::: 0 . Such V are called €v -compactly bounded in [Tru] . It follows from Nash's inequality that (1. 15) yields (5.11) with '{) = n!2 {3 + � ; the converse is also true, provided v > �, and is deduced using a localization argument. In the critical case '{) = � , (5.11) holds if and only if V E L00 (Rn ), while for 0 < v < � ' it holds only if v = 0 . Necessary and sufficient conditions for ( 1 . 1 5 ) , or equivalently (5.11) with v = �{3 + � (see [MV5] ) , can be formulated in terms of Morrey-Campanato spaces
398
L
E. VERBITSKY
using mean oscillations of the functions f and rems 5.1�5.4.
1
which have appeared in Theo
5.5. Let V E V'(Rn), n 2: 2, and let 0 < f3 < +oo. (i) Suppose there exists t:o > 0 such that ( 1.15) holds for e11ery t: E (0, t: 0 ) . Then V can be represented in the form THEOREM
(5.12)
V
=
div f + ')',
where f = - V' ( l - �)-1 V E Lf0c (Rn) n and 1 = ( 1 - �)- 1 V E Lf0c (Rn) . Moreover, there exists So > 0 such that (5.13)
{
}Bo(xo)
lf(x) - ffi£� (xo) (f) l2 dx :S c Sn� 2 �+i ,
0 < S < Jo,
r b(x) l dx :::; c sn� ff!-r ' 0 < 8 < 6o , }Bo (xo )
(5. 14)
where c does not depend on xo E Rn and So . Furthermore, f E L�nif(Rn) n if (3 2: 1, and f E L= (Rn) n if 0 < f3 < 1. (ii) Conversely, if V is given by (5.12) where f E Lf0c(Rn)n, 1 E Lfoc (Rn) satisfy (5.13), (5. 14) for all 0 < S < So then there exists Eo > 0 s·uch that (1.15) holds for all 0 < t: < t:o . REMARK 5.6. (a) In the case (3 = 1, it follows that (5. 13) holds if and only if f E bmo(Rn) n . In other words, V E bmo_ 1 (Rn), where bmo_1 (Rn) can be defined as the space of distributions f that can be represented in the form f = div § where § E bmo(Rn)n. We ohf;erve that bmo�1 (Rn) = F.:.•{"' (Rn), where F!;,·q stands for the scale of inhomogeneous Triebel�Lizorkin spaces (see, e.g., [KT], [T] ) . (b) I n the case 0 < f3 < 1, (5.13) holds if and only if f is HOlder-continuous: lf(x) - f(x' ) l :S c l x - x'l i3Tl , -
-
1 - iJ
lx - x' l
1, (5. 13) holds if and only if f lies in the inhomogeneous Morrey space /3-1 C 2 , n� 2 13 + 1 ( Rn t , i.e., - )l �Bo (xo) lr(x
2
dx
� c bn- 2 .!L::l
ll+ l ,
0 < o < Oo .
•
These statements follow from the known characterizations of Morrey spaces. Note 213 that, according to (5. 14) , 1 E £1 , n � P+I (Rn). REMARK 5.7. (a) An immediate consequence of Theorem 5.5 is that, for all (3 > 0, (1.15) is equivalent to the following localized energy condition:
1 1 (1 - � ) - � ( 17b, x0 V ) l l l2(Bo ( :ro ) ) :S c on- 2 ��i , 0 < o < Oo, Xo E Rn, where 1/Q, x0 (x) = ry (o-1 (x - x0) ) ; here ry is a smooth cut-off function such that 17 E c= (B1 (0) ), 0 :S TJ :S 1, and ry = 1 on B (0) . i (b) A similar energy condition, is
1 1 ( 1 - � ) - � ( TJ&, xo V ) l ll2 (Rn) :::; c on - 2 �+� ' 0 < 0 < Oo, Xo E R", sufficient, but generally not necessary in the case n = 2.
WEIGHTED NORM
399
INEQUALITIES
We next state a criterion for the multiplicative inequality (1. 18) to hold, which is equivalent to a homogeneous version of (1. 15) with t:o = +oo and p 13�1 . =
E V'(Rn), n 2: 2, and let 0 < p < 1 . (i} Suppose that (1. 18) holds. Then V can be represented in the form
THEOREM 5.8 . Let V
(5. 15) where f (5. 16)
=
V1.6. - 1 V,
V
=
div f,
and one of the following conditions hold:
f E BMO(Rn) n
if
p
=
�;
f E Lip1 _2p (Rn)n
f if(xW dx -::: c n+ 2- 4v, jBs (xo) where does not depend on x0 E Rn , 6 > 0. (5. 17)
if 0 < p < �;
if � < p < 1;
(ii) Conversely, if V is given by (5.15) where f E L?oc ( Rn )'� and satisfies (5.16), (5.17), then (1. 18) holds. c
REMARK 5.9. In Theorem 5.8, the "antiderivative" f = V1.6. -1 V can be re placed with (-.6.) - � V . Furthermore, a corollary we deduce that ( 1 .18) holds if and only if V E BM0_ 1 (Rn) = P:·� (Rn) for p = � ' where BM0_ 1 (Rn) is defined as above in the caBe of its inhomogeneous counterpart bmo_I (Rn). (See, e.g., [KT) where this space is thoroughly studied in the context of Navier-Stokes equation�:�.) For 0 < p < �' ( 1 . 18) holds if and only if V E B�2;' (Rn) for 0 < p < �- Here .fr:;.,q and B�,q are homogeneous Triebel-Lizorkin and Besov spaces respectively (see aB
[T] ) .
In the case p ! , statement (ii) of Theorem 5.8 (sufficiency of the condition BMO) is equivalent via the 1-£1 - BMO duality to the inequality Vu E Ctf (Rn). (5. 18) l l u Vui iHl (R" ) 'S c l l u i i £2 (R") 1 1Vui i £2(Rn) , Here 1-£1 (Rn) is the real Hardy space on an ([St]). The preceding estimate yields the following vector-valued inequality which is used in studies of the Navier-Stokes equation, and is related to the compensated compactness phenomenon [CLMS) : (5. 19) div it = 0, l l (it · V ) itl l1il (R" ) 0, x E Rn. A complete characterization of the class of admissible measures wt� 2 (Rn ) can be expressed in several equivalent forms discussed above. These criteria employ var ious degrees of localization of w, and each of them has its own advantages depending on the area of application. We now otate the main form boundedness criterion [MV6] . For A = (aij ) , let At = (aji) denote the transposed matrix, and let Div : D'(Rn)nx n --> D'(Rnt be the row divergence operator defined by
n
Div(aij ) = (L: Oj aij)'?=l ·
(6. 10)
j= l
6. 1 . Let L = dh· (A V' · ) + b V' + V, where A E D' (Rn)nx n , b E D'(Rn)n and V E D' ( R" ) . n 2 2. Then the following statements hold. (i) The sesquilinear form of L is bounded, i.e., (6.4) holds if and only if � (A + At ) E Ux:; (Rn )nxn , and b and V can be represented respectively in the form THEOREM
·
b = c + Div F,
(6. 1 1)
V = div h,
where F is a skew-symmetric matrix field such that
(6.12)
F - � (A - At ) E BMO( Rnt x n ,
whereas c and h belong to (6 . 13)
Lroc(Rn)n,
and obey the condition
lcf + lhl 2 E
wt� 2 ( Rn) .
(ii) If the sesquilinear form of L is bounded, then c, F, and h in decomposition (6. 11) can be determined explicitly by h = V' (D.. - 1 V), c = V'(A- 1 div b) , (6 .1 4)
(6. 15) where (6.16) and
(6.17)
402
I. E. VERBITSKY REMARK
placed with
Condition
6.2.
in statement (ii) of Theorem
(6.16)
may be re
6.1
(6. 18) which ensures that decomposition (
REMARK
� (A + At ) E
In the case n
6.3.
L00 (R2 ) 2 x 2 ,
REMARK
6.11) holds. = 2, we will show that (6.4)
b - � Div (A - At)
holds if and only if
E BM 0 _ 1 (R2 ) 2 , and
V
=
div b =
0.
Expressions like \7(�-1div b) , Div ( � - 1 curl b') , and \7(�-1 V)
6.4.
ru;ed above which involve nonlocal operators are defined in the sense of distributions . This is possible,
as
is shown in
[MV6] ,
since �- 1div b, �-1curl b, and �-1 v
can be understood as the limits in the sense of the weak BMO-convergence (see
[St] , p. 166) of, respectively, � -1 div (1/JN b) , � -1 curl ('1/JN b) , and � -1 ('1/JN V) as N --+ + oo . Here ¢N is a smooth cut-off function supported on {x : l xl < N}, and the limits above do not depend on the choice of It follows from Theorem
1/JN .
that £ is form bounded on
6.1
L1· 2 (Rn)
x
L 1• 2 (Rn)
if and only if the symmetric part of A is essentially bounded, i.e . , � (A + At) E
L "" (Rn) n x n ,
and b1 ·
\7 + V is form bounded, b1
(6.1 9) if
=
-
-
b-
In particular, the principal part
where
2 Dtv(A - A ) . ·
I
Pu
=
t
div (A \lu)
is form bounded if and only
t oo nxn � (A + A ) E L (Rn ) (6.20) , Div �(A - A1) E B:M0 1 (Rn )n. (6.21 ) A simpler condition with � (A - At) E BMO(Rn ) n xn in place of (6.21) but generally not necessary, unless n :::; 2.
is sufficient ,
Thus, the form boundedness problem for the general second order differential
operator iu the divergence form
(6.22)
£
=
b
·
\7 + V,
As a corollary o f Theorem
i.e., for all
u, v E CQ" (R") ,
l l,
(6.23)
(6.3)
b E D'(R")",
6.1,
b
holds, where � - 1 curl b E
div b
=
Jlfx-yi 0, V = 0.
r
x
=
combined with
E
b \7 + V ·
is form bounded,
l
\7 ( � -1div b) + Div (L:l-1curl
BMO(R")nxn,
and
[ I V' (� - 1 div EW + IV' (�
E Rn,
in the case
We observe that condition
by L:l - 1curl b
E D'(R").
(b · \lu v + Vu v) dx :::; C l l u l l u. 2 (R") I Iv i i£1. 2 (R") •
(6.24)
for all
V
deduce that, i f
we
then the Hodge decomposition
(6.25)
is reduced to the special case
L:l-1 V E
(6.25)
n
:;:::
b)
- l vW l dy :::; canst rn-2,
3;
in two dimensions, it follows that
is generally stronger than � - 1 div b E BMO
BMO, while the divergence-free part of
BMO, for all n :;:::
2.
b is
characterized
403
WEIGHTED NORM INEQUALITIES
A close sufficient condition of the Fefferman-Phong type can be stated in the following form:
1
(6.26)
lx-yl. V7 e+ i>. ' =
where the gauge .-\ is a real-valued function which lies in L��? (Rn). The problem of choosing an appropriate gauge is known to be highly nontrivial. In the present paper, ..\ is picked in a very specific form:
..\ = r log (Pw), 1 < 2T < n�2 , n ;::: 3, where is a constant, and Pw = ( -.6.)- 1 w is the Newtonian potential of the equi T
librium measure w associated with an arbitrary compact set e of posiLive capacity. We will verify that , with this choice of A, the energy space L1• 2(Rn) is gauge invariant, and the irrotational part c V (.� -1div b) of b obeys =
1 IC1 2 dx
s; const cap ( e) ,
404
I.
E. VERBITSKY
where the constant does not. depend on
JcF
E
9Jt� 2 (Rn ) .
to BMO, and
b = c + Div F.
e.
This is known to be equivalent to
Jn addition, a careful analysis shows that
F = �-I curl b belongs
These conditions combined turn out to be necessary
and sufficient for (6.29 ) .
[MV6]
In
applications are given t o the magnetic Schrodinger operator
M
de
V + JeW
and a V' are form bounded. Thus, the form boundedness criterion for M can be
fined by (6.2) .
It
is shown that
M
is form bounded if and only if both
·
deduced from Theorem 6 . 1 .
These results are extende d to the Sobolev space
sary and sufficient conditions are given
([MV6] ,
C : Wl, 2 (R " ) _,
lV1• 2 (Rn ) . In particular, neces
Theorem
5.1)
for the boundedness
of the general second order operator
This solves the
w -1 , 2 (Rn ) .
M, with respect to the Laplacian on L 2 (Rn). Th e
relative form boundedness problem for £,
magnetic Schrodinger operator
and consequently for the
proofs involve an inhomogeneous version of the div-curl lemma (see
[MV6] ,
Lemma
5. 2 ) . Other fundamental properties of quadratic forms associated with general dif ferential operators, e.g., relative compactness, infinitesimal form boundedness, in equalities of Trudinger's and Nash's type can be characterized using a similar ap
proach ( see
[MV6] ) . References
D.R. Adams and L.I. Hedberg, Function Spaces and Potential Theory, Springer-Verlag, Berlin-Heidelberg-New York, 1996 M. Aizenman and B. Simon, Brownian motion and Harnack inequality for Schrodinger [AS] operator�, Comm. Pure Appl. Math. 35 ( 1982) 209-273. [COV1] C. Cascante, J.M. Ortega, and I.E. Verbitsky, On imbedding potential spaces into Lq(dw), Proc. London Math. Soc. 80 (2000), 391-414. [COV2] C. Cascante, J.M. Ortega, and I.E. Verbitsky, Nonlinear potentials and two weight trace inequalities for general dyadic and radial kernels, Indiana Univ. Math . .J . 53 (2004) , 845-882. [COV3] C. Cascante, J.M. Ortega, and I.E. Verbitsky, On LP - Lq trace inequalities, J. London Math. Soc 74 (2006), 497- 5 1 1 . [ChWW] S.-Y.A. Chang, J.M. Wilson, and T.H. Wolff, Some weighted norm inequalities con ceming the Schrodinger operators, Comment. Math. Helv. 60 ( 1985), 217-246. [CLMS] R. Coifman, P.L. Lions, Y. Meyer and S. Semmes, Compensated compactness and Hardy spaces, J. Math. Pures Appl. , 72 (1993), 247-286. [EE] D.E. Edmunds and W.D. Evans, Spectr·al Themy and Differential Operators, Clarendon Press, Oxford, 1987. [F] C. Fefferman, The uncertainty principle, Bull. Amer. Math. Soc., 9 (1983), 1 29-206 M. Frazier and I.E. Verbitsky, Global exponential bounds for Green's function and the [FV] cond1tional gauge, preprint (2008). [H] K. Hansson, Imbedding theorems of Sobolev type in potential theory, Math. Scand. , 45 (1979), 77-102. K. Hansson, V .G. Maz'ya and l.E. Verbitsky, Criteria of solvability for multidimensional [HMV] Riccati's equations, Arkiv for Matern. 37 (1999), 87-120. [HW] L.I. Hedberg and Th.H. Wolff, Thin set� in nonlinear potential theory, Ann. Inst. Fourier (Grenoble) , 33 ( 1983), 161-187. N.J. Kalton and I.E. Verbitsky, Nonlinear equations and weighted norm inequalities, [KV] Trans. Amer. Math. Soc. 351 (1999), 3441-3497. R. Kerman and E.T. Sawyer, The trace inequality and eigenvalue estimates for [KS] Schrodinger operators, Ann. Inst. Fourier (Grenoble) 36 (1986), 207-228.
[AH]
WEIGHTED NORM INEQUALITIES
(KiM a) [KT)
[Lj
[M aZi) [Mlj [M2] [M3]
[M4] [MN] [MSh]
[MVl)
[MV2] [MV3]
[MV4] [MV5] [MV6]
[NTV) [PV1]
[PV2]
[PV 3) [RS ] [RSS]
[S ] [Sch] [Sim] [ StW]
T. Kilpel ainen and J.
405
Maly, The Wiener test and potential estimates for qua.siline.a.r
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SCHOOL OF �ATHEMATICS, UNIVERSITY OF BIRMINGHA M,
BIRMINGHA:-.1, B15 2TT,
UNITED
KINGDOM
E-m.n.il address : I . E . Verbitsky«lbham . ac . uk
DEPARTMENT OF MATHEMATICS, UNIVERSITY OF M ISS OURI , COLUMBIA, �0
E-mail address: igorl!lmath .missour i . edu
6521 1,
USA
Proceedings of Symposia in Pure l\1athematics
Volume 79, 2008
THE MIXED PROBLEM FOR HARMONIC FUNCTIONS IN POLYHEDRA OF JR3 MOISES VENOUZIOU AND GREGORY C. VERCHOTA
ABSTRACT. R.
Brown's
theorem on mixed Dirichlet and Neumann bound special case of polyhedral do mains. A (1) more general partition of the boundary into Dirichlet and Neu mann sets is u�ed on (2) manifold boundaries that are not locally given as the graphs of functions. Examples are constructed to illustrate necessity and other implications of the geometric hypotheses . M.
ary conditions is extended in two ways for the
1 . INTRODUCTION In [Bro94] R. M . Brown initiated a study of the mixed boundary value problem for harmonic functions in crea.-;ed Lipschitz domains n with data in the Lebesgue and Sobolev spaces £2(00) and W1 •2 (an) (with respect to surface measure ds) taken in the strong pointwise sense of nontangential convergence. At the end of his article Brown poses a question concerning a certain topologic geometric difficulty not included in his solution: Can the mixed problem be solved in the (infinite) pyramid of JR3, \X1 \ + \X2 \ < Xa , when Neumann and Dirichlet data are chosen to alternate on the faces? In this article we avoid the geometric difficulties of what can be called Lipschitz faces or facets and provide answers in the cm;e of compact polyhedral domains of JR3 . Some other recent approaches to the mixed problem for second order operators and systems in polyhedra can be found in [MR07] [MR06] [MR05] [MR04] [MR03] [MR02] and [Dau92] . Consider a compact polyhedron of JR3 with the property that its interior n is connected. n will be termed a compact polyhedral domain. Suppose its boundary an is a connected 2-manifold . Such a domain n need not be a Lipschitz domain. Partition the boundary of n into two disjoint sets N and D, for Neumann data and Dirichlet data respectively, so that the following is satisfied. (1.1) (ii)
(i) ]\l is the union of a number (possibly
D = an \ N is nonempty.
zero
) of closed faces of 80.
(iii) Whenever a face of N and a face of D share a !-dimensional edge as boundary, the dihedral angle measured in n between the two faces it-> less than Date :
2000
7r.
June 9, 2008.
Mathematics Subject Classification.
35J30,35J40. second author gratefully acknowledges part ial support provided Foundation through award DMS-0401159 E-mail address: [email protected] The
407
by
the National Science
@2008 American
Ma.thematica.l Society
408
MOISES VENOUZIOU AND GREGORY C. VERCHOTA
The L2-polyhedral mixed problem for harmonic functions is (1 .2) Given f E Wl.2 (8D) and g E L2 (N) show there exists a solution to l:::,. u = 0 in n such that (i) u -tn.t. f a.e. on D. (ii) 8,u -+n.t. g a.e. o n N. (iii) 'Vu* E L2(8D ) . Here 'Vu* is the nontangential maximal function o f the gradient of u . Generally for a function w defined in a domain G For a choice of a > 0
by
( 1.3)
w* (P)
=
XH(P)
sup lw(X) J , P E 8G.
nontangential approach regions for each P E fJG are defined
r(P) = {X E G : IX - PI < (1 + a)dist(X, 8G)}
Varying the choice of a yields nontangential maximal functions with comparable LP(fJG) norms 1 < p � oo by an application of the Hardy-Littlewood maximal function. Therefore a is suppressed. In general when w * E LP(fJG) is written it is understood that the nontangential maximal function is with respect to cones determined by the domain G. The outer unit normal vector to n (or a domain G) is denoted v = Vp for a . e . P E an and the limit of (ii) is understood as r(P) 3X--+P
lim
vp
·
'Vu(X) = g(P)
and similarly for (i) . A consequence of solving ( 1 .2) is that the gradient of the solution has well defined nontangential limits at the boundary a.e. In addition, as Brown points out, solving the mixed problem yields extension op erators W1•2(D) --+ W1•2 (8D) by f �--+ u l an where u is a solution to the mixed prob lem with u iD = f . Consequently problem (1 .2) cannot be solved for all f E W1•2(D) when D and N are defined as on the boundary of the pyramid. For example, since the pyramid i::; Lipschitz at the origin so that Sobolev functions on its boundary project to Sobolev functions on the plane, solving (1 .2) implies that a local W1•2 function exists in IR2 that is identically 1 in the first quadrant and identically zero in the third. Such a function necessarily restricts to a local W ! •2 function on any straight line through the origin. But a step function is not locally in w! ·2 (JR). The boundary domain D (and its projection) 0 has continuous boundary values avg - limt!O g(x, y, t)/t at every point of D for which y > ¢(x) .
:=
(ii) JD (avg)2ds = + x . (iii) atg (x, y, 0) = 0 at every point of N \ {0} . (iv) avg E L 2 (oZ).
Proof. (i) follows by Schwarz reflection while (iii) follows by the symmetry in t of 0 and g. The maximum principle shows that the Green function for Z dominates from below the Green function for 0, gz :::; g :::; 0. On az both Green functions vanish so that av9z 2: avg 2: 0 while av9Z is square integrable there, establishing (iv). D. S. Jerison and C. E. Kenig's Rellich identity for harmonic measure ([JK82] Lemma 3.3) is valid on any Lipschitz domain G that contains the origin. It is (n - 2 )wa (O ) =
{ (avga (Q))2v lac
·
Qds(Q)
with respect to the vector field X. Here ga(X) = F(X) + wc(X) is the Green function for G, and F is the fundamental solution for Laplace's equation. Denote
4 18
:tvlOISES VENOUZIOU AND GREGORY C. VERCHOTA
by w.,. , w and Wz the corresponding harmonic functions for the n.,. , n and z Green functions respectively. By Z ::> Z \ D 11 ::> fl.,. and the maximum principle =
8.,g.,.
S
av9Z on an.,. \ D \ D.,.
and (3.1)
For Q E D and 11 = 11Q the outer unit normal to 11.,., 11 (Q - u3 ) = r , while for 0. Formulating the Rellich identity with respect to the Q E Dr , v · (Q - T€3 ) vector field X - re3 and using these fact!> (n = 3) •
=
wT (Teg )
so that
=
r
r
lan�\D\D�
laz
(a,_,gT )2 v . (Q - T eg ) ds + T
( fJ.,gz ) 2 v (Q - Te3 )ds + T ·
r (8,_,gT )2ds
JD
$_
r (avg)2ds = Wz (re3 ) + T JDr (avg?ds
JD
w (re3 ) - Wz (T€3 ) < Wr (Te3 ) - Wz (Te3 ) $_ T T and (ii) follows from (3. 1 ) and r 1 0.
{ (8.., g)2ds
}D
D
For o > 0 define smooth subdomains of 0
Go = {g
0}. Then D u N c 8Z+ n {t = 0}.
=
{(x, y, t) E
Lemma 3.2. . Suppose 6u = 0 in Z+, \7u* E L 2 (8Z+ ) , 8vu ---+ n.t. 0 a . e . on N , and u ___.n .t. 0 a .e . on D. Let Y c Z be a scaled cylinder centered at Po with dist(fJY, 8Z) > 0. Let Y+ be the corresponding half-cylinder. Then u E C(Y + ) ·
Proof. The hypothesis on \7u* implies u* E L2 (8Z+) so that u and V'u have non tangential limits a.e. on fJZ+ [Car62] [HW68] . Extend u to the bottom component of Z \ D \ N by u(x, y , t) = u(x, y, -t). By the vanishing of the Neumann data on N, D.u = 0 in the sense of distributions in the domain 11 = Z \ D and then classically. Fix d > 0 and suppose X E Y + is of the form X = (x, y, d) for y � ¢(x) - Md. Denote 3-balls of radius and distance to D comparable to d by Ed. Denote 2-discs in fJZ+ with radius comparable to d by b..d and let
f denote integral average. Then
THE MIXED PROBLEM FOR HARMONIC FUNCTIONS IN POLYHEDRA OF by the mean value theorem, the fundamental theorem of calculus, the of u on D and the geometry of the nontangential approach regions
l u (X)I �
f
Bd(X)
l u i � Cd (
f
R3
4 1 \J
a. e. vanishing
\lu*ds)
�d (x,y+2Md,O)
C depends only on M . By absolute continuity of the surface integrals and \lu* E £2 there i� a function TJ(d) ----? 0 as d ----? 0 so that J� J \lu*)2ds � TJ(d) for all �d c az+ . Consequently the Schwarz inequality now yields l u(X)I ::; C7J(d) . Suppose now X is of the form X = (x, (x) - Md, t) for 0 � t � d. Because u
where
has been extended
l u(X) I �
f
lui � (
B.t(X)
and
f
f
l ui ) + d(
Hd(x,.p ( x ) - Md,d)
\lu*ds)
�d(x,¢(x)-Md,O)
lu(X)I � 2Cry(d). The lemma follows.
Partition 8Z+ by N+ = N, D+ = 8Z+ \N and 8Z+ let zr be the scaled cylinder centered at p0 of width the corresponding half-cylinders Z+ with
0
= N+ uD+· For 3/4 > r > 0 2r and length 81vfr. Define
N� = N+ n az� (not a scaling of
N+) and D� = 8Z� \ N�
Z+ is called a split cylinder with Lipschitz crease. By (3.2) and the Fubini theorem, g E W1•2 (8Z+ \ {t = 0}) for a.e. r.
With this partition
3.3. Let 9 be the Green function for n Z\D For almost every > r > 0 there exists no solution u
Proposition
with pole at the to the L2 -mixed problem (1.2) in the split cylinder with Lipschitz crease Z+ 71Jith boundary values u ----? n .t. g E W1•2 (D+ ) and Ov U -+ Ov9 = 0 on N'; . origin.
�
=
Proof. Suppose instead that there is such a solution u with \lu* E £2(8Z+). Then the first paragraph of the proof of Lemma 3.2 applies and, in particular, u extends to zr \ D evenly and harmonically across N+ . The Dirichlet data that n takes a.e.
on D+
is a continuous function, as is the Dirichlet data that u t akes (continuously) u takes a. e. on 8Z+ will be shown to be a continuous function if it can be shown to be continuous across the boundary 8N� of the surface N� . Lemma 3.2, scaled to apply to the split cylinders here, shows that the Dirichlet data is continuous across the Lipschitz crease part of 8N.f- . The same argument used there works on the other parts: Suppose dist(X, azr) = d for X E N'; . Let be a disc approximately a distance d from X + de3. Then Ll d c azr n on
N� . The Dirichlet data
D+
ln(X)
-f gds l � I f �d
Bd( X )
u(Y) - u (Y + de3)dYI + I
f
Bd (X+ de3)
u(Y)
-f
gds l
� C17(d)
t:.. d
and the continuity across 8N'; follow� from the continuity of g and 7J(d) ----? 0. Thus t h e data u takes a . e . on 8Z� i s a continuous function. Since also u• E L2(8Z+) it follows t hat u E C(Z.f- ) . The evenly extended ·u is then continuous on zr , harmonic in zr \ D with the same Diriclllet data as g on 8( Z" \ D) . The maximum principle implies u = g.
420
lv!OISES VENOUZIOU AND GREGORY C. VERCHOTA
Let 9r denote the Green function for zr \ D with pole at a point {P} of N+ . Again 9r is cont inuous in zr \ {P}. Let B c B c zr \ D be a ball centered at P. Then by the maximum principle cg 2: 9r on zr \ B for some constant c. By this domination, the vanishing of both g and 9r on D+ n { t = 0} and ( ii ) of Lemma 3.1 applied to Yn it follows that o.,g which is not in L2 (D) can neither be square integrable over the smaller set D+ n { t = 0}. Since u = g this contradicts the 0 a..o:;sumpt.ion on the nontangent ial maximal function of the gradient . The nonsolvability of the L2-mixed problem in the split cylinders can be extended to nonsolvability in any polyhedron that has a Lipschitz graph crease on any face by a globalization argument. Let g and r be as in the Proposition. By using the approximating domains zr n Ga as 6 -+ 0, the Green s representat ion '
g(X)
=
{
lazr
o.,Fx g - Fx o.,gds -
{
_ Fx dft0, X E zr \ D
JDnzr
can be j �stified where J-L0 is harmonic measure for n = Z \ D at the origin and F is the fundamental solution for Laplace's equation. Let x E C0 (IR3) be a cut-off function that is supported in a ball contained in zr centered at Po, and is identically 1 in a concentric ball B r with smaller radius. Then define
u(X)
= -
f _ FxxdJ.lP JDnzr
harmonic in R3 outside supp(x) n D. Similarly g(X) - u(X) is harmonic inside Br. Consequently V'u* � L 2 (Br n D ) by applying a scaled (ii) of Lemma 3. 1 to g again. Also (3.3)
u(X) = -
{ _ Fx (Q) (x(Q) - x(X)) dp,0(Q)
lvnz··
- x(X) {
lazr
ovFx9 - pX Ovgds + x(X)g(X)
The last term has bounded Neumann data on N and vanishing Dirichlet data on D. The Cauchy data of the middle term is smooth and compactly supported on D U N . For any X � D the gr-adient of the first term is bounded by a constant , depending on X, times
l lnzr
I
�
Fx (Q ) dJ-to ( Q ) :::; - Fx (o) + g x (o) :::; 47r X I (negative ) Green function for D = Z \ D with pole
Here gx is the at X. Thus the first term is Lipschitz continuous on D+ U N+.. Altogether u has bounded Neumann data on N and Lipschitz continuous data on D while \i'u* � L 2 (D) . Finally V'u E Ltoc(lR3 ) by (3.3) since this is true for X9· Thus whenever a split cylinder Z+ can be contained in a polyhedral domain so that az� n {t = 0} is contained in a face and so that the Lipschitz crease is part of the boundary between the Dirichlet and Neumann parts of the polyhedral boundary, then the harmonic function u just constructed is defined in the entire polyhedra domain. Its properties suffice to compare it with any solution w in the class \i'w* E L2 by Green's first identity J I V'u - V'wl 2 dX = J(u - w) o,_ (u - w)ds. Regardless of the nature of the partition away from Z+. , when w has the same data as does u it must be concluded, as in Proposition 3.3, that it is identical to u . This establishes
THE MIXED PROBLEM FOR HARMONIC FUNCTIONS IN POLYHEDRA OF JR3
421
Theorem 3.4. Let n
c JR3 be a compact polyhedral domain with partition an = D u N. Let :F be an open face of an such that :F n D is a Lipschitz domain of :F with nonempty complement F n N. Then there exist mixed data (1.2) for which there are no solutions u in the class V'u* E L 2 (an ) .
3.2 . Whenever a face of N and a face of D share a !-dimensional edge boundary, the dihedral angle measured in n between the two faces is less than Continue to denote points of JR3 by X = (x, y, t ) . Define D to be
as
the upper half-plane of the xy-plane. Introduce polar coordinates y r cos 0 and t = r sin 0, let 1r � a < 21r and define N to be the half-plane e = a. The crease is now the x-axis. 1r.
=
Define
b(X) = r 2"c. sin( � e) 20' for X above D U N. These are Brown's counterexample solutions for nonconvex plane sectors (Bro94] . The Dirichlet data vanishes on D while the Neumann vanishes on N, and V'b* tfi L2. These solutiollB are globalized to a compact polyhedral domain with interior dihedral angle a : Denote by e the intersection of a (large) ball centered at the origin and the domain above D U N. Then b(X) is represented in e by b(X) =
{ OvFxbds - { Fxav bds lae\D lae\N
Let x E Ctf (IR3) be a cut-off function as before , but centered at the origin on the crease. Define u(X) = 8v Fx xbds Fx xovbds
i
fv
As before, u is harmonic everywhere outside supp(x) and V'u * rf. L2 (supp(x) n (D u N)). Also
(3.4) u(X) = {
JNne av F _
x (Q)
- lne
n
(D U N)
(x(Q) - x (X ) ) b (Q) ds (Q)
pX ( Q)
(x(Q) - x(X)) av b ( Q) ds ( Q)
- x(X) {
lae\ N\D avFxb - F
x 8v bds
+ x(X)b(X)
Again the boundary values around the support of x are the issue. The last two terms are described just as the middle and last after (3.3). The gradient of the second term is bounded because the integral over D can be no worse than, for example, J; dx J� v'xl+r2 � < oo for any /3 < 1 (e.g . /3 1 - : ) . 2 For a 8�. derivative define tangential derivatives (in Q) to any surface with unit J normal v by a; = ViOj - VjOi · Then by the harmonicity of F away from X and the derivative of the first integral equals the sum in divergence theorem in e, the .7 i of f ai Fxa; ( (x - x(X ) )b) ds =
at
}Nne
_
plus integrals over ae \ N \ D ( b vanishes on D) that will all be bounded since X is near the support of X · When the tangential derivative falls on b the integral is
42 2
J>,1QISES VENOUZIOU AND GREGORY C. VERCHOTA
bounded like the second term of (3.4) . The remaining integral has boundary values in every LP for p < oo by singular integral theory. (In fact, it too is bounded by a closer analysis, thus making it consistent with the example from Section 3.1.) Finally V'u E Lfoc(JH:3 ) by its now established properties and the corresponding property for b. The argument using Green's first identity as at the end of Section 3.1 is justified and
The solutions u can now be placed in polyhedral domains that have interior dihedral angles greater than oT equal to 1r and provide rnixed data for which no L2 -solution can exist.
4. POLYHEDRAL DOMAINS THAT ADMIT
0:-JLY TIIE TRIVIAL MIXED PROBLEM
Consider the L2-mixed problem for the unbounded domain exterior to a compact polyhedron. \Vhen the polyhedron is convex the requirement of postulate (iii) of (1.1) eliminates all but the trivial partition from the class of well posed mixed problems. In this C&':ie we will say that the exterior problem is rnonochrornatic. The mixed problem for a compact polyhedral domain can also be monochro matic for the interior problem. An example is provided by the regular compound polyhedron that is the union of 5 equal regular tetrahedra with a common center, a picture of which may be found as Number 6 on Plate III between pp.48-4U of H. S. M. Coxeter's book (Cox63] . An elementary arrangement of plane surfaces that elucidates the local element of this phenomenon is found upon considering the do main of ffi'.3 that is the union of the upper half-space together with all points (x, y, t) with (x, y) in the first quadrant of the plane, i.e. the union of a half-space and an infinite wedge. The boundary consists of 3 faces: the 4th quadrants of both the xt and yt-planes and the piece of the xy-plane outside of the 1st quadrant of the xy-plane. The requirement of postulate (iii) is met only by the negative t-axis. But no color change is possible there because any color on either of the 4th quadrants must be continued across the positive x or y-axis to the 3rd faee of the boundary. On the other hand, a color change is possible for the complementary domain and i::; possible for the exterior domain to the compound of 5 tetrahedra. Is there a polyhedral surface with a finite number of facp.-; for which both interior and exterior mixed problems are monochromatic? (Agm65] Shmuel A gmon , Lectures
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(Car62]
THE MIXED PROIJLEM FOR. HARMONIC FUNCTIONS IN POLYHEDRA OF IR3
[Fol95] [Gla70] [GT83]
[HW68] [JK81J (JK82] [Lan72]
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[MR03]
[MR04]
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[MR06]
[MR07]
423
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[RS72j
[St e70) (Ver84]
(Ver01 j (VV03) [VV06]
___
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