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0. For any e > 0 there is a constant > 0 such that
/
If we choose 0 < e < a and p> r + e, then integration of (1.32) results in i,,.
(2.39)
1. Decompositions of functions
16
introduced in Definition 2.6(i), is a quasi-Banach space. Fur-
Then
therowre
= {f E
:
2 bkm
Qk,
(2.53)
mEZ"
with bkm
=
j
= g2-.kn
(2.54)
I. Decompositions of functions
18
keN0, mEZlz}
(2.55)
with Akm =
(2.56)
We may assume Ill
(2.57)
VA
(equivalent quasi-norms), where the right-hand side is given by (2.8), (2.6) with A in place of A. We comment on (2.57) in Remark 2.10 below, where we also give references. We take (2.57) for granted. Let x e S(RTh),
=
=1
if
C SUJYp tpk, .SUPP
C Qk,
(2.58)
where k E N0. We multiply (2.53) with Xk and extend it by zero from Qk to Then we have = (2.59)
x — m) m€Z"
=
Akm
C
xv (2kx — m),
x
W',
mEZ"
where we used (2.54) and (2.56). The entire analytic function x"(x)
E
can be extended from R" to C's. By the Paley-Wiener-Schwartz theorem we have for some c > 0, any b> 0 and an appropriate number Cb,
lx"(x+iy)I
(2.60)
with (s) = (1 + see e. g., [Trio], 1.2.1, p. 13. where x C and y E Iterative application of Cauchy's representation theorem in the complex plane yields
xv(z1. =
.
.
J
J
.
(2.61)
. .
where zk E C. By (2.60) we obtain
(2.62)
2. Spaces on R": the regular case where is independent of r e with f3 E and E Let and be the functions introduced in Definition 2.4. We expand the function in (2.59) at the point where I N is fixed. (It is our intention to prove the converse of (2.50) for these numbers which is sufficient.) We have — 1)
m)
—
=
— m)(x —
xV\12—Q1 — m1
=
— 1)
(2.63)
— 1)
By (2.15) we have
1=
—
—
1) =
(2.64)
—
lo 1EV'
1EV'
is taken over all lattice points = where j = 1,..., n. As indicated in (2.64) we concentrate on the term with lo = 0, where +••• stands for the remaining terms which can be treated in the same way. Then we obtain by
where the finite sum with respect to
Z" with 0 < Ioj
0. This follows in a somewhat 2.4.1, in particimplicit way from the technique of maximal functions in ular Corollary 2 on pp. 108/109, and may be found more explicitly in [FrJ9O], [FJW91], Ch. 7, and [Dm95], [FarOO] (where the two latter papers deal also with the anisotropic case). Some basic ideas of the above proof for the simwith (2.39) may be found in [Trio], 14.15, pler case of the spaces pp. 101—104. For the spaces with (2.41) there are additional technical complications which forced us to break down the resolution of unity in (2.64) . into n partial sums and which resulted m the factor 2 n in (2.72) and m the additional shifting factor in (2.71) withareferenceto (2.103) in 2.15 below. A full proof of the anisotropic generalization of Theorem 2.9 may be found in [FarOO]. To overcome the indicated technical problems connected with the F-case, W. Farkas used more directly the technique of maximal functions .
than we did here (but it is behind all the technical assertions mentioned in 2.15 below). 2.11
Unconditional convergence, estimates of constants, and optimal coefficients
with with (2.39) and the spaces By Theorem 2.9 the spaces (2.41) coincide with those spaces usually denoted in this way. The main point
of the approach described here is the representation (2.24) with (2.23) and (2.27), respectively. By the discussion in 2.7 the series in (2.24) converges for some r, 1 ( r co. This absolutely and unconditionally at least in justifies writing (2.31),
/=
x
R",
(2.73)
fl,v,m
with the abbreviation (2.74) fl,v,m
Pr0
which will be used in the sequel. Here (flqu)vm(x) are the (s,p)-fl-quarks, which, of course, depend on 8 and p. At least in the above context, s and p are considered to be fixed. This may justify the omission of s and p in the notation of 8-quarks. Looking at the interplay between individual elements I E and the above function spaces, one may adopt two points of view, which are the two sides of the same coin:
fl (i)
I. Decompositions of functions
Characterize all f E S'(R") which belong to a given space, say,
or
and Fq(R"), to which a given (ii) Characterize all spaces, say, belongs. distribution f E Our preference, so far, is the point of view (i). On the other hand, Step 2 of the proof of Theorem 2.9 is essentially in the spirit of the point of view (ii). To make clear what is meant we introduce the notion of optimal coefficients. Let / E S'(RTh) be given. Then one asks for a constructive procedure
uEN0, mEZ'1,
(2.75)
which allows us to decide whether f can be represented by (2.24) with (2.23) or (2.27). Optimality means that in addition there is a number c > 0 such that for all f in question, IIA(f)
C
(2.76)
flA(f)
cli!
(2.77)
or
Here we used the notation introduced in Definition 2.6, now indicating the dependence of the coefficients on f. Since by definition, the converse of (2.76) and (2.77) is obvious, these inequalities can be re-written as equivalences, IIA(f)
il/
(2.78)
Hf
(2.79)
and 1IA(f)
where the related equivalence constants are independent of f. Furthermore, we used in writing (2.76)—(2.79) the fact that all the quasi-norms in Definition 2.6 and Theorem 2.9 are equivalent to each other and that there is no need to distinguish between them in the sequel. The dependence of the optimal coefficients constructed in Step 2 of the proof of Theorem 2.9 on given a, p, reduces, as we shall see, to the above normalizing factors and an additional mild influence of For our later purposes it is useful to fix some of these dependencies of the above constants on e. We assume that and hence r with (2.14), (2.15), and from 2.8 are fixed. In particular we and with respect to such a fixed take the quasi-norms in A(f) be the optimal coefficients and their sequences Let again constructed in Step 2 of the proof of Theorem 2.9. We claim that ii!
iIA(f)
s c2
2? Il/
(2.80)
2. Spaces on R": the regular case
23
where c1 and c2 are independent of with Q > r + 1. Recall = min(1,p,q). The left-hand side follows from (2.50) (here one needs o r + 1), whereas the right-hand side is covered by (2.72). Finally we need later on the interplay of Let I be given by (2.43) with p, say, with p> r+E for some e > 0, and E are the above optimal coefficients. (2.44) where we assume that = Then we have by (2.47) and (2.80),
0 is sufficiently large and c' > 0 is independent of Then (2.78) and (2.79) can be strengthened by Cj
Ill
Cj
If
I)'(f)
Ill
(2.89)
and IfpqIIp
(2.90)
respectively, where c1 > 0 and c2 > 0 are independent of Summarizing the main observations in Section 2, especially Theorem 2.9 and Corollary 2.12, we arrived at the goal outlined in 1.5: The constructive Weierstrassian approach (2.73), (fiqu)vm(x),
I=
X E ]R",
(2.91)
(3,i',m
to the function spaces
(W') and
(RTh) with the optimal Cauchy coeffi-
cients (2.82),
= where the
(f, are given by (2.86).
(2.92)
2. Spaces on R": the regular case 2.14
25
Generalized quarks
We return to Step 2 of the proof of Theorem 2.9. In the reformulation (2.65) of (2.59) we used (2.15) (resolution of unity) and no other specific properties. Then we obtained the /3-quarks (/3qu)k+p,j in (2.66). The estimates of the coefficients which followed afterwards and which resulted finally in (2.72) have not very much to do with these constructions. In other words, without substantial changes we can modify this part of the proof of Theorem 2.9 as follows. Let k E N0 and let :
m
(2.93)
C
be a sequence of approxAmate lattices such that there are two positive numbers c1 and C2 with —
Xk.m21
Cl
2—k
keN0,
m2,
m1
(2.94)
and
=
U
C2
2k),
k e N0
(2 95)
s. We need the following additional assumption for the above functions i,bk,m : there is a constant c3 > 0 such that
E ir, k E N0,
E N0 with L max(—1, [a,, — s]) be fixed. Let s) and be (s,p)'-'-13-quarks with respect to a be (a,p)-/3-quarks and Let p> r, where r has the same meaning as in (3.3) and (3.9). fixed flAnction which can be represented as Then ts the collection of all f
f
(3.12)
($qtL)pm + f3,v,m
with II?lIbpqII0+
r + e and
3.8
E
Generalizations
By (2.101) or (2.102) one can replace fpq in connection with Definition 3.4 and Theorem 3.6 by with c> 0. Furthermore, the arguments in Step 2 of the proof of Theorem 3.6 are based on liftings which reduce the problem to spaces covered by Section 2. Then it is quite clear that also in this context the /3-quarks (13qu)k,,(x) can be replaced by the generalized quarks in (2.99). The arguments in Step 1 of the proof of Theorem 3.6 apply without any changes to this more general situation. Then one gets an obvious analogue of the end of 2.14. 3.9
Discussion
We introduced the general spaces and in Definition 3.4 and Theorem 3.6 via the representations (3.12) with the respective conditions for the coefficients. It is well known that moment conditions of type (3.7) cannot
I. Decompositions of functions
34
be avoided for general values of a. The simplest way to incorporate them is to apply powers of the Laplacian as has been done in (3.6). Hence, in general, the A-terms in (3J2) are indispensable. What about the 'q-terms in (3.12) ? Of course they cannot be totally omitted. Otherwise one would have moment conditions of type (3.7) for the whole function f. This is not the case. But it comes out that one needs only the ij-terms with ii = o. This will l)e discussed in a somewhat different context in Section 8. We refer in particular to Theorem 8.7. From the point of view of the above spaces we deal there only with special spaces. But the arguments given there can be extended to all of the above spaces. From an aesthetical point of view, the outcome, for example (8.43). looks better than (3.12). But (3.12) is more stable and hence in many applications more useful: If one multiplies f in (3.12) with functions in connection with poiutwise multiplier problems, or if one wishes to use such representations a.s a starting point to study function spaces in domains, on manifolds, or on fractals, then the full in (3.12) is very helpful. This is also the case in connection with the question to what extent the support of f is reflected in these representations. We refer to [Tri6], Corollary 14.11. p. 99. For this reason we prefer here the above version. But we refer to Section 8 for (so we hope) elegant complements.
4 An application: the Fubini property 4.1
The Fubini property
Let I < p < oo and a = k E N0 . Then spaces mentioned in (1.4). Let n 2, and
Let I E
are the classical Sobolev
x3=(xi,...,x3_1.x,+1
and s..,
= f(x),
x E
R for any fixed x3 E has the so-called Fubini property
If
W(R)(j
it is well known that
.
(4.2)
where (in the usual measure-theoretical a.e. interpretation) the inner norm is with respect to x3. This is and the taken with respect to essentially a Fourier multiplier assertion, [Tri/3], 2.5.13, p. 114, and Theorem
4. An application: the Fubini property
2.5.6, p. 88. Of course, if k = 0, then (4.2) is essentially the classical Fubini theorem, from where the name comes. The extension of this assertion to the Sobolev spaces with 1 < p < 00, s > 0, according to (1.8), (1.9), and to the special Besov spaces with 1 p 00, 8 > 0 is due to Stricliartz, [Str67j and [Str68), respectively. An extension of this assertion to the special Besov spaces B,,(R'1) with
O<poo, tilay be found in
(4.3)
2.5.13, p. 115. The Fubini property for the spaces
with
l 0, and let Xvm' (x') be the corresponding characteristic functions. If c is suitably chosen then we may assume — m') Xvm'(X'). This applies also to the x1-variable. By the onedimensional version of (2.28) with (2.8), modified by (2.101), we have
I
00
Step 3 Since Fubini property. In this case the above arguments can be extended top = 00.
with 0 < p < 00, s > a,, is known,
But the Fubini property for 2.5.13, p. 115.
Step 4 It remains to disprove that the spaces have the Fubini property. Let
=
with (4.6) and p
q
=
. . .
with a compact support near the origin be a non-trivial C°°-function in and with the indicated product structure. Let
f(x) =
—
mi))
(4.21)
,
= (j,... ,j). We have by Definition 2.6(i),
where a3 E C and
c(
Ill
/00
\i=i
(4.22)
)
/
(usual modification if q = oo) for some c > 0 which is independent of a3. Although it is almost clear that the quarkonial decomposition (4.21) of I is optimal and that (4.22) is an equivalence, we give a detailed proof. Let again be the usual differences according to (1.12). Let M > s. Let
where jEN0.
:
Then I
/00
If
Ill
+
sup
)
(4.23)
/
\i=i
is an equivalent quasi-norm. This is the discretized version of Theorem 2.5.12 on p. 110 in [Trif3]. Let h E K,. We may assume that the support of is near the origin. Then we have by the supports of the functions involved 00
(. —
ILP(Rn)V
(4.24)
I. Decompositions of functions
for some c> 0 (again obviously modified if p = oo). Inserting this estimate in (4.23) we obtain the converse of (4.22). Hence, (4.25)
If
where the equivalent constants are independent of a3 (they may depend on has the Fubini property, hence by Definition
Now we assume that 4.2,
hf
=
By the assumed product structure
fX'(x) = f(x) =
>aj
(4.26)
.
II
we can rewrite (4.21)
(2(x' — m'3))
—
j)) (4.27)
The counterpart of (4.25) is given by 00
- m'i)))
IB;q(R)hI
(4.28)
where we assume that for fixed x' at most one term on the right-hand side of (4.28) is different from zero. But then it is clear that application of with respect to x' results in I hIfx'(xi)
.
(4.29)
I
Of course this applies to any of the terms on the right-hand side of (4.26). Together with (4.25) we get
too
I
\q A
(4.30)
\i=i
/
(usual modification if p = co and/or q = oc) where the equivalent constants are independent of a3. This proves p = q.
5. Spaces on domains: localization and Hardy inequalities 5 5.1
41
Spaces on domains: localization and Hardy inequalities Introduction to Sections 5 and 6
Let U 1w a domain in Then the spaces and can be intro(bleed by restriction of and on respectively. This applies in particular to the special cases considered in 1.2: Sobolev spaces (classical and fractional). Hölder-Zygmnund spaces, classical Besov spaces etc. These concrete spaces. hilt also the general scales have been studied in and great (letail. We refer to the books listed in 1.1. It is the aim of this section
and the following one to contribute to this theory in the spirit of Sections 2 and 3. where we discussed corresponding spaces on R'1. \iVe describe our intentions by looking first at a simple case, where all assertions which are of interest for us are known. Let be a bounded C°° domain in R". Let I <J) < and a E N0. Then =
{f
there is agE = inf
Il/I
with
f}
,
(5.1)
(5.2)
I
such that its restriction
where the infiininii is taken over all g E
to U coincides in D'(fl) with 1. Furthermore, D(U) =
=
is the completion of
and
in
=
{f E
1111
considered as subspaces of
: suppf c = Hf w;(R")II,
,
(5.3)
(5.4)
It is well known that ,
Ill
(>
ILP(fl)IIP)
(55)
= W(1l)
(5.6)
faIs
norms) and
iiorins). Let
d(x) = dist(x, Ofl),
x E R",
(5.7)
I. Decompositions of functions
42
be the distance of a point x e to the boundary well-known Hardy there is a c> 0 such that
j
of ft We have the
f EW(Ifl.
I
(5.8)
As a consequence one obtains easily the following localization assertion: Let K3 be balls of radius rj > 0 in R'3 with
UK3=CL dist(K,,ôfl)—.r,,
(5.9)
such that at most N of them have a non-empty intersection, where N E N is a suitable number. Assume that there is a subordinated resolution of unity with
E
sUppçokCKk ifkEN,
(5.10)
and (5.11)
for some
> 0. Then
I
1111
(5.12)
One can introduce corresponding spaces, say, and their B-counterparts. it is the aim of this section to study their interrelations, where we are especially interested in counterparts of (5.12). But there is a significant difference between the F-spaces and the B-spaces. Let be the same function as in Definition 2.4 with the resolution of unity (2.15).
Put
=
where in E
Let 0
restriction (5.58). We describe an example of (5.79) which is not covered by (5.78). First we recall: The chanzctenstw function of the bounded C°° domain is a pointwise multiplier in if
11
O<poo,
1
\
1
(5.102)
ifp=oo). We refer to [RuS96J, 4.6.3, p. 208, for the final version, which in turn is based on [Ka185J, [Fra86J, and the forerunners mentioned there. As a consequence we have in these cases
= F;q(cz).
(5.103)
Combining this observation with Proposition 5.7 we obtain the following assertion: Let
O 0 is suitably chosen. Afterwards one can find a resolution of unity (5.117) with (5.115), (5.116). 5.14
Theorem
Let il be a bounded C°° domain in :
be
and let
j EN0; r= 1,...,N,}
the above resolution of unity. Let
O<poo,
(5.119)
on domains: localization and Hardy inequalities
5.
61
(with q = DC if p = DC). Let be the spaces introduced in Definition arid notationally complemented by (5.18). Then f E L1(1l) belongs to if. (2fl(l only
I
s we have I
/ pa2'
If
—
I
A
+ c (M 1=0
(x),
(5.137)
0
where x E bK,,.. Recall N—K> 8 and that the left-hand side of (5.137) is zero if x bK,,.. Hence, the sum over the pth power of (5.137) is the mtegrand on the left-hand side of (5.130). By the equivalent quasi-norm (4.15) the left-hand side of (5.130) can be estimated from above by
Ill
+ (M Iv;8f(v)Ii (.) Since p> u we can apply the Hardy-Littlewood maximal inequality. Hence the second term in (5.138) can be estimated from above by .
(5.138)
5. Spaces on domains: localization and Hardy inequalities
65
Together with (5.79) we obtain (5.130). We prove (5.131). From (5.132) and the above maximal function follows I
(I'a2i
t
0 such that as
c
(5.141)
(Jo'
for all
suppg C B1 = {y E
gE
< 1).
:
(5.142)
This follows by standard arguments from the compact embedding of, say, where B2 has the same meaning as in (5.126), in all spaces L,-(B2) with 0 < r p* for some 1 0, then it would follow by the usual compactness argument that there is a non-trivial function g E with compact support in B1 and = 0 for ahnost all x E and hi 1. But this is a contradiction. Now let f be given by (5.126). We apply (5.141) to g(x) = f(Ax). For the ball means we have
(x) =
dh)
We insert (5.143) in (5.141) and obtain (5.127).
=
(Ax).
(5.143)
I. Decompositions of functions
66
5.15
Corollary
be a bounded C°° domain in and let d(x) be the distance of x E as in (5.15). Let p, q, 8 be given by (5.119) (with q = 00 if p = oo). be the ball means introduced in (4.14) with u < min(1,p,q) and s < N E N. Then f L1(cl) belongs to if, and only if, Let to Let
>4)jr(X)f
(6.30)
j=0 r=1
(absolute and unconditional convergence in
and
(6.31)
/
\3=Or=1
I
(equivalent quasi-norms). We start, say, with and expand this function according to (6.28) with f in place off, ignoring possible translat ions. This results in some sequences of (irregular) lattice points near where, say, k = 0,1,2,. . ,with the required properties, in particular (6.4). Then we do the same with, say, f. We get new lattice points
opt iniall
.
1. Decompositions of functions
76
They might interfere with }. But the new lattice points can be chosen in such a way that are again approximate lattices according }u Now to 6.2, especially with the counterpart of(6.4), near
one can continue this procedure step by step. If K E N0 is fixed, then only those lattices with j 0, and > x2. Proof Step 1 By iterative application of the lifts in Theorem 7.15 we may assume, without restriction of generality, 8i > 0p,q,
and
>
(7.63)
Then we can apply Theorem 7.10. Let 6 0 and 2 x2. Now the continuous embedding (7.62) follows from (7.38) and (7.39), (7.61) with 5 2 0, together with the monotonicity of the 4-spaces. Next we prove that the C be the resolution embedding (7.62) is compact if S > 0 and xj > x2. Let
7. Spaces on manifolds
95
of unity described in 7.7 and let .1
A!,
JEN.
(7.64)
j=O
a cut-off function. By the pointwise multiplier property 5.17, as used in Step 2 of the proof of Theorem 7.10, it follows that be
(M,
!Ico.jf
(M.
)II C
)II
(7.65)
and 11(1 —
< X2)
0, and the compactness of the embedding (7.61) it follows that the map
where C,
f i—p çajf
(7.67)
.—÷
:
is compact. Together with > and (7.66) one gets the compactness of the embedding (7.62). Step 2 We prove the converse assertion. Let B be an (open) ball with C If we have the continuous (or conipact) embedding (7.62), denoted with id, then it follows by
= reoido ext
id (F;1kq1(B)
(7.68)
that also the embedding (7.69)
C
is continuous (or compact). Here ext is a (linear and bounded) extension Operand re is the restriction operator from (B) into (M, ator from Then S 0 in the continuous case and 5 > 0 in onto x2 if the embedding the compact case follow from 7.16. Next we prove (7.62) is continuous and > x2 if this embedding is compact. Let
jEN0,
(7.70)
1. Decompositions of functions
where Q, is given by (7.21), and c > 0 is small. Let function with compact support near the origin. Let
be a non-trivial C00
a,EC.
(7.71)
.1=0
We
may assume that the balls B' and the function
are chosen such that we
can apply (7.38) as follows, (Al, gXI )II hf
/00 f
\j=0 /00
".'
(
aiim
)
(7.72)
,
/
= 2—2 in the same way as in connection with (7.39). We have the same equivalence with the index 2 in place of 1. Hence, if the embedding (7.62) is continuous then where in the latter equivalence we used (5.147) with
/00
\P2 )
/
/00 0. We rewrite (7.88) as
= 2—xk
(7.91)
—
and put temporarily
= These (s, p. Fp8q(M,
(23y
—
tJi3:(y),
(7.92)
/3 E
— /3-quarks are normalized building blocks for the spaces from Theorem 7.10: Under the restriction (7.37) we obtain by
(7.38)
++
I(/3qu),z
(7.93)
for some m = m(j, 1), where ++ indicates a few other terms neighbouring k
and rn. By (7.39) and (7.90), (7.31), (7.80) it follows that
c($),
(7.94)
independently of j and 1. As for the dependence of c(/3) on and (7.84) in the same situation as in 6.4. We obtain
we are by (7.78)
E
(7.95)
for some c> 0 which is independent of i, I and /3. Recall
that L1(ftgT) =
{f
normed in the obvious way. Furthermore, 7.22
0. Hence,
(8.18)
for fixedpwith S'(Rtm)=
(8.19) 8 r where r has the same meaning as with respect to a fixed function in (8.20). Then is the collection of all I which can be represented as
I=
(8.38)
with IA
IbpIIQ 0. Then, by Proposition 8.5, any
/E
= C3(IRTh, (x)s)
(8.62)
I. Decompositions of functions
112
can be represented by
f(x) =
—
m)
(8.63)
i=0
with, for some c> 0,
P E N0, m e Zn.
+ mi)3
(8.64)
Again p> r and c depends on p. This coincides with (1.23) and is the best that can be expected. If the weight (x)3 is replaced by (Z)a with a E R, then one has to correct only the coefficients in (8.63), (8.64). If f E then moment conditions cannot be avoided and again (8.59), (8.60) seems to be the simplest possible case.
8.11 A remark on optimal coefficients In 8.6 we added a remark on optimal coefficients in connection with the spaces covered by Proposition 8.5. The proof of Theorem 8.7 and, hence, of Corollary 8.9, is reduced by lifting to Proposition 8.5. Hence, there are optimal coefficients in (8.43) and (8.59) which depend linearly on f. However this applies only to those / for which we have the a priori information (8.45) for some s, which influences the calculations, and not uniformly to all tempered distributions. 8.12
Tempered distributions in domslinR
be a bounded C°° domain in As above, D(1l) is the collection of all complex-valued C°° functions in with compact support in Il. Its dual D'(Il) is the collection of all complex-valued distributions in Let S(Q) be the collection of all E C°°(fl) such that for all multi-indices 'y E Let
= 0 if y E
(8.65)
With obvious interpretation S(1l) can be identified with the subspace E
:
C
}
(8.66)
k E N0.
(8.67)
of .9(R?z), furnished with the norms
=
hik
sup xEfl
8. Taylor expansions of distributions
One gets a complete locally convex space. Its strong topological dual is denoted
With the usual
by S'(f) and called the space of tempered distributions in interpretation we have the dense embeddings
c
c S'(Q) c 1Y(cl).
(8.68)
The situation is similar to that in 8.2. In particular we are interested in the counterparts of (8.13), (8.15) and (8.19). Let 1 p oo and s E R. As in (8.5) we put for brevity
B(R') = We denote the completion of
(8.69)
We have
by
in
= B(W') if 1
O
one can interpret
in the context of the dual pairings
and
D'(Il)) and (5(d), S'(d)).
(8.77)
If
s>0,
and
(8.78)
= B7(dI)
(8.79)
then
2.10.5,4.8.1, as far as 1
!+k_1.
(8.87)
maps
{IEB;(cfl
(8.88)
isomorphically onto If 1
onto
(8.90)
Then we have the isomorphic map from
8. Taylor expansions of distributions
Step 2 We prove part (1). Let p and By (8.80) we may assume
with (8.84) be fixed. Let I E
with
fE
117
(8.91)
s = a — 2k be
the inverse of the
(Ilqn)vm(x)
(8.92)
for some k E N. whore a is given by (8.89). Let mapping (8.90). Theii
=
E
can be expanded according to Theorem 6.6 by oc
v=Onz=1
with the (s + 2k, p) — fl-quarks
=
2
(x —
Wi,pi(X)
(8.93)
and (8.83). where the latter coincides with (6.21). The series in (8.92) converges gives (8.85) with Application of absolutely and unconditionally in the — fl-quarks (8.82) where
L+1=2k=o—s> —s. in particular L[—sl. The series converges unconditionally in S'(!l).
Step S Part (ii) follows from the discussion in 6.8. iii particular in connection with (6.48)—(6.51). Here one needs L 2 [—s}. which is equivalent to L+s+1 >
8.16
Corollary
(Taylor expansions of tempered distributions in domains) be the same family of domain in R't. Let Let be a bounded approximate resolutions of unity according to 6.2 as in Theorem 8.15. For any E S'(Il) there are a natural number K and a positive number c such that
f
=
[(x
(8 94)
v=Oin=1
(unconditional convergence in S'(cl)) with
E C and
kmI C
(8.95)
I. Decompositions of functions
118
Proof
We may choose in Theorem 8.15 and (8.82),
p=oo and L+1=2K,
(8.96)
for some K N. Then we have a = K — in (8.91), assuming that f belongs Then we have (8.85), (8.86) with p = oo. This coincides with to (8.94), where we incorporated some factors from (8.82) in the newly-defined coefficients 8.17
m
Comment on Taylor expansions
The above corollary is the counterpart of Corollary 8.9. With the necessary modifications, the discussions from 8.10 and 8.11 can be extended to the situation considered in Corollary 8.16. In particular, in 1.3 we described in (1.18) the classical Taylor expansion for holomorphic functions adapted to our purpose with the coefficients (1.19), (1.21).
The representation (8.94) with the coefficients (8.95) might be considered as the approprzate extension of Taylor expansions from holomorphic functions to tempered distributions. There is a new phenomenon, expressed by
in (8.94), compared with (1.18).
It reflects the nature of singular distributions in relation to (holotnorphic) functions or regular distributions. Then one needs elementary building blocks satisfying some moment conditions, and A" might be considered as a simple way to incorporate them. 8.18
Generalizations
Theorems 8.7 and 8.15. and also Corollaries 8.9 and 8.16 deal with global Tayand in bounded C°° domains. lor expansions of tempered distributions in The considerations are based on the representations (8.19) and (8.80), respec-
and D'(l) tively. It is well known that the topological structure of is more complicated. In particular there are no representations of type (8.19) and (8.80) and nothing having the same elegance as Corollaries 8.9 and 8.16 can be expected. But there is a more or less obvious weak substitute. Let, for Let be given by (7.21), example. Q be an arbitrary bounded domain in and let {çoj be a resolution of unity, =
1
if x E
(8.97)
8. Taylor expansions of distributions
119
One might think that S0j is given by
XE ci,
(8.98)
according to (7.29)—(7.31). If f E D'(fl) then
=
fcp3
One can apply, for example, Corollary 8.9 or Corollary 8.16, to I çaj. Then one gets via
f=
convergence in
D'(ci),
(8.99)
something like a Taylor expansion for f. The outcome might be of some use in applications, but it is not satisfactory from an aesthetical point of view. 8.19
Final remarks
The first steps in the direction of Taylor expansions of tempered distributions were taken in [Trio], 14.13, p. 101. The above Corollary 8.9 might be in considered as a more detailed version of the remarks made there. Our considerations are based on some special weighted spaces introduced in 8.2. But this can be done on a much larger scale. As discussed briefly in 7.3 one can replace in 8.2 by weights w with (7.17), (7.18). Then one gets spaces of type and
(8.100)
One has always the property that I '—'
w(•)f
(8.101)
is an isomorphic map from these weighted spaces onto the corresponding unweighted spaces. This can be done in the framework of Details may be found in [HaT94a], [HaT94b], and [ET96], Chapter 4. In particular any quarkothai assertion for unweighted spaces can be transferred via (8.101) to weighted spaces of the above type. This applies to the theory developed in Sections 2 and 3, but also to Theorem 8.7 and Corollary 8.9. On the other hand, if w(x) is, for example, of growth C±IXI' with 0 < x < 1, then one needs ultra-distributions. The related theory of the spaces (8.100) may be found in [ST87], 5.1. Again (8.101) gives an isomorphic map onto the related unweighted spaces. Hence, also in this case, the unweighted theory can be transferred to these spaces of
I. Decompositions of functions
120
ultra-distributions. It is somewhat surprising that this observation can even be extended to spaces with exponential weights of type The theory of the related spaces of type (8.100) has been developed in [Scho98a] and [Scho98bl, including the remarkable fact that (8.101) again provides an isomorphic map onto the related unweighted spaces. Hence even in the case of exponential weights one has counterparts of the corresponding assertions from Sections 2 and 3, and of Theorem 8.7 and Corollary 8.9. As for spaces on domains, including Theorem 8.15 and Corollary 8.16, we relied on Sections 5 and 6, and hence on [Tri99a]. We followed [TriOOa]. There one finds also further results. In particular, in this paper we developed first the theory of Taylor expansions in bounded C°° domains. Then we used these results
combined with localization assertions of type (5.13) to prove corresponding results for tempered distributions on We refer to [TriOOaj, Theorem 2.3.4. The outcome is somewhat different compared with Theorem 8.7 and Corollary 8.9.
9
9.1
Traces on sets, related function spaces and their decompositions Introduction
In connection with fractal elliptic operators which will be considered in Chapter III we are particularly interested in d-sets and their perturbations, called
A compact set r in
(d,
is called a d-set (in our notation) or a in with F = suppp such that
(d, 'If)-set if there is a Radon measure
r))
rd,
rd W(r),
r))
or
0
1 then we have also
One could even define the trace operator trr as the dual of idj' under the above circumstances and r> 1. But we prefer the above version, completion of (9.14), since it makes sense also for a wider range of the parameters involved, for example as in (9.7).
We transformed the problem (9.15) into the equivalent problem (9.19). We deal with the latter question in terms of local means. Let
00.
(9.23)
Let
f)(x) =
j
—
y)) 1(y) dy,
u E N0.
(9.24)
9. Traces on sets, related function spaces and their decompositions
125
in the usual interpretation as dual pairings
be the local means of f in Then f E
if, and only if,
belongs to I
0 with
Iltrrf ILr(1')II 0 with C2 hg ILr'@')hi,
9 E Lr'(r).
(9.42)
9. 'flaces on sets, related function spaces and their decompositions
129
But it is not clear whether one can take any advantage of this observation. On the other hand, the intimate relations between mapping properties of Riesz potentials (9.4) and also Bessel potentials and maximal inequalities including measures are well known. We refer to [AdH96], Section 3.6, and [Ver99], Section
4, and the literature mentioned there. 9.6
Corollary
(Necessary conditions)
L.etp, q, r, a and the measure j.t 8
—
be
=—
as in Theorem 9.3. Let
for some d E R.
(9.43)
If trr exists according to (9.28), then there is a number c> 0 such that
yEN0, Proof Let ii E N0 and m E with characteristic function of and
f(i') =
(9.44)
> 0. Let again Xvm
;tr(Qtirn) Xvm('Y),
E r.
be
the
(9.45)
Then, by (9.26),
If ILr'(I')lf = 1
9.fld
fL'm
=
(9.46)
Inserting (9.46) in (9.29) one gets (9.44). 9.7
Remark
If s>
then the space (R") consists of continuous functions. Then one has pointwise traces and (9.44) is obvious. Hence one may assume s in (9.43). If d> n then it follows from (9.44) and measure properties that p = 0, which is always tacitly excluded. Finally, d 0 makes sense, but as said, (9.44) is trivial. Hence, a and 0 < d < n are the natural restrictions in (9.43). 9.8
Corollary
(Sufficient conditions)
Let p, q, s and the measure p be as in Theorem 9.3.
I. Decompositions of functions (i)
Letpr 0 which is independent of v. Now part (iii) is a consequence of (9.29) and (9.54). 9.9
Theorem
(Necessary and sufficient conditions) (i) (D. R. Adams) Let p. q, r, s. and the measure p be as in Theorem 9.?.
Let, in addition, r > p, and 7)
1'
with O 0 with p(2Q,,m)
V E
N0.
in E Z".
(9.56)
= Let p, q, s, and the measure p be as in Theorem 9.3. Let + as the range of the continuous mapping Then the trace space trr
(ii)
L1(r).
trr
1.
(9.57)
according to (9.38) exists if, and only if.
< oc. inEZ"
(9.58)
I. Decompositions of functions
132
Proof Step 1 We prove (i). By Corollary 9.6, condition (9.56) is necessary for the existence of tn- with (9.28). By Corollary 9.8(u) condition (9.56) is sufficient in case of the strict inequality (9.48) in place of (9.55). The limiting case (9.55) is much deeper and essentially covered by the following famous beautiful observation of D. R. Adams in the context of mapping properties of Riesz potentials: Under the above circumstances, 00> T > p> 1, (9.55). and (9.56), there is a number c> 0 with (9.3) for all f E where the Riesz potential is given by (9.4). For proofs we refer to [Mar85], Theorem 2 on p. 52, and [AdH96], Theorem 7.2.2 in combination with Proposition 5.1.2, pp. 193, 131. The original proof goes back to [Ada7lI and [Ada731. One can see easily that the Riesz potentials can be replaced in (9.3) by the Bessel t f. This follows also from a Fourier multiplier assertion potentials (id — in with 1 < p < 00. But (9.3) with the Bessel for + potential in place of the Riesz potential is equivalent to
Iltrrf ILr(F)II 0 are of interest. 9.13
Proposition
be a compact d-set in with 0 < d n and let be a related Radon measure with supp = I' and (9.67) (which is uniquely determined up to
Let r
equivalences). (i)
Let p, q, 8
be
as in Theorem 9.3. Then the trace space
ac-
cording to (9.38) and Theorem 9.9 (ii) exists if, and only if, 8> (ii)
Let
1 0 ifo < r < 1 and = 0. Then r satisfies the ball
Let
I. Decompositions of functions
140
condition according to 9.16 if, and only if, there are two positive numbers c and A such that
for oil uE N0 and xE N0.
(9.85)
Proof Step 1 We assume that r satisfies the ball condition. Let with i/ E N0 be a cube with side-length centred at some point E I'. We subdivide naturally in 2" cubes with side-length where x E N. If x is large then at least one of these sub-cubes has an empty intersection with I'. We apply this argument to each of the remaining 2"'' —
1
=
(9.86)
cubes with side-length 2"". By iteration, r n Q" cubes with side-length
can be covered by Switching to balls and using (9.84) we
have
c1 p.(B('y, 2")) c2 p(B('y, 2_v—ad))
0 (independently of the couples (ii, m) involved). We may assume that all these balls have pairwise disjoint supports (or that there is a number
N N such that at most N of these balls have non-empty intersection). By the discussions in 3.8 and 2.15 one can use in the sequence spaces in (3.17) the characteristic functions of the balls Bvm instead of, say, the characteristic
functions of Q,,m. We refer in particular to (2.102). Restricted to the above couples (ii, m) with (9.98) the related quasi-norms of fpq are independent of q (equivalent quasi-norms). This applies to Ii with the following outcome. Let we obtain 0< qo qi 0,
di8t(x,r)
:
(9.102)
be the E-neighbourhood of r. Let k E N0 and let
: m = 1,.. .
C r and
,
:
m = 1,.. . ,
(9.103)
be approximate lattices and subordinated resolutions of unity with the following properties: There are positive numbers c1, C2, c3 with —
kEN0, m1
c1
m2,
(9.104)
and Mk
r'Ek c
2_k)
C2
,
kEN0,
(9.105)
Here B(x, c) has the same meaning as in (2.96): a ball centred at x E and of radius c> 0. Furthermore, (x) are non-negative C°° functions in with where Ek = C3
c
,
kEN0, in = 1,...,Mk, (9.106) kEN0, m=1,...,Mk, (9.107)
9. Traces on sets, related function spaces and their decompositions
for all a E
and suitable constants
145
and
Mk
kENO, XE rek.
= 1,
(9.108)
We always assume that the approximate lattices and the subordinated resolutions of unity can be extended to such that one gets apand related resolutions of unity according to proximate lattices 2.14 with (2.93)—(2.98), (2.100) (with sufficiently large K in (2.100)). In other in R" and subordiwords, of interest are those approximate lattices according to 2.14 which are adapted near I' nated resolutions of unity in the way described above. Let again p be a finite Radon measure in with compact support r = sup'p p. By Corollary 9.8 and Theorem 9.9 it is quite clear what type of conditions for p might be helpful in connection with traces of, say, on r. By Theorem 9.21 and Remark 9.22 it is also quite clear that it is reasonable to switch now to The latter spaces are not only technically simpler from but they produce also a richer scale of trace spaces on r. 9.25
Definition
Let p be a finite Radon measure in R" with compact sizpport 1' = suppp. Let Bkm
2_k+I)
=B
,
k E N0,
m = 1,.. . ,
(9.109)
and t0.
(9.110)
be the above balls with (9.105). Let
Then 00
£
Mk
V
=
(9.111)
) (with the obvious modification if u and/or v are infinity) and, with
= sup{t 9.26
0.
(9.163)
By Hilbert space arguments (which have nothing to do with the specific situation considered here) it follows that there is an isomorphic map, denoted by ext, from H' onto the orthogonal complement of
{f
trrf = o} =
(9.164)
which results in the Weyl decomposition
=
t2(Rfl)
ext H'(r).
(9.165)
Expanding f E according to Corollary 2.12 optimally and linearly and reducing these quarkonial expansions via id = r.o ext to H'(l') as in the proof of Theorem 9.33, one gets optimal quarkonial decompositions of 9 E H'(r) where the coefficients depend linearly on g (but they are not universal in the above sense). If is not a Hilbert space then the situation
I. Decompositions of functions
158
is less favourable. By the above arguments one finds optimal coefficients in the quarkonial decomposition (9.132) which depend linearly on g if there is s+n—t If into Bpq a linear and bounded extensions operator from N then, by [J0W84], Theorem 3 on p. 155, and EJon96] there is such a linear and bounded extension operator, o < s
ext :
(9.166)
in (9.132) which depend linearly on Hence there are optimal coefficients g (but they are not universal). Other cases may be found in [BriOlJ. Again let F = sup'pjz be a compact d-set according to (9.1) with 9.34(viii) o < d < n. Let 1
0 such r) centred at E I' and of radius r, 0 < r < 1, there that for every ball r) such that the ball inscribed in the convex hull Ern are n + 1 points conv('yi,. .. ,'yfl+i) has a radius not less than cr. 9.34(ix) Again let p be a finite Radon measure in R" with compact support r = suppp. By Theorems 9.9(u) and 9.33 we have satisfactory answers to the questions under which conditions trace spaces exist. Excluding q = oo in is dense in (9.142), it makes sense to ask under what conditions
= {i E
Ba"
:
trrf = o}
,
(9.171)
where all the notation has the same meaning as in Theorem 9.33. Problems of this type have a long history and in the classical case when F is smooth, for one has final answers example 1' = Oil is the boundary of a C°° domain in 4.7.1, p. 330. If F is non-smooth or even which may be found e.g., in the situation is much more complicated. In the context an arbitrary set in of spectral synthesis, where conditions are expressed in terms of capacities, one has deep and definitive answers. We refer to [AdH96], Theorems 9.1.3 and 10.1.1, pp. 234, 281. There one finds also historical comments. This theory goes back to Hedberg, Netrusov and their co-workers. Our aim here is different. We in some spaces return to the above problem about the density of of type (9.171), later on in connection with the Dirichlet problem in fractal domains, see 19.5. 9.34(x) A quasi-metric on a set X is a function X x X '—* [0, oo) with
forall xEX, yEX, Q(x,y) =
0
g(x,y)c(p(x,z)+Q(z,y))
if, and only if,
x = y,
for xEX,yEX,zEX,
(9.172) (9.173) (9.174)
for some c > 0. By [CoW71J a space of homogeneous type (X, p, is a set X equipped with a quasi-metric (generating a topology) and a positive locally finite diffuse measure p satisfying the doubling condition in analogy to (9.84). On spaces of such a type one can develop a substantial analysis, based on Calderón's reproducing formula and Littlewood-Paley techniques. Homogeneous spaces of type and with i5p, q oc, si e, on (X,p,p) were introduced on these bases in rHaS94]. This theory has been gradually extended to homogeneous spaces with p < 1 in [Han94], [Han98], [HaL99], and to inhomogeneous spaces in [HLY99a] and [HLY99bI. This includes atomic characterizations, Ti theorems, and Calderón-Zygmund integral operators. We do
160
I. Decompositions of functions
not go into detail and refer to the quoted papers. If, as always in this section, X = r = suppp is the compact support of a finite Radon measure p in naturally equipped with a metric (and maybe satisfying the doubling condition (9.84)), then we have at least three possibilities to define spaces of type and the quarkomal approach according to Definition 9.29, Theorem 9.33 (maybe complemented by an the Jonsson-Wallin approach briefly mentioned in 9.34(iii), and the just indicated possibility. It would be of interest to study the interplay of these diverse possibilities more closely than has been done so far.
Chapter II Sharp Inequalities 10
Introduction: Outline of methods and results
Let again
be euclidean n-space and let
O 0 such that
IE
Ill
(11.3)
(continuow9 embedding). On the other hand we do not use the word embedding in connection with inequalities of type (10.12). for the full scale of parameters and The spaces
O 0, the left-hand side of (11.56) is finite if, and only if,
Jexp {(AIf(t)I)P'} dt
0.
(11.68)
In this version, (11.56) is due to R. S. Strichartz, [Str72], including a sharpness assertion. Corresponding results for the classical Sobolev spaces
l 0, the continuity envelope function
some
(t) = Then 6c
t)
(R'1)It
:
1},
0 < t <e.
(12.77)
is a positive, continuous, unbounded function on the interval
(0,r1 with
j=J.J+1.... , (where the equivalence constants are independent of j). Furthermore, is equivalent to a monotonically decreasing function, and for any is a number > 0 such that
0 0, we have the non-limiting embedding
'.
[TriI3J,
(12.80)
Finally, for given in, 1 >
w(f,t) o0 and allfEA1,,1"(R") if, andonlyif,uv_ 0 by
=00.
(13.55)
We get a contradiction. This proves (13.7). Similarly one obtains (13.9). 13.3
Inequalities
The above theorem covers all cases of interest (excluding borderline situations
according to (13.2)). It describes in a rather condensed way very sharp inequalities. It seems be reasonable to make clear the outconie. We use Example 2 in 12.4, Definition 12.8 and Proposition 12.10. Let 0 <e < 1. 13.3(i) The B-spaces Let x(t) be a positive monotonically decreasing function on (0,E]. Let 0 < u cc. Let p and q be given by (13.6). Then I.
(I
(13.56)
for some c> 0 and all f if, and only if, is bounded and q u 00 (with the modification (13.59) below if u = cc). in particular. if 1
(13.74)
near the origin. This is now essentially covered by the function f given by (13.38) with, say, b =
j=2,3
(13.75)
Then b E 4, and hence we have on the one hand (13.39) especially for HJ (IR"), and (13.44), especially for On the other hand, if lxi 2_k, then it follows by (13.41),
1(x) .
(13.76)
At the same time it is now clear that the functions in (13.38) improve the earlier developments in [Tri93] and [ET96]. The equivalence (13.39) in Step 4 of the above proof coincides essentially with [EdT99bJ, Theorem 2.1. This paper might be considered as a forerunner of Theorem 13.2, restricted to (IR") and Even worse, we used there (11.57), going back to [Has79] and
[BrW8O], as a starting point and derived the corresponding inequality for this means (13.56) with u = p = q and K = 1, via non-linear interpolation from (11.57). Otherwise the sharpness in [EdT99b] is on the x-level as described in 13.3. All other parts of Theorem 13.2 and its proof are new and published here for the first time. Especially the concept of growth envelopes in the above context came out very recently in collaboration with D. D. Haroske, [HarOl]. Finally we mention the extension of the related results in [Tri93] and in [ET96], Theorem 2.7.1, to spaces with dominating mixed derivatives in [KrS96}, including optimality results via extremal functions.
II. Sharp inequalities
216
13.6
Spaces on domains
Let be a domain in The spaces and have been introduced in Definition 5.3 for all admitted s, p, q. The concept of the growth envelope and
the growth envelope frnction according to Definition 12.8 and the notational agreement (12.60) can be carried over under the same natural restrictions as there to the respective spaces We denote them by (13.77) = In the critical case, considered in this section, Theorem 13.2 can be extended
to spaces on domains: If p, q are given by (13.6), then
=
=
(13.78)
=
(13.79)
and, if p, q are given by (13.8), then
=
To justify these assertions we remark first
6c,nA,q(t) s
0 < t S e,
(13.80)
as a more or less immediate consequence of the definition of spaces on doOn the other hand, the mains by restriction of corresponding spaces on construction of extremal functions in Steps 4 and 5 of the proof of Theorem 13.2 is strictly local. Hence the arguments in Steps 6 and 7 of this proof can be carried over from to Then one obtains (13.78) and (13.79). 13.7
The space bino
We always exclude borderline situations. In our context, described by Theorem 11.2, this means in general (13.2), and with respect to the critical case, (13.3). Furthermore, we excluded in all our considerations so far the spaces If 1 0 and all f E
(R") if, and only if, x is bounded and
q 0 and all f E q 5 u 5 oo (again with the indicated modification if u = oo). In particular, if then 1 0,c1 the modification as in (14.30) if u = oo). As in connection with (14.28) one does not need for the sharpness assertion in (14.30) that is monotone. t) fol14.3(111) Explanations The above inequalities with respect to 2 in 12.4, Definition 12.14 and the modified low from Theorem 14.2, and t) in place of Proposition 12.10 with and I respectively. Or in other words, they simply describe what is meant by a continuity envelope. Furthermore, (14.26), (14.27), and (14.33) are covered by Step 1 of the proof of Theorem 14.2. The only point which is not immediately clear by the above theorem and its proof is the boundedness of in (14.24) and (14.31). By (12.14) this question can be reduced to sup
x(t)IVfI*(t)
(14.34)
O 0 and all f E
"(Rn), where
O 0 finitely many functions
Si with j= 1,...,M(6), such that for any f with Of
if w(f
—
t)
"(f)ii
5.
1,
uniformly for
0 0,
where j=1,...,J.
(15.17)
We insert (15.17) in (15.15). Since (15.16) is an atomic decomposition we get
/J \i=i
/j \q
1
.1
0 and c1 > 0 are independent of J. But this is a contradiction. In in case of the F-spaces we assume v
0 and all f E with the modification sup
O 0 and all I in place of when u = oo). Also the u oo (modified by (15.20) with other assertions for the B-spaces after (15.20) have obvious counterparts, in particular the two end-point cases (15.9) according to (12.26). (Ii) were introduced in 11.6(i). The above theorem and The Lorentz spaces the explanations just given can be reformulated in terms of natural and sharp and with (15.4) into We embeddings of the spaces complement these assertions by looking at corresponding optimal embeddings into Zygmund spaces Lr(logL)a(Ic) according to 11.6(u). By (11.46) the original definition (11.44) can be reformulated in terms of rearrangement. Optimal
means here that for given r in (15.4) and in (11.46), (11.54), one asks for all numbers a for which we have the desired embedding, again formulated in terms of inequalities. Corollary
15.4
Let p, q, s be given by (15.4) and let 0
0 and all f E (iii) Let, in addition, r < 1
q
1
r
q
(15.23)
if, and only if, a 0. oo. Then (15.2.Y) holds if, and only if, a
Osuch that
j
f
dmu(t)
dt
(16.1)
for all
uES(IR)
with
d'u
for j=0,...,m—1.
In the years after, and especially in the last decades, hundreds of papers and dozens of books have appeared dealing with numerous variations of inequalities of this type. The reader may consult [OpK9OI and the references given there. As far as this book is concerned we refer to 5.7—5.12, making clear how different natural inequalities for F-spaces and B-spaces might be. Of special interest in this section is the following consequence of the previous results. Let be a
bounded C°° domain in R". Let
f=O1, be
D(x)=dist(x,I')= infix—yl, XEW', y€
(16.2)
the distance to r and, for e > 0, r'e = {x
E
:
D(x) <e}
(16.3)
be a neighbourhood of I'. Let
O 0 the space (—e, E) is continuously embedded in C(—e, e) (in obvious notation and with a reference to, say, 2.7.1) one has an immediate and rather I obvious counterpart of (16.26) with (R") in place of and with —1'• an arbitrary positive integrable function in place of Then, in this special case, (16.21) with n 2 is obvious. On the other hand, the above arguments depend on the special structure of r in (16.24) and on the possibility to apply the Fubini Theorem 4.4. But this is not the case if I' is a general d-set or an arbitrary fractal. In other words, the problem arises under which geometrical conditions for r the inequality (16.21) is substantial and sharp. Finally one can use (16.26) to complement, our considerations in 5.23 and also of (16.5). We formulate the outcome. 16.7
Corollary
Let
be a bounded C°° domain in (16.3). Let 0
and let F,
< < 1 and let K(t)
and D(x) be given by
a positive monotonically decreasing function on (0, eJ. Let p, q be given by (16.7). Then
[ x(D(x))f(x) logD(x)
JFE
be
0 and all f if, and only if, x is bounded. Proof This follows from (16.27) and standard localization arguments. 18.8
Remark
If p, q, a are given by (5.104), then we have the sharp Hardy inequality (5.105). If now p, q are restricted by (16.7) and s = then
f(s) 1
+ IlogD(x)I
p
dx
C )
This is an itrimediate consequence of (16.30).
.
(16.31)
17. Complements
243
Proposition
16.9
Let p be a finite Radon measure in IR't and let F = suppp be compact. Let p, q be given by (16.7), 0 < e < 1, and
= where
B(x,
J B(x,e)
p(dy)
is a ball centred at x E
f Ip,e(x) If(x)17'dx
E
and of radius e. Then c
for some c > 0 and all f E Proof Let xe be the characteristic function of Ke given by (16.6). Let Then it follows by (16.8) that
I
(16.32)
iogix—yd Xe(x—')
(16.33)
E F.
(16.34)
Integration with respect to p and application of Fubini's theorem results in (16.33). 16.10
Remark
As mentioned at the end of 16.4 the Propositions 16.5 and 16.9 are far from final. This is also clear from the discussion in 16.6 and the more satisfactory assertions in 16.7 and 16.8. We mainly wanted to make clear that there might be a sophisticated interplay between the geometry of irregular fractal sets F and the singularity behaviour of functions belonging to spaces and near r. We restricted ourselves in the course of this discussion to the critical case extending Theorem 16.2. But of course one can deal in the same way with the sub-critical case as considered in Theorem 16.3. 17 1T.1
Complements Green's functions as envelope functions
Looking at (13.7) or (13.9) one may ask whether there are functions I belonging to or F2 (1W') such that f (t) is equivalent to log or log respectively. If q < oc in (13.7), then it follows from (13.57) that this is impossible since in such a case the middle term diverges. Because always p < co,
244
11.
Sharp inequalities
(1W'). Similarly one has by (13.62) a corresponding argument for the spaces for the sub-critical case according to Theorem 15.2 and (15.7). Corresponding questions can also be asked for the super-critical case considered in Theorem 14.2. If q = oo then the situation is different. We deal first with the critical case as covered by Theorem 13.2 and by 13.3. Let S be the usual 5-distribution in 1W' with the origin as the off-point. Then
SE
where
0< p
0223"
.
(19.72)
kEN.
(19.73)
functions in fZ\1', The eigcnfunctions uk(x) are (classical,) harmonic
if
(19.74)
Then
if, and only if,
Uk E
0,
if,andonlyif,
(19.75)
(19.76)
The eigenfunctions u1 (x) have no zeroes in
= cu(x) with
c
C and u(x) > 0 if x E IL
(19.77)
Step 1 The necessary explanations of what is meant by the operator been given in 19.2, 19.3 with a reference for further details to [Trio],
UI. Fractal elliptic operators
266
especially Sections 28 and 30. In particular, the above theorem might be conof Theorem 30.2 in [Trio], p. 234. By this theorem, B sidered as an is a non-negative compact self-adjoint operator in H' (f') with null-space
N(B) = {i E
:
trrf = o}
,
(19.78)
generated by the quadratic form (19.70) and with (19.73). By (19.27) or Fig. 19.1, and Proposition 19,5, (19.78) coincides with (19.69). Hence it remains to prove that the largest eigezivalue m is simple and all assertions concerning the eigenfunctions, including (19.77). Step 2 First we prove
where l<poo, k€N,
(19.79)
with the special case k E N.
tLk E
(19.80)
This is essentially a matter of spectral invariance. Let 1 n—d l 0 if x E and 'y E 1', it follows that there is a neighbourhood of some point r where w is strictly positive.
Then, by (19.96) we have w(x) > 0 for all x E ft Hence v(x) < 0 for all x and u = —v satisfies (19.92). Next we assume that w(x) in (19.95) is identically zero. Then we have v(x) 0. By the same arguments as above we get v(x) > 0 in Q and, hence, u = v satisfies (19.92). We proved a little bit more than stated: If v is a real eigenfunction with respect to the largest eigenvalue
then v is in ci either strictly positive or strictly negative.
Step 6 We prove (i) and (iv). In Step 1 we mentioned that (19.73) is known. is simple. Let us assume that p is not simple. Then by We prove that = Step 5 there are two real '(ci)-orthogonal eigenfunctions v1 (x) and v2(x) with
=
=
= 0.
(19.97)
By the end of Step 5 we may assume that v1 and v2 are strictly positive continuous functions in ci and hence on I'. But this contradicts (19.97). Hence p is simple and we have by Step 5 also (19.77). The proof is complete. 19.8
Two comments
We comment on two points of the above proof. First, as we have noted, (19.84) is known. But the two embeddings follow also from the quarkonial representations for B-spaces. By (19.83) the normalizing factors for the po)-13-quarks and the (sj,p1)-$-quarks in (2.16) are the same. Then the first embedding in
(19.84) follows from (2.7), Definition 2.6(i) and the monotonicity of the 4As for the second embedding we remark that one needs .-.' balls of radius to cover ci in connection with the (s1,pj)-/3-quarks in (2.16). Furthermore, by Holder's inequality applied to )q E C, spaces.
P1
(19.98)
for some c' > 0. By Definition 5.3 quarkonial decompositions for B-spaces on ci can be restricted to the above-indicated balls. The first factor on the right-hand side of (19.98) compensates the normalizing factors for ,Pi )-f3quarks and (si,po)-f3-quarks in (2.16). Then one gets the second embedding in (19.84) as above as a consequence of (2.7) and Definition 2.6. Secondly, we add then the right-hand side of (19.96) is well a comment on (19.96). If x E
19. Spectral theory for the fractal Laplacian
271
defined, since both G(z. and (trr are continuous and the OUt(OflW COincides with the left-hand side: The latter follows by definition for the Sobolev mollifications (trrw)h (x). The rest is a matter of completion. However these observations remain also valid if x E 1'. Let n 3. Then G(x.
—
near
x.
This singularity is well compensated by (18.1), since d > n — 2. Similarly if n = 2. We return to this point in greater detail in Section 20 in connection with single layer potentials. 19.9
Discussion
As said above. Theorem 19.7 is the continuation of Theorem 30.2 in [TriS], p. 234. We described in 19.3 what was known so far. We repeated the corresponding assertions in Theorem 19.7 with references to [Triol. Detailed discussions and interpretations connected with the physical background described in 19.1 may be found in 30.3. 30.4. pp. 235—236, including the very few references dealing with problems of this type. We took over the crucial equivalence (19.73). Its proof in [Trio] is based on quarkonial representations. entropy numbers and approximation numbers. At that time the attempt to prove assertions of type (19.73) was the decisive impetus to develop the theory of quarkonial decompositions for function spaces. We return to this technique in the later parts of this section and also in some other sections of this chapter. In comparison with the spectral theory for, say. the (Dirichiet) Laplacian in bounded domains with smooth or fractal boundary one may ask whether (19.73) caii be strengthened by
Qk=ek'r(1+o(1)), k€N,
(19.99)
where e > 0 is a suitable constant and o( 1) is a remainder term tending to zero if k tends to infinity. But this cannot be expected. There is even a counterexample. We refer to [TriO], 30.4, PP. 235 236, and the literature mentioned there. 19.10
Nullstellenfreiheit
Part (iv) of Theorem 19.7. including the simplicity of the largest eigenvalue is the fractal version of Courant's classical assertion for the Dirichlet Lapladan in
(1924). Courant's strikingly short elegant proof on less than one page
may be found in [CoH24], pp. 398 -399, where the title 'Ciiarakterisierung der ersten Eigenfunktion durch ihre Nullstellenfreiheit' indicates what follows in a few lines. Based on quadratic forms Courant relies (in recent language) on
III. Fractal elliptic operators
272
H'-arguments. But lie did not bother very much about the technical rigour of his few-lines-proof. Problems of this type have a long and rich history at lea.st since that time. More recent versions may be found in [Tay96), pp. 315-316. We refer also to ITai96i for generalizations and to IReS78I, Theorem X1I1.43, for an abstract version.
19.11
Singular perturbations; the case n =
1
First we (leal with n =
1. Then we have by Theorem 19.7 for B given by (19.68) and d-sets F with 0 < d < 1,
kEN.
and
(19.100)
Of course, we have = —t/'(x) in this case and (19.74) means that the eigenfunctions uk(x) are linear in In this case, 13 makes sense for any finite Radon measure z with, say,
=
rc
= (—1,1).
(19.101)
First we remark that (19.102)
1
again + = 1. The first embedding is Holder's inequality, the last embedding follows again from [Tri[3j, 2.7.1, p.129. As for the middle embedding where
we refer to (9.9)-(9.11), extended to L1(f') (complex measures). All spaces in the shaded region in Fig. 19.2, now applied to n 1, are continuously embedded in C(1), the space of continuous functions on the interval (—1, 1). Hence there are pointwise traces for all spaces of interest in Theorem 19.7 and its proof. We have for the eigenfunctions Uk,
We have a quick look at the case F=
with
— 1
0,
and
k
00
if,
and only if,
ri
= 0.
We do not go into detail. We only mention that these observations can be proved using entropy numbers for estimates from above and approximation numbers for estimates from below. The abstract background will be described in 19.16 below and used later extensively. However this phenomenon is not new. It can be found in a larger context in [BiS74] and in [Bor7OJ.
If n 2 then it is not possible by our method to deal with arbitrary finite Radon measures with compact support. On the other hand, in the slightly different but nearby context of quantum mechanics, it is of interest to study, say, ± S in R", or related operators with strongly singular measures. This attracted a lot of attention since the 1960s. The state of the art may be found in [AIKOO] with almost a thousand refereines. The methods there and here are different. Nevertheless the question arises of whether they can complement each other.
III. Fractal elliptic operators
274
19.12
Fractal drums in the plane
In 19.11 we discussed fractal strings, where n = 1. Also the cases ii = 2 and n = 3 are of physical interest. As explained in 19.1, if n = 2 then one might think of vibrations of a drum C R2 with a fractal membrane 1' C Let be a bounded C°° domain in the plane R2, let r be a compact d-set with F C fI and 0 < d < 2 and let B be given by (19.68). Then we have by Theorem 19.7,
Qk"-'kt
and
ukECd(Ifl,
kEN.
(19.106)
In the plane, —1 is the classical Weyl exponent concerning the distribution of cigenvalues of the inverse of the Dirichiet Laplacian (—s) in bounded smooth domains. We discussed this problem in greater detail in fTriöl, 26.1 26.3 and 28.9-28.111 pp. 199-202 and 230, respectively. In particular, the expo-
nent —1 in (19.106) is independent of d. This observation can be immediately extended (based on the techniques developed in [Trio], explained and used later on in this section and in the following sections of this chapter) to the following situation: Let fl he a bounded C°° domain in the plane, let (—s)-' be the inverse of the related Dirichiet Laplacian, let F3 where j = 1,. . , N be pairwise disjoint compact dy-sets with 0 < d3 0 such that 2
conditions
dx)
=c for all
E
(19.110)
Then, by completion, the trace operator trr,
trç
L2(r).
:
(19.111)
exists, where L2(I') must be understood with respect to the given measure p according to the notation (9.14). A refined version of what is meant by traces has been given in Step 1 of the proof of Proposition 19.5. Recall the classical duality assertion =
H'(fl)
(19.112)
in the understanding of the dual pairing (D(fl), 4.8.2, p. 332. Then we have by for the identification operator idr.
idr Let again (19.16) that
(19.113)
be given by (19.23). Then we obtain by (19.111), (19.113) and
B=
otrr
'—i
:
(19.114)
is a bounded operator and as indicated in 19.3 the generator of the quadratic form (19.70). To make clear what is going on we remark that we have for any finite compactly supported Radon measure p in the plane
idr L2(r) c
c
(19.115)
This follows from (9.9), (9.12). extended to complex measures p (in particular = the spaces in (19.113) and (19.115) f L2(T)). Since differ only by the third index. This makes clear that we have a rather delicate
276
III. Fractal elliptic operators
limiting situation. In particular by 13.1 or Theorem 11 .4(u), especially (11.23), always with n 2, there is no continuous embedding of H' (1k) in C(Ifl. As a consequence a measure p with (19.111) must be diffuse (or non-atomic; as for
notation we refer to 9.17). By Theorem 13.2 with n = 2 the measure p with (19.111) must compensate the singularity behaviour of functions belonging to H' (1k) expressed by the growth envelope
=
(19.116)
.
On the other hand, by Theorem 9.3, we have a necessary and sufficient criterion under which circumstances the trace operator (19.111) exists. Let f,,,,1 be given by (9.26). Then trr in (19.111) exists if, and only if,
fvm M. The crucial observation in our context is Garl's inequality
kEN,
(19.129)
proved in [Carl8l] and [CaT8O] for Banach spaces and extended to quasiBanach spaces in [ET96J. We followed [Trio], Section 6 and [ET96]. 1.3. There one finds further results and references to the extensive literature. Furthermore we need the approximation numbers which are defined as follows. Let A. B be complex Banach spaces and let T L(A, B) . Then for all k E N approximation number ak(T) of T is defined by the
ak(T) = inf{hIT— Lu
:
L E L(A.B). rankL < k}
(19.130)
where rank L is the dimension of the range of L.
Of course ai(T) =
11Th. Let
now H be a Hilbert space and T
L(H) be a
compact self-adjoint operator. Let be the sequence of all eigenvalues of T, repeated according to their geometric multiplicity and ordered so that IA1(T)I
.
. .
.
(19.131)
III. Fractal elliptic operators
280
Then
IAk(T)I=ak(T),
kEN.
(19.132)
This is a well-known classical assertion. References, further information, and also comments about the relations between ek(T) and ak(T) have been given in [ET96], 1.3, P. 7—22, and [Trio], 24.3—24.7, p. 191—192. We do not repeat these assertions with exception of some rather elementary inequalities which are needed later on.
Let A, B, C be complex Banach spaces, let S E L(A, B), T E L(A, B) and R E L(B, C). Let hk be either the entropy numbers ek or the approximation numbers ak. (i) 11Th = h1(T) (ii)
h2(T)
For all k E N, 1 E N,
hk+1l(R oS)
hk(R) h1(S),
(19.133)
hk(S) + hz(T).
(19.134)
and
hk+,_j(S + T)
To avoid a misunderstanding we remark that above either all h3 are entropy numbers or all h, are approximation numbers (no mixed inequalities). 19.17
Theorem
A finite, compactly supported, strongly diffuse Radon measure in the plane 1R2 as defined in 19.15 (ii), is a Weyl measure according to 19.18. Proof Step 1 We begin with some preliminaries. Let be a bounded C°° domain in the plane R2 with
supp,z=r c
(19.135)
is a finite Radon measure with (19.124), (19.125). We may assume We must prove that the right-hand side of (19.70) is a bounded quadratic form in H1 (i)). Then, by the previous considerations, the generator B, defined by (19.70), is non-negative and seif-adjoint and can be represented by (19.68) in the interpretation of (19.114), based on (19.113), and the explanations given in 19.3. Hence first one has to prove that where
,z(r) =
1.
trr
L2(r)
(19.136)
19. Spectral theory for the fractal Laplacian
281
is bounded. As before, L2(r) is the L2-space with respect to the given measure We begin with a closer look at the interplay of ji, the geometry of its support r = supp ft and the quarkonial set-up in 9.24. Step 2 Let be the above measure with i(F) = 1 and the compact support r = suppji. We use the quarkonial set-up in 9.24, where we now replace the
j&.
balls
by the open cubes Qk,m
(19.137)
E r and with side-length c2 with sides parallel to the axes, centred at This is immaterial for the quarkonial approach described there. Otherwise we use the same notation as in 9.24. We may assume in addition that :
m=1
'1(k) c
in
= 1,... ,Mk+l} ,
kEN0, (19.138)
and that for any cube Qk+lm there is a cube Qk,z with Qk+1,mCQk.1,
kEN0.
(19.139)
The additional assumption (19.138) does not cause any problems. If (19.139) is not satisfied one can replace all squares Qk,m for all admitted k and rn by It follows by geometrical 2Qk,m, centred at and of side-length C2 reasoning that these modified cubes have the desired property. Hence, we assuine that also (19.139) holds from the very beginning. Again this modification is immaterial for the quarkonial approach in 9.24 and what follows afterwards. We may aLso assume that only one cube with k = 0 is needed to cover r. Let K2 be the number of all cubes Qk,m with
j E N0.
(19.140)
We wish to estimate the number of these cubes. Let x = with L E N in be a cube with (19.140) and let Qk÷L,1 be a sub-cube of 19.15(u). Let in iteration of (19.139). Then by (19.125) and (19.140), (19.141)
p(Qk+L.1)
be the number of cubes with k N0 and (19.140). By the above construction (19.139) and (19.141) these cubes are disjoint for different values of k. Together with the controlled overlapping of cubes with the sanic k it follows that Let
0 is independent of j, the sum is taken over all k and in where the respective cubes have the property (19.140). Similarly one can estimate with I = 0, 1,. . , L — 1, being the number of cubes QkL+I,m with the property .
(19.140). We get K5
=
K,,1
= N,
(19.143)
for some c> 0 which is independent of j. Step 3 After these preparations we first observe that there is a number D > 0 (diffusion number) with
j E N,
(19.144)
for some c> 0: If L has the above meaning and (19.145)
then, using (19.139) and iteratively (19.125),
(
li(Q,,m)
(19.146)
with D = L1. We apply Corollary 9.8(11) with
p=r=n=2, .s=1 and d=D>0,
(19.147)
and obtain by (9.28) that
trr
H'(R2)
:
L2(F)
(19.148)
is a bounded operator. Since F C Q this is the same as (19.136). Let t = > 0 he the typical number introduced in Definition 9.25, where the t2 l)alls I3krn in (9.109) can be replaced by the above cubes Qk,m (since we have the doubling condition it (loes not. matter whether we choose or Here D has the same meaning as in (19.146). We apply Theorem 9.33 to = and p = q = n = 2 and obtain
=
Ht(F).
(19.149)
We denote temporarily the embedding operator from Ht (F) in L2 (1') by id,
H'(F) :: L2(r).
(19.150)
19. Spectral theory for the fractal Laplacian
can be estimated by
We wish to prove that the entropy numbers of
ek(idt)ek2,
283
kEN,
(19.151)
for some c> 0. We rely on the quarkonial representations in Definition 9.29. By Definition 9.27 the (t, 2)-fl-quarks, responsible for Ht(r), are given by = 2k1131 — (19. 152) E 1',
now with respect to the above cubes Qk,m in place of the balls
By
Definition 9.29,
Ht(r) with
gE
IHt(T)II
(19.153)
1
can be represented as oo
MA,
=
(19. 154)
k=0rn1
E C with
where the fl-quarks are given by (19.152), and
/00
MA,
sup /iEN0
Here
\k=Orn=l
/
2.
(19.155)
0 has the same meaning as in 9.29 and can be chosen arbitrarily large.
By Remark 9.30 and Proposition 9.31 all representations of type (19.154), (19.155) converge absolutely and unconditionally and can be rearranged as one wishes. In particular, one can re-organize the collection of all cubes by
: k€No; m=1 (19.156)
where Qjm are the cubes with (19.140), and Hence, g can be represented as
call be estimated by (19.143).
30
gey)
(19.157)
= k=O m=1
where
are the 13-quarks related to Qk,m and
coefficients with
/30 (
Isk
I
are corresponding
III. Fractal elliptic operators
284
We prove (19.151) by factorization through sequence spaces. For this purpose we introduce some notation in modification of Sections 8 and 9. Let
5ER, ,>0,
and LkENwherekENo.
(19.159)
Then
1p 0. Finally, Tin (19.166) > is given by
Tx=
(19.167) k=Oin=1
19. Spectral theory for the fractal Laplacian
285
where a is the (19.161) with Lk = Nk, terms with Kk < m Nk are simply neglected, and are the /3-quarks in (19.157). First we have a look at T. For fixed j3 E we have K,
1L2(r) k=O m=1 2
Kk
krO
if >
ji(d'y)
m=1
/14
00
k=O
2
2_k)
(
.
(19.168)
\m=1
The first estimate is simply the triangle inequality. The second estimate comes from (19.140). the support properties of the involved cube Qk,m described in
Step 2 and r > 0 in 2n181 is the same constant as in 2.4 and 2.5 adapted to the above more general situation (19.152), and c> 0 is independent of We always assume
Xl >
>r>0
(19.169)
in (19.164) (19.166) is linear and l)ounded. We have (19.163), where S is bounded (not necessarily linear) and
T is linear and bounded. As for idt we can apply [Thö], Theorem 9.2, p. 47, and obtain
kEN.
(19.170)
Here we use (19.143). Now (19.151) follows from (19.170) and (19.163). Step 4 We summarize what is known so far. We always assume that r and ci are related by (19.135). By (19.149) and the compactness of ide, (19.150), (19.151). it follows that
trj'
:
L2(r)
(19.171)
is compact. We have (19.113) for the identification operator idr and hence,
tr"=idrotrr :
(19.172)
is compact. Together with (19.16) it follows that
B=
ot?'
is compact in
I°f'(fl).
(19.173)
III. Fractal elliptic operators
As discussed in 19.14 we have as in (19.70), based on (19.29), (19.174)
and the counterpart of (19.78),
: trrf = o}
N(B) = {i E
(19.175)
for null-space. Hence B is a non-negative, compact, seif-adjoint operator in By the usual Weyl decomposition and (19.149) we have
= N(B)
= N(B) ® trr
Ht(r).
(19.176)
with and We identify the orthogonal complement of N(B) in is a positive seif-adjoint denote the restriction of B to Ht (1') by Br. Then compact operator with the same eigenvalues, if
k—'oo,
(19.177)
as B, and the related orthonormal eigenfunctions Uk,
BrUk=PkUk,
kEN,
UkEHt(I'),
(19.178)
span Ht(1'). Fltrthermore, by (19.174),
IHt(r)II = Ill
(19.179)
and {uk} is also an orthogonal system in L2(f). By construction, Dir, the restriction of D(R2) on r, is dense in Ht(r). Since is a Radon measure, is also dense in L2(r). This follows from the proof of Theorem 3.8 in [ThöJ, p.7 (with a reference to [Mat95J, Theorem 1.10, p. 11). Hence any p can be approximated in Ht(r) and consequently also in L2(r) by linear combinations of {uk}. In particular, {uk} is a complete orthogonal system in L2(r). Let Br be the extension of B' to L2(JT). It has the same eigenvalues and (19.150) we and eigenfunctions. By (19.179) with ..JBj in place of have in obvious notation, (19.180)
By (19.129) and (19.151) we get
kEN.
(19.181)
According to Definition 19.13 it remains to prove the converse inequality.
19. Spectral theory for the fractal Laplacian
287
Step 5 We use again the coverings of r by cubes as constructed in Step 2. By the doubling condition (19.124) there is a number C> 0 such that for any two cubes with (19.139),
C/t(Qk+1,m),
k E N0.
(19.182)
j E N0,
(19.183)
Let j E N0. We ask for all cubes Qk,z with
2'
such that there is no larger cube in which Qk,1 is contained according to the hierarchy (19.139) having this property. We denote these largest cubes by Qm• \Ve claim
rcUQ;,m.
(19.184)
Let -', E I' and
This follows from E Qk,m. If k is large then JL(Qk,m) (19.144). Stepping iteratively from k to k — 1, then by (19.182) there must a nunhl)er k with (19.183). There might be even smaller k's with (19.183). But in any case there is a largest (or several largest) cubes with this property. This (19.184). Let Pj be the number of cubes Q,m• By (19.184) we have ? e23 for some c> 0. Two such cubes = Qk,m with different values
of k arc disjoint by construction. For the same level k we have the above controlled overlapping. In any case there are C°° functions
= 0 if
fl
C 1
and
1L2(r)II
24
with
m,
1
(19.185) (19.186)
,
where all equivalence constants are independent of j. In the last equivalence we used again the doubling condition. We obtain for linear combinations of t IleSe functions. F,
Pi
19.187 •
and.
(19.174), P;1
1k1(I)D P2
(EIAJ.112)
.
(19.188)
III. Fractal elliptic operators
288
We wish to estimate the approximation numbers ap,
defined by (19.130).
Let L be a corresponding operator in (19.130) with rankL < Pj. Then one finds a non-trivial linear combination as used in (19.187), (19.188) with Lço = 0. Inserted in (19.130) one gets
iEN. Using P,
(19.189)
and (19.132) we obtain for some C> 0,
kEN.
(19.190)
The proof is complete.
Problems and comments: Weyl measures
19.18
19.18(i)
B=
We proved that
otr1'
exists and that
Ok ".'
k'.
k EN,
(19.191)
for its positive eigenvalues Ok according to Definition 19.13 if p is strongly diffuse. This assumption is isotropic (there are no distinguished directions). In Section 30 and in [FaT99] we dealt also with anisotropic and nonisotropic measures p in the plane and related fractals (ferns, grasses etc.). For corresponding fractal drums and related operators B we obtained only estimates,
kEN, 0
(19.200)
\Ve equip r with the Radon measure p = b(x) PL and ask, in our previous notation, for existence, boundedness, compactness, spectral properties, of
B=
o
tr" as an operator in
(19.201)
Recall that (—s)-'
is the inverse of the Dirichlet Laplacian in Specializitig 19.14 to the above situation, (19.110) reduces to the Hardy inequality
/
b(x) 14r)12 dx
c EJ
dx,
q
D(fl).
(19.202)
III. Fractal elliptic operators
292
If there is a number c> 0 with (19.202) for all D(1l), then B is a bounded operator in ill which, in our context, and with an obvious interpretation can be written as ob,
B=
(19.203)
justified by
=
f
=
o
(19.204)
E D(cl). The boundedness of B is naturally and intimately E D(cl), related to the sharp inequalities in the critical case as described in Theorem 13.2 and detailed in 13.3. Let b be the rearrangement of b, and let for some where
c>0, b*(t) I
log
t2'
0 0. Now it is clear that the embedding of If' (Il) in the space related to the left-hand side of (19.211) is compact: the first term on the right-
hand side creates a x(S)-net and for the second term one has the classical compact embedding of H'(fl) in L2(1l). This completes the proof of (19.209).
There remains the problem under which circumstances for b in general, and is a Weyl for b given by (19.207) with x(0) = 0 in particular, = b(s) measure according to Definition 19.13, and hence
kEN,
(19.212)
for the positive eigenvalues of B, given by (19.203). This is not so clear so far then we have (19.209), but (to the author). IfS> 0 and K(t) =
ii =
I
log lxi
12L,
lxi
(19.213)
III. Fractal elliptic operators
294
is not strongly diffuse according to (19.125), and hence we cannot apply Theorem 19.17. On the other hand, jL
e >0,
= lxi
(19.214)
lxi
is strongly diffuse. Hence, by Theorem 19.17, it is a Weyl measure, and we have (19.212).
In tET961, Chapter 5, we developed a spectral theory for degenerate elliptic operators, especially the examples on p. 211 of [ET96] are related to the above considerations. There one finds also further references.
20 20.1
The fractal Dirichiet problem Introduction
where, temporarily, n 3, and let Let fl be a bounded C°° domain in r = Ofl be equipped naturally with a Radon measure equivalent (or equal) in R" to I'). The to ir (the restriction of the Hausdorif measure single layer potential G,
(Gh)(x) =j
xE
lx
(20.1)
makes sense both in RTh and on r (using the same letter G) if, for example, h is bounded. Since 1' is a compact C°° manifold,
H8(I') =
s E IR,
(20.2)
can be introduced in a canonical way via local charts. It turns out that C (restricted to F) makes sense for some spaces H8(I'). In particular,
=
(20.3)
is an isomorphic mapping. This has the consequence that the uniquely determined solution u(x) of the (almost) classical Dirichlet problem
xEIl, uEH1ffl),
(20.4) (20.5)
for given g, can be uniquely represented by (20.1) as u = Gh with some hE
It is the main aim of this section to extend these observations
20. The fractal Dirichiet problem
295
to fractals. We rely on the techniques developed in Section 19. We describe what can be expected. First we remark that the boundary F = of the above C°° domain is an (n — 1)-set according to (18.1). In particular, the singularity x y E F, is well compensated by (18.1) with d = n — 1. In addition it is quite clear that this argument applies to any compactly supported d-set F with d> n — 2. One obtains the generalization
n—2 0.
(20.13)
Then, by (9.143),
8>0.
=
H8(F) =
(20.14)
Some further information is given in 19.2, including references to [Trio], Theorems 18.2 and 18.6 on pp. 136 and 139. The identification operator has been introduced in (9.16). We rely in particular on the duality assertions (9.20), (9.21).
Let w be an arbitrary bounded domain in RIL. Of special interest for us are
H'(w), and (20.15) Recall that H'(w) and H'(w) are defined by restriction of H'(]R") and respectively, on As usual, p1(w) is the completion of D(w) in H'(w). We have in H1(]R") the explicit norm (1.4) with 8 = 1 and p = 2. Then it follows by standard arguments that for any bounded domain the space can be equivalently normed by
In Ill (with a corresponding scalar product). As usual, (complex-valued) distributions in w, the dual of
2\2I )
(20.16)
/
is the collection of all
20. The fractal Dirichiet problem 20.3
Proposition
Let
be
297
an arbitrary bounded domain in IR". Then
= H1(w)
(equivalent norms)
(20.17)
with respect to the dual pairing (D(w). D'(w)). Proof This &ssertion is well known if w is smooth and also if w is replaced by IR",
(H'(R"))'
(20.18)
with respect to the dual pairing
trary domain. Let yE By the explicit norm (1.4) of
Now let w be the above arbi-
according to the dual pairing (D(w),D'(w)). we have
II.qII 11w"' (w)II =
Interpreting I°J'(w) as
a
II
E D(w) .
.
(20.19)
closed subspace of H'(R"). we find by (20.18) an
with
element h E g(w) = (Ii. w)
IIh
1H' (R't)lI
E
D(w)
.
(20.20)
With G = hlw (restriction to w) we have = (G.w),
where c > element h
IIGIH'(w)U
is independent of q. Conversely, let G E H '(w). There is an H_1(Wl) with G = hlw and
0
lhlH'(R")ll Let
(20.21)
=
IIGIH'(w)II.
(20.22)
if ço E D(w). Then we have
=
2 Since u 20.8
J°I'(w) it follows that
=
= 0.
(20.54)
= 0 with j = 1,... ,n and finally u = 0.
Comments and the variational approach
The above proof of uniqueness is based on Proposition 19.5, which, in turn, used the substantial Theorems 9.1.3 and 10.1.1 in [AdH96], pp. 234, 281 (here Theorem 9.1.3 dealing with Sobolev spaces is sufficient). We refer to 19.6 for further comments. The proof of the existence and the representation 20.8(i)
20. The fractal Dirichlet problem
303
of a solution u with (20.47)—(20.49) is based on the techniques developed in connection with Theorems 19.7 (with a reference to [Tri6], Sections 28 and 30) and 19.17. Basically one shifts the problem from fi to r and proves that the resulting operator B" is an isomorphic map as indicated in (20.41). The rest is interpretation of this observation. If r is a smooth or Lipschitz-contirnious (maybe the boundary of a respective domain), then d = n — 1. surface in Hence we have by (20.41)
B"
'—÷
H4(fl,
(20.55)
as indicated in (20.3). As will be seen below in Corollary 20.10 and its proof, the single layer potential on the right-hand side of (20.49) can be replaced by the single layer potential according to (20.1). Furthermore, after this modification (20.6) corresponds to (20.41). 20.8(11)
If one looks only for the existence of a solution u with (20.47),
(20.48), and not for its representation, one can use the very classical variational Then by definition there is a approach as follows. Let again g E function
ve Let again
'(&l)
with
v = g.
(20.56)
= 1l\F. Then we ask for a function h with h E J°f'(w)
and
—
=
E H'(c&).
(20.57)
This is the point where one needs Proposition 20.3. It follows that the righthand side of (20.58)
is a linear and continuous functional on H' (w) and, hence, can be represented by the left-hand side with some h E (ci.,). Then we have (20.57) and u = v+h is a solution of (20.47), (20.48).
We concentrate in this section as in Section 19 on the Dirichiet Laplacian —is. But it is quite clear that many of our considerations can be 20.8(iii)
generalized as indicated in 19.4 with a reference to [Trio], Sections 27—30. This
applies also to Theorem 20.7. Similarly the variational approach outlined in 20.8(u) can be used for wider classes of elliptic operators to prove the existence of weak solutions of the Dirichlet problem for elliptic equations in non-smooth domains. The corresponding uniqueness assertions can be based on density
III. Fractal efliptic operators
properties a.s mentioned at the beginning of 20.8(1). Both together (variational approach and density assertions) result in existence and uniqueness assertions for Dirichlet problems in non-smooth domains. This has a long history and goes back to [Sob5O] (and Sobolev's original papers in 1936- 1938). These problems have also been considered in some detail in [AIH96], Corollary 9.1.8, Theorem 9.1.9, PP. 236, 237, including Sobolev's original problems connected with the poly-harmonic operator. 20.9
The Dfrichlet problem in dom*mi with a fractal boundary
and let its boundary 1' = I be a d-set. Let Il be a bounded domain in Then we have necessarily d n — 1. We assume n — 1 < d < n (with d> 0 in case of n = 1; then 11 is a disconnected bounded open set). By (20.14) and in modification of Theorem 20.7 we ask for solutions u of
uEH'(IZ),
(20.59)
where g is given. Let K be an open ball with C K and let GK(x, y) be the Green's function for the Dirichlet Laplacian with respect to K. We apply Then one finds an element Theorem 20.7 to K in place of and to r = n— d hE H-'+r(r) such that
u(s) =
(20.60)
solves (20.47), (20.48), with K in place of ft We have in particular (20.59) as a solution. As for the uniqueness we recall first our point of view: By (20.12) all spaces on 11 are defined by restriction of the corresponding spaces on R". This avoids problems of extendability and traces on F = for intrinsically defined
spaces. As for traces we discussed this problem in some detail in 9.34(vi) (but this is hardly needed here from our point of view). Hence the uniqueness problem can be formulated as follows: Let u E and let
and trru=0.
(20.61)
Then one has to prove that u(s) = 0 in Ii By Proposition 19.5 it follows that D(R"\I') is dense in {v E
Let
E
H' (W') :
v = o}
.
(20.62)
Then by (20.61),
J
(20.63) )
20. The fractal Dirichlet problem
305
Since u belongs to the spaces in (20.62) it can be approximated in H' Then it follows from (20.63) by standard 0 in Il. Hence we have existence and uniqueness for the Dirichlet problem (20.59). But the representation (20.60) is unsatisfactory since it depends on K. However at least in case of n 3 this awkward description can be replaced by a more natural one. by functions belonging to
arguments, first u(s) =
20.10
in c and afterwards u(s) =
Corollary
where n E N, and let I' =
be a bounded domain in
Let n—
c
1
0 is sufficiently large then + pid
H8(I')
s E R.
(20.79)
is an isomorphic Imiap. Some information concerning spaces on Riemannian manifolds may be found in 7.2 (although in a soiriewhat different context); otherwise we refer to [Tri5]. 1.11 and Chapter 7. As in the eticlidean ease (or the n—torus) the fractional powers of the positive—definite self—adjoint operator
308
elliptic operators
III.
+ Qid with pure point spectrum have the expected mapping properties. In particular,
ER.
+ Qid :
(20.80)
combined with (20.78) one gets (20.81)
:
as an isomorphic map. This sheds some light on the relations between the Laplacians on Il and r = Oil. 20.13
with boundary r = [Xl. The quality Again let il be a bounded domain in of the boundary r has a strong influence on the Dirichiet problem as treated above. If F is C°° then one has a complete theory in all reasonable spaces even with p < 1, where the final results go back to [FrR95J. and which may also be found in [RuS96], 3.5. Looking for an Lu-theory for nonsmooth boundaries there is apparently a big difference between F E C' and F E Lip1. We refer to [Ken94], [JeK95], Theorem 5.1. p. 191, and iii particular to [FMM98]. The last paper deals also with mapping properties of single layer onto for some s potentials of type (20.69) on F E Lip' from 6 for some 6 > 0, where 6 depends oii with 0< s < 1 and p with — the Lipschitz constant of F, [FMM98I, Theorems 3.1 and 8.1. We refer also to [ZanOO]. This seems to suggest that in the fractal case as treated in Corollary
20.10 there might be little hope to step from £2 to
(or Br). Further
information and additional references concerning layer potentials, boundary integrals and Dirichlet problems for Laplacians (on Riemannian niauifolds) in Lipschitz domains may be found in (MeMOO] and in Chapter 4 of We refer in this context also to the survey [JoW97], where the authors summarize their contributions to boundary value problems of the above type. Let again P" be the operator P given by (20.69). restricted to the d-set F. We have the mapping properties as described at the end of 20.10. One may extend given by these considerations to Riesz potentials on d-sets F in
(P[h)(A) = where
/
A E F, IA
d> a > 0. Mapping properties of these operators and related spectral
problems have been studied recently by M. Zähle. LZahOO]. in the framework of an L2-theory.
20. The fractal Dirichlet problem 20.14
309
Classical solutions
be a bounded domain in where n 2 and let 1' = Oil be its boundary. be the space of all continuous functions on r, obviously formed, and let. be the space of all Lipschitz continuous functions on r. normed in analogy to (10.17). Let g E or, more restrictive. g E Lip(r). The classical Dirichlet problem for the Laplacian asks for harmonic functions u in il with Let Let
x—*',Er.
and
(20.82)
The first decisive step for arbitrary bounded domains Il goes back to Perron, [Per23J. and is known as the method of subharmonic and superharmonic functions. One gets always a harmonic function u in 11. The main problem is to clarify for which e r or for which boundaries F one ha.s the desired pointwise boundary behaviour as indicated in (20.82). The final solution goes back to Wiener and can be described as follows. Let !IC = R"\il be the complement of il in R" and let C1.2(K) be the (1,2)-capacity according to (19.44). this means with respect to the above classical Sobolev space Ht(Rn). Then one has (20.82) for a given point y e F and g E C(1') if, and only if,
/
C1,2
(B('y,r)
dr = 00,
(20.83)
r) is again a ball centred at E F and of radius r. In particular there is a natural connection to the theory of weak solutions. this means H'where
solutions. We do not go into detail. A description of the method of subharmonic functions may be found in [GiT77], 2.8. pp.23-27. As for the Wiener criterion (20.83) we refer to [Ad1196], Theorem 6.3.3. p. 165, and to [Ken94j, p. 5. In our context it is of interest under what additional conditions for F the solutions IL from Corollary 20.10 are classical solutions with respect to Perron's method. By [Ken94]. p.5 (restricted to our situation) one has the following assertion: The Wiener criterion is necessary and sufficient such that for every Lip(F) the (weak) solution u in (20.6.4) is continuous in Il (and hence a g classical solution).
We discuss (20.83) in connection with the domains in Corollary 20.10. First we remark that always
092>°
large,
(21.26)
such that Theorem 7.22 can be applied. Furthermore x = x1 > 0 (recall that = 0) has the above meaning. The numbers and are unimportant. We may assume P2 (the F-spaces are monotone with respect P1 and to the q-index). Crucial for the estimate of ek(Id) is the knowledge of Er. Afterwards one gets (21.24) factorizing id in (21.19) via Id in (21.25) in a
21. Spectral theory on manifolds
315
similar way as in Step 3 of the proof of Theorem 19.17. First we estimate Er. Let I E be given by (7.98), where ((3qu)3i(x) are (sj,pi,x)-flquarks according to (7.88). Comparing this expansion with a corresponding representation in (M) (recall = 0) one gets
1=
= j=0 1=1
j=O 1=1
(21.27)
where (/3qu)1(x) denote temporarily the corresponding (82, P2, Here 5 is given by (21.18) and k by (7.89). If r E N0 then we have to estimate the number Er of balls K3, given by (7.78) with dist(K31,i91Z)
for
m.=0,...,j+c
(21.28)
and
rK=mx+(j—m)t5. (21.29) (7.90). Forfixedmwithm =0,...,r+c'
Hereweusedm
2'
we have to estimate the number of balls K31 of radius —' which, say, intersect u1rn given by (7.21) with (7.24), and where j is calculated by (21.29). Hence
2md 2(r_m)?,
2—vn(n—d) 2jn
(VOlIlm)
(21.30)
where we used again (7.24). Summation over m results in (21.31) ,n=0 We
obtain
if d>
Er
if d
0. independent of r. In case of (21.52), (21.32) we have
r EN0. In case of (21.52), (21.33) we put a = ec2ro(id)
(21.70)
and get
=
,
r EN0,
(21.71)
where we used (21.54). Here c> 0, e' > 0, are independent of r E N. But this disproves (21.49) with (21.50). As said above, this is sufficient to prove the converse of (21.46), (21.47).
III. Fractal effiptic operators
320
21.4 Thecasep=oo In Step 2 of the above proof we excluded so far the case p2 = oo (and hence with 0 < s < 1 = oo and let C8(Q) = also = oo). Let now be the Holder-Zygmund spaces, normed by first differences as in (1.11)—(1.13)
and according to Definition 5.3. By Theorem 7.10 and direct arguments one gets
O 0 as in (21.80), which coincides with (21.4). On the other hand locally, near the nucleus, the Coulomb potential in M (say with n = 3) should be Hence something like Izi;' looks reasonable. Then one gets, similar as in with the additional specification x = 1, the operator in (21.8). We call such operators hydrogen-like. But we could not find in the literature anything like a quantum mechanics in hyperbolic spaces providing a better justification of operators like in (21.8) than the above vague arguments. We look at operators of type (21.8) from a slightly more general point of view expressing local singularities by b L,. ( M). Then we have first to complement
Theorem 21.3 in the operator version of (21.87). This will be done in 21.9. Afterwards we extend Theorem 21.7 to potentials with local singularities. We introduced the weighted Sobolev spaces H(M,gM) in (7.42) with the weighted Lebesgue spaces
1 0 such that for the related entropy numbers ek = ek
(2_n)
kEN, if
0 such that
W(xx) 0. This proves (22.10). Step 2 It remains to prove the right-hand side of (22.12) for an admissible decreasing function 'I'. Let
<x
0.
By the mass distribution procedure and (22.35) we
obtain
=
=
j EN.
=
(22.40)
Using (22.12) and (22.11) we have C1
where ci >
0
C2
and
'11(r3),
(22.41)
and C2 > 0. Hence,
j EN. By (22.39) and (22.11) we have
--' r3.
(22.42)
Then (22.18) is a consequence of
(22.42). 22.9
Function spaces: Preliminaries
As indicated in Section 18 and in greater detail in 22.1 we are now interested in spaces of type (22.7) tailored to (d, '11)-sets according to Definition 22.4. Since (d, '11)-sets might be considered as perturbations of d-sets, one can expect someand thing similar for the spaces in (22.7) compared with This is largely the case and one could follow the Weierstrassian approach to the latter spaces developed in Sections 2 and 3. But this will not be done here. We restrict ourselves to a summary. Then it might be better to begin with the Fourier-analytical definition and to say afterwards how atoms, /3-quarks etc. appear. The first steps have been carried out in [EdT98] and [EdT99a) under
III. Fractal elliptic operators
340
the restriction to the B-scale and the range of the admitted parameters. By the recent work of S. D. de Moura, we refer in particular to [MouOlb], the full theory for B-spaces, F-spaces and all parameters is now available. As a compromise we follow mainly [EdT99a}, complemented by some assertions taken from [MouOlbI, avoiding full generality. As said above, no are given. Brief outlines may be found in [EdT99al (under the indicated restrictions),
full details are presented in [MouOlb] and [MouOlaJ. We use the notation introduced in 2.8. In particular, the functions have the same meaning as in (2.33)—(2.35). First we define the counterparts of (2.37) and (2.38). 22.10
Definition
Let
sER,
(22.43)
and let 'I' be an admissible function according to 22.2.
is the collection of all f E
Then
(i)
such that 1
=
Ill
(22.44)
11(w21)"
(with the usual modification if q = oo) is finite. (ii) Let in addition S'(RTh) 8uCh that
p < oo. Then
is the collection of all I E I
=
hf
(22.45)
(with the usual modification if q = 22.11
00
) is finite.
Comments
If 'I' = 1 then
has the
usual Fourier-analytical definition of the spaces On this basis we developed in [Tri/3] and [Th'y] the theory of these spaces in detail. Many properties obtained there can be carried over to the above spaces and In particular, they are one
and
independent of the
chosen
resolution of unity {cok}. Furthermore they are
22. Isotropic fractals and related function spaces
341
quasi-Banach spaces (Banach spaces if p 1 and q 1). We refer to [MouOlbJ, 1.2 and 1.3. where one finds detailed proofs. In particular characterizations in terms of local means as developed in [Tk17j, 2.4.6, 2.5.3, for the spaces Fq (IR?') and can be extended to the above spaces; Theorems 1.10 and 1.12 in [MouOlb) include also some improvements even in case of 'I' = 1. By (22.12) it is quite clear that a remains the main smoothness and W stands for an
additional finer tuning. 22.12
Atomsandf3-quarks
In [Triö], Sections 13 and 14, and in Sections 2 and 3 of this book we developed
the theory of atomic and quarkonial decompositions for the spaces and (R"). There one finds also the necessary references to the literature. By [MouOlbJ, 1.4, there is a full counterpart of this theory for the spaces introduced in Definition 22.10. In particular the simultaneous proof of Theorems 1.18 and 1.23 in [MouOlb], covering atomic and quarkonial decompositions. fits in the scheme of the above Sections 2 and 3. We do not describe the full
theory here. We wish to provide an understanding of the necessary modifications needed now, compared with the above Sections 2 and 3, where it is sufficient for our later purpose to restrict the considerations to those spaces where no moment conditions for atoms and [3-quarks are necessary. Otherwise we use the notation introduced in Section 2. Let i[' and with i3 e be the same functions as in Definition 2.4. Let
a E R and 0
0.
ifyE R,
(25.29)
According to [Trio], Theorem 13.7 on p. 75, xE
1(x) =
(25.30)
j=O is
(R). In particular,
an atomic decomposition in
fE
1+1
(IR)
for all
0
Let either
l = F,;,. In the last equivalence we used s — i) and
This proves (25.132). Next we prove (25.133). By (25.129), (25.130) and the localization property from [ET961, Theorem 2.3.2, pp. 35-36, we obtain that .1
11111-
1)
1B(R)II
=
(Because of the required support properties in [ET96]
(25.135) we
do not obtain the
desired equivalence immediately.) But (25.135) is also a consequence of atomic
decompositions. Hence it remains to prove that there is a constant c> 0 such that 11111
for
—191 B;(]R)II
First let, in addition, we have that
—
i) <s
O}.
.
In specification of Definition 2.4 we assume that
is
(26.1)
a non-negative C°° func-
tion in R" with C {y E (R÷)" :
(26.2)
liii
= have the same for some r > C) and with (2.15). Otherwise the By (26.2) we have now that meaning as there, where /3 E
0
for all
xE
and
/3 E
(26.3)
were introduced in Definition 2.4. In The related (8, p) — fl-quarks be the spaces particular they are also non-negative. Let and according to Definition 2.6. We are not interested in the most general cases. and complement the results So we concentrate mainly on the spaces obtained by a look at the Sobolev spaces
11(R") =
(26.4)
In other words, in what follows we always assume that n E N and that
O r. Hence, the right-hand side of (26.11) converges absolutely By (26.9) with = max(p, 1) and, hence, unconditionally in in
it follows that there is a constant c> 0 such that
fE
(26.13)
in place of Hence, Q is (and an obvious counterpart with a bounded operator in all these spaces. But it has much better continuity properties than the operators T and (m real spaces) considered in the previous section. Obviously, we have that If(x)I 26.4
(Qf)(x),
x E Re',
fE
(26.14)
Theorem
Let n E N and let p, q, s be given bij (26.5). Let Q be the operator according to Definition 26.2. (1)
Thereisaconstantc>Osuchthat IIQf
for all f (ii)
c
— Qui
and all g
—g
(26.15)
(Lipschitz continuity).
Let, in addition, p < oo. Then there is a constant c> 0 such that DQI
c
—
(26.16)
for all f E H(R') and g E H(RTh). Proof After the above preparations the proof is simple. We prove part (i). By (26.10), (26.11), and the obvious counterparts with g in place off, we have
that Qf — Q9 =
—
(26.17)
27. Semi-linear equations; the Q-method
389
This is an admitted quarkonial decomposition of Qf — Qg in accordiiig to Definition 2.6 and the above considerations. Using the linearity of the (listinguishe(l coefficients we have that
I
I
=
—
By (26.9) and the discussion there the sequence )t(f
—
—
g)
g)I
.
(26.18)
is optimal. Then it
follows that IIQf for
c
—
some c > 0 and all f
(26.19)
—g
and g
This proves (26.15).
The proof of (26.16) is the same. 26.5
Remark
be a truncation couple according to Definition 25.6 and
s) E
Let
Then (26.14) can be complemented by
(25.40). Let 0 < q
< (Tf)(x) < (Qf)(x), x E R'2, f E
(26.20)
By the above theorem and Theorem 25.14 the continuity properties of Q are much better than the continwty properties of T and T+. This will be the decisive observation in the applications given in the following section. We followed [TriOlJ.
27 27.1
Semi-linear equations; the Q-method Semi-linear integral equations
Let K E L1(R") be real and let h be real, where 1 p oo. Then one can apply Banach's contraction theorem to find a real (unique) solution u
f
K(y)u+(x — y)dy + h(x),
x E W',
(27.1)
provided that IlK is sufficiently small. This follows from the obvious fact t hat the truncation operator T+, given by (25.5), is Lipschitz continuous in But what can be said if h belongs to some real spaces as, for example. in (26.8)? Since a) is is involved the assumption that a truncation couple accorchng to Definition 25.6 and Theorem 25.8 seems to be natural, or at least reasonable. If one has no further information for the
27. Semi-linear equations; the Q-method 27.2
Definition
Ltr/
> 0. Then
is the collection of all (complex-valued) functions
f(.r. y) with x E
and y E
such that
=j 27.3
391
dy
flf(•,y)
0. Recall that the real spaces
in (26.8) are specifications of
and
(25.1). (25.2). 27.4
Theorem
Lt is E N.
0
(27.36)
With the modifications indicated one can follow Step 1 of the proof of Theorem 27.4. Hence there is a (uniquely determined) solution u E 13w' (R") of
u(x) =
c
+ id)'
o
+ H(x).
(27.37)
The first term on the right-hand side of (27.37) belongs even to Since H E we get finally u E 2 The proof of part (ii) is the same. Now we can choose E [0,2] in such a way that s +.\ avoiding the exceptional cases in Theorem 25.8 concerning the boundedness of T+.
27.7 Two comments We are more interested in presenting the Q-method than in the most general applications. This applies both to Theorem 27.4 and, to an even larger extent. to Theorem 27.6. In connection with (27.24) we relied on the properties (27.27)
398
IV. Truncations and semi-linear equations
for its Green's function. Here C E L1 (R'2) follows from the exponential decay of C(y) if oc. But these properties of Green's functions, positivity and exponential decay, are well known for a large class of second order elliptic differential operators in R'1. All that one needs may be found in [Dav89]. This
applies also to some functions of these operators, in particular to fractional powers. In other words there is a good chance to replace + id in (27.24) by some classes of elliptic pseudodifferential operators.
Secondly, we add a comment on the supersolution technique. It goes back to 0. Perron, [Per231, in connection with boundary value problems for the Laplacian in bounded domains in R". This method was used later on, especially in the 1970s and 1980s, in connection with nonlinear equations of type
Lu = f(x, u)
and
Lu = f(x, u, Vu),
(27.38)
where L is a second-order linear elliptic differential operator. The underlying function spaces are preferably Holder spaces, but occasionally also classical Sobolev spaces. In contrast to our very special nonlinearities 1(u) = lul and the admitted nonlinearities in (27.38) are described in qualitative 1(u) = terms, by smoothness and growth conditions. Details of the use of the supersolution and subsolution technique in the just-outlined context may be found in [Ama76], [AmC 781, [KaK78], [FKY87], to mention only a few typical papers. At this moment it is not so clear whether the techniques developed in these and related papers can be combined with the methods presented in this chapter. But it might be worth consideration. As an indispensable ingredient one needs sharp mapping properties of composition operators of type u f(z, u) in function spaces of type We refer to [RuS96] where one finds and far-reaching assertions in this direction. 27.8
Local singularities
As just said there might be many possibilities to apply the Q-method. We have a closer look at how to incorporate local singularities in the right-hand side of (27.24), which then looks like
+ id)u(x) = E(1 + b(x) 0. To find out which type of singularities can be admitted one may consult Fig. 27.2. In the proof of Theorem 27.6 we used suitable points on the vertical line segments of length 2 characterized by o, say the point C representing for example H (R"). However, instead of going down from A to C on the vertical line, one may use sharp embeddings from A to D and afterward Holder inequalities at level a to step from D to C. This is the point where b
27. Semi-linear equations; the Q-method
399
I___ Fig. 27.3
in. The underlying theory has been developed in [SiT95] and may also he found in [ET96J. 2.4 and [RuS96], 4.8.2, pp. 238—239. We describe a special case given in [ET96], p. 54, formula (17). Let
1
