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d sequence ~ = {(j } ~ 1 of complex numbers there is J E HOO(D) such that J(Zj) = (J for 1 :::; J < 00 In 1958 Carleson gave an affirmative answer and moreover characterized all sueh interpolating sequences in the disk. Implicit in Carleson's solution, and explicitly realized by Shapiro and Shieldh in 1961, is the equivalence of this problem with eertain Hilbert hpace a.nalogues which we now describe Let H2(D) be the Hardy space consisting of all holomorphie functions J(z) = ~:'=O anzn in the unit disk with IIJIII-[2(I» = V~:=o la n l2 < 00. The Hardy space H2(D) can be identified with the closed subspaee 1-£2('11') of £2('11') given by
Indeed, simply associate each J(z) integrable fUIlction
=
~:'=O a"z" in the unit disk with the square
J* on the circle satisfying
Pen) =
{ao'
~~~ ~ ~ ~
. The
inner product on H2(D) is then defined by that on 1-[2('11') inherited from £2('11')
2.3. Carleson's duality proof of Interpolation
13
if I(z) = 2:~=oanzn and g(z) = 2:~=obnzn. Cauchy's Theorem applied to polynomials, and then followed by a limiting argument, f:ohows that
~
I(z) =
2m
(
f*(e i9 ) d(e i9 ) =
Ie e,9 -
Z
~ 271"
f*(e~9)
(
Ie 1 -
e .9 Z
dO = (f, kz)
where the "reproducing kernel" k z E H2(lIJl) satisfies kz«() = l!z( and k;(ei9 ) = ,g. We note in passing that if I(rkz) is a rapidly converging sequence in H2(D) (where the radii r k tend to 1 from below), then f*(e i9 ) = limk-+oo l(rkei9) for a e. O. Moreover, Fatou's Theorem shows that we actually have f*(e i9 ) = linlr -+ 1- l(re i9 ) for a.e 0, but we will not need either of these facts for the moment. The above computations show that for fixed ZED, the linear functional I -> I(z) is continuous on H2(lIJl) with norm Ilkzll = v'(kz , kz) = v'kz(z) = (1-lzI 2)-L
l-i..
Thus the map I -> {/(z)V1 -Iz) 12}~1 is bounded with norm 1 from H2(lIJl) to eOO(Z). The relevant quef:otions that Shapiro and Shields then asked were these. When is this map onto (respectively into) the smaller Hilbert space £2(Z)? In other words, when is the ref:otiction map RI = {/(Zj)}~l onto (respectively into) the weighted space
£'(p)
~ {~~ {~;}~, ' 11,11,,(,) ~ ~ 1,;1' ";
..f(Zi))2 = (I, kZj - >"kZi /
1
< IIfl\~2(][})
(1 - IZj 12) {llkzj 1I~2(J[)) + 1>..1211kzi \1~2(J[) - 2Re (kzj , >"kz.) }
< C
C (kz j C
if we choose
,
A
1
Ilkz] - >"kz.II~2(][})
~Zj)) { kZj (Zj) + 1>"1 2 kZi (Zi) - 2 Re >"kZi (zJ) }
{2 _2Jk
=
Ik zi (zj)1 } zj (Zj) Zi (Zi)
Jk
Ik •. (Zj)1 k. Zj U· h'd . ') ( ) . smg tel entIty k •• (Zj k •• Zi \ Zi - ZJ
[2
l-zjzi we have 1
and hence
-
(1-lziI2)(I-\ZjI2) Ikzi (zJ)\2 11-zjzil2 = kZi(Zi)kzj(zJ)'
-I
Zi - ZJ 12 1- ZjZi
(2.6)
~ (1 _ ~ )2, 2C
I:~:;;;. I~ c > 0 for all i #- j, which easily yields the corollary.
ReIIlRrk 2.3 The useful identity (2.6) arises very often in function theory on the disk, and generalizes to other situations, e.g. (6.19) on the unit ball. We note that if R maps HOO(l!)) onto lOO(Z), then the open mapping theorem easily shows that Z is separated. Indeed, there is c > 0 such that R (unit ball in HOO(l!))) :J cxunit ball in £OO(Z). Thus for fixed i #- j there is f E HOO(l!)) of norm at most one with f(Zi) = c and f(zJ) = O. If CPw(z) = l~~z is the involutive automorphism of the disk l!) that interchanges 0 and w, then 9 = ! 0 CPz. maps l!) to l!) and 9(0) = !(Zj) = O. Thus the Schwarz lemma implies that (recall CPZj 0 CPZj is the identity) ]
which shows as above that Z is separated. Thus separation is a necessary condition for either £2(1-') C R(H2(l!))) or £OO(Z) C R(HOO(lJ))), and we may as well assume it from the outset. Now we can state Carleson's Interpolation Theorem with contributions from Shapiro and Shields.
2.3. Carleson's duality proof of interpolation
Theorem 2.4 Suppose that Z
15
= {Zj}~l
C
JDl is separated, i.e.
I ?:. c > 0 for all i i= j, I1Zi- ~j ZjZi let R be the restriction map Rf = {f(zj)}~l and f.1z the following conditions are equivalent: 1. R maps HOO(JDl) onto £OO(Z), 2. R maps H2 (JDl) onto £2 (f.1 z), 3. R maps H2(JDl) into £2 (f.1z) , 4. f.1z(T(I) :'S C III for all arcs leT. 5. infj
IT iofj
(2.7)
= 2:}:1(1-lzjI2)8zr Then
I t~;:~ I == 6z > O. J
,
Here the tent T(I) over the arc I is the convex hull of I and ZI = re ifJ where eifJ is the midpoint of I and r = 1 - ~~. Marshall and Sundberg identified the crucial interplay between the spaces HOO(JDl) and H2(lIJ) here, namely that HOO(lIJ) is the multiplier algebra MH2(JfJJ) of the Hilbert space H2(lIJ). This will be exploited in the proof below Definition 2.5 If R maps HOO(lDl) onto £OO(Z) with kernel N, then Ii : HOO(lDl)/N ---4 £OO(Z) is invertible by the open mapping theorem (Theorem A.29 (p. 158) in Appendix A). Denote by Mz the norm of the inverse map.
The proof below will show that
~ 6z
< Mz < C~ 8z
(1 + ln~) 6z
.
There are examples on page 284 of Garnett's book [20] of finite sequences Z with M z ~ (j~ In (j~ for arbitrarily small 6z > O. Remark 2.6 We have stated the theorem for the unit disk lDl, but in the course of the proof we will switch back and forth between the disk and the upper half plane 1U, using the setting most convenient for a given argument. In fact, arguments involving 5 are easiest in (a bounded region of) the upper half plane, while arguments not involving 5 are better handled in the disk. Here is the reason. Using the Cayley transform Z = i that takes lDl one-to-one and onto 1U, we see that the Blaschke product in l!J with zeroes Z = {i, ... , i} U {Zj }~1 C 1U\ {i}, where i has multiplicity k ?:. 0, is given by
i+:
B(Z)=(Z-~)krrOO Z + 2 J=l .
!l+z;!z- z
J,
1 + zJ~ Z - Zj
and that the corresponding Blaschke condition is 00
'~ " j=l
where
Zj
=
Xj
+ Yj·
YJ 2 1 + IZjl
< 00,
In the special case that Z is bounded, then the Blaschke 00
condition becomes simply 2:}:1 Yj
< 00, and the infinite product IT j=l
:=:' converges
uniformly on compact subsets of 1U without the compensating factors
J
I~::~
I. Thus
16
2. The Interpolation Problelll
we can redefine the Blaschke product in this case to be simply B(z) =
00 n :=~, 3=1
3
and with little loss of generality we will do this in the proof that follows. Proof. Conditions 1, 3, 4 and 5 are equivalent in more general situation~ where 2 is no longer sufficient Thus we begin by showing the implications 1 =?5 -{=> 4 -{=> 3 =?- 1 00 (1 implies 5) Let B(z) = ~=~ be the Blaschke product in the upper half
n
j=l
J
space llJ associated with Z and set
B;(z)
Z-z
= __ J B(z) z - Zj
By the open mapping theorem, the one-to-one map f --+ {f(Z)}ZEZ from HOC IN onto £00, where N is the kernel of the restriction map, has a bounded inverse If Afz is the norm of this inverse, and fJ E HOC solves fJ (Zi) = 8j with IIf} \\00 :=::; Mz + c, c > 0, then by Corollary C 7 (p 187), 9} (z) = ~~~~) satisfies jig} 1100 we obtain
\Bj~Zj)\ = I~;~;/)I = \9;(Zj)\ ~ \\9jIl00 = > ° Thus {}z = inf; \Bj(zj)\ ~ Jz > °
IIfJlloc
= II Ii \\00
and
~ M z +€,
for every € (5 implies 4) We continue to work in the upper half plane First we ~how that 5 implies the following restricted Carleson condition for the associated measure p, = p,z = L~l YjOz j where zJ = Xj + iYj·
p,(T(Zk» =
L
Yj:=::; CYk = C x height of T(Zk),
(2.8)
zjET(z.)
where the tent T(Zk) is the equilateral triangle with vertex Zk and opposite side on the x-axis Fix k. If Bk(Z) = TIj #k ;=~ denotes the Bla..o;chke product with zeroes Z \ {Zk}, t.hen from - In t ~ 1 - t for t > (which follows since In is concave and t - 1 is t.he t.angent line to the graph of In at (1,0», we have that
°
2ln
~ Oz
> -In\Bk(zk)\2 = - LIn 1Zk - ZJ 12 Zk -
}#k
>
L
)#
(1 -1
Zk Zk
(2.9)
z)
=ZJZJ 12) = LNk \Zk4Yk~ - z,\
2
For future reference we note the converse inequality -In \Bk (Zk)\2 :=::; (1
+ 21n ~) L
4Yk~ 2' z)\
c 3'# \Zk -
where c is the separat.ion constant in t.he upper half plane analogue of (2 7)
I
z1 -
Zi
Zj -
Zi
I~
c
> 0,
i
i- J.
This follows from the following reverbe tangent line inequality -In t
~
(1 +
2ln ~) (1 - t)
for c2 < t
} 2Nr ::; C 2Nrc 2
II Wr,f:,NII =
-7
O. (220)
Indeed, the natural approximation /-tr,c,N
L
=
/-t(K)8zJ
j KjnLd'
obviously tends to /-t weak star while the error W.,f:,N tends to 0 strongly MOIeover, the measures /-t.,f:,N have Carleson norms bounded by 2 if IIw. c.NII ::; r. Indeed,
/-tr,f:,N(Q) ::; /-t(Q}
+ W,.f: N(Q)
::; £(Q)
+ W" c,N(Q} ::; 2£(Q),
since /-tr,f:,N(Q) -=I- 0 implies that £(Q) ;?: r. Momentarily fix such a meabure v = /-tr,f:,N with (2.21) (which using (2 20) can be accompli&hed by choosing N laIge enough depending on £}, and note that v has the form
r and
where 0 ::; kj ::; N. Note also that v(Q) ::; 2£(Q) for all dyadic cubes Q on JR.. Consider a new sequence {(j}f=I' such that each point in Z = {Zj}f=l occurs exactly kJ times in {(j}f=1 Then with (j = ~1 + irJj we have 1 00 v = -N~ "rJ·8r )"3 )=1
We now decompose the sequence {(j}f=l into 6N subsequences Zl, . ,Z6N each of which has separation constant at least ~ and Carleson norm at most 4. For this we first note that if T( Q) denotes the top half of a square Q resting on the x-axis, then
#{j : (j
E
T(Q)} ::; 3N.
(2.22)
2.4. Peter Jones' constructive proof of interpolation
Indeed, we may assume that l(Q)
#{J : (j
E
T(Q)}
~
27
r, and we then have
=
where the final inequality is (2.21) Order the points (J = ~j + iTJJ (with the necessary repetitions) in the strip Sn = {( = ~+2TJ' 2- n - 1 < TJ:::; 2- n } accordingtonondecreasingrealcomponent~j' and partition them into new sequences Yt, 1 :::; £ :::; 2N, by putting every 3Nth point in order into a given Yt. From (2.22) we see that the points in YtnSn have separation constant !. Thus if we define Z2t = Un evenYt n Sn and ZU-l = Un oddYt n Sn for 1 :::; £:::; 2N, then each of the sets Zk has separation constant at least ~. It remains to show that the measure f-£YI = L(,EY, TJjb zj corresponding to Yt has Carleson norm at most 4. However, if Q is any dyadic square of side length £(Q) resting on the x-axis, then we have /-Ly,(Q)
L L
=
TJj
o,N->oo Ur,e,N as a solution to (2.18) in some suitably useful way. Of course we will let r, IS -+ 0 and N -+ 00
28
2. The Interpolation ProbleIIl
so that both (2.20) and (2.21) hold. At this point it is convenient to pass from the unbounded upper haIf plane to the compact disk. Then Ur,g,N is a rational function in the disk and by Green's Theorem (or the residue theorem) we have for each J E Hl(]]}),
1. -2 7r~
iTf J(z)urg' ,N(z)dz
1 () ~l
-1. 27r~
J(z) ~U1 g N(z)dZ Adz
J[JI
uZ
J(z)d/1-r.g N(Z)
Since the restrictions of the function:> U r g N to the cir de lie in a fixed norm closed ball in L=('JI') = Ll ('JI')*, Corollary A.45' (p. 167) in Appendix A shows that there is a sequence of these restrictions that converges weak star to a bounded function U on 'JI'. We may assume (2.20) holds ab well for the sequence, and in fact the stronger assumption that IIw Nil - t 0 From this we then have that IIwrg.Nllcarleson - t 0 and so by Corollary C 9 (p 187), y
:
We now claim that the theory of the Poisson integral implies
(223)
To see this we use Corollary C 8 to write! = gh with g, h E H2 and Iigli H 2 IIhIl H2 • Now if zJ is the center of K J , then for Z E K J ,
1I!IIHl
2.4. Peter .Jones' COnstructive p~f' of' Interpolation
29
since the K j have hyperbolic diameter about c. Of course we also have !g(z)\ IlPg(z)! ~ IP !g! (z), with similar estimates for h. From these estimates we have
lin
-1, 9(Z)h(Z)dI!r,t;,N(Z)!
g(z)h(z)dp,(z) =
L (
lK
J
~L
(
=
{[g(z) - g(Zj)]h(z) + g(Zj)[h(z) - h(Zj)J}dp,(z) j
{\g(z) - g(Zj)! Ih(z)!
+ !g(Zj)llh(z) -
1K3 ~ Cc ~ [j IP Igl (z)1P Ihl (z)dp,(z)
h(Zj)l}dp,(z)
1
1
~ Cc ( l (IP Iglfdp,) ~
1
( l (IP !hJ) 2dp,)
2
Now we use the nontangential maximal function NG(ei6 ) == sUPzEre !G(z)l, where r6 i& the convex hull of {ei6 } U ~1Dl, to obtain that (see the proof that 4 implies 3 of Theorem 2.4 above)
l (IP Igl) 2dtt
~C
i
~C
NIP Igl (e i6 )2dfJ
i Ig!2
=
C
IIgll~2'
and similiarly for h Thill>
IlI(Z)dIL(Z) -
10 I(Z)dI!.,t;,N(Z) \ ~ Cc IIgll H2 IIhllH2 = CC II/11H1 ,
and we now obtain (2 23) as c ~ 0 Altogether then we have that u
tz
~
{ I(z)u(z)dz =
lT
2n
Now suppose that dJL(z) below we see that
= JL in the sense of Green's Theorem: 11"
I(z)dtt(z).
~ ( G«() d( 1\ d(
in (-
211"~ satisfies F E Cl (1Dl) n CO(jj) and solves tzF 211"~
(
lID
= G(z)dxdy is a Carleson measure. Then from (3.21) F(z) =
~
~
iT( l(e
i6 )[F(ei6 ) -
Z
= tt Thus by Green's Theorem
u(e i6 )]d(ei6 )
~[/(z)F(z)]dZ 1\ dz - ~11" io( I(z)dp,(z) lD uZ 1 -2 l(z)G(z)az 1\ dz - ~ I(z)dtt(z) = 0, n 11" n
1. { = -2 1I"t
=
·1
11"~
1
for all 1 E Hl(IDl), so that F(e i6 ) - u(e iO ) = h(ei6 ) on l' where h E HOO(IDl) Thus if we extend u to the disk by
u(z) == F(z) - h(z),
z E 1Dl,
then u E Cl (1Dl) n c°(ij) is a solution to (2.18) that is bounded on the circle l' with supremum bound controlled by the Carleson measure norm of tt.
30
2. The Interpolation problem
2.4.2 A continuous version. Jones also constructed in [22] a bounded (on the boundary) solution to (2.18) directly from a kernel modelled on the linear operator of interpolation in (2.19). Here are the details in the setting of the complex upper half plane C+ = lR.~. Define
K o(u,Z,()=2i( 7r
z-
~~(Z -
()exp{j {
Jlmw:$Im(
for 17 a positive Carleson measure on C+ and z, ( Carleson measure on C+, set
(z=i_+(~_)du(W)}, w w E
C+, and for 11 a complex
Theorem 2.10 Let 11 be a complex Carleson measure on C+. Then w~th 80(11) as above we have 1. 8 0 (11) E Lioc(C+) 2. :Z80(11) = 11 in the sense of distributions
3.
'
f fc+ IKo( II~WL, x, () Id 1111 «) ~ C 111111 Car for all x
E lR.
OC+, so
~ C 111111 Car· Proof: We first prove assertion 3. We have
1180 (11) lIu">(IR) Re
(_~_. ) = (-w
Re (i(, -W») = Im( + Imw < l(-wl 2
if Imw ~ Im( It follows that if 17
Rei (
= ~I~I ' ~
_ i du(w)
Jlmw:$Im( ( -
1(-wI 2
w
0 and all i -=1=], and the Carleson condition 4 above can be rewritten oc 1 ---28zj is a Carleson measure for H2,
L
J=I
JJkzJ
where kz«) = I~ 1 remains unsolved for all other Besov-Sobolev spaces and their multiplier algebras.
CHAPTER 3
The Corona Problem The continuous functions G(K) on a compact set K provide the prototypical example of a commutative Banach algebra Gelfand's theory shows that an arbitrary Banach algebra X has a natural homomorphism~: X -+ G(M) with kernel radX (the intersection of all maximal ideals) into G(M) where M is the maximal ideal space of X,
M = {
'> -
Z
and
- / /
O~1fJ) az A dz
- / F1fJdz = 0 Using (3.20) with the above and Fubini's Theorem we thus obtain
~~ (z)1fJ(z)dz A az
/ /
- / / F(Z/';: (z)dz
A
_//{_1 / ill>f ( -
az
G«()d(Adz'}O::(Z)dzAaz
21rZ
Z
- / iDf G«(){~// 21rZ =
/
uZ
~(Z)dzAaz}d(Ad(
(- z
/ G( ()1fJ( ()d( A de·
Since this holds for all1fJ E C~(D) we conclude that ~f = G in D, i e (3.21). Now every solution b E C 1 (D) n cOi) of (3 12) has the form b = F + h where h is analytic (~; = 0) and lies in the disk algebra A (D) = H= (D) n C (D). The minimal norm of such solutions b to (3.12) is therefore given by inf
hEA(D)
lIF + hll£'x>(1r) = sup{
/1
211"
0
I
dO Fk-. k 21r
E
HJ(D), Ilklll S; 1}.
(323)
Indeed, for any Banach space X with closed subspace Y, the Hahn-Banach Theorem gives the isometric embedding (XjY)* ~ y.L (Proposition A.41 (p. 166», i.e for x E X and x* E X*, inf{lIx + yll . y E Y}
= sup {I(x, k)1 . k E y.L, IIkll
S; I}.
(324)
Now take X = Cpr), Y = A(T) and (I,p,) = f~1I" J(O)dp,(B) in (3.24) The F. and M. Riesz Theorem C.lI in Appendix C shows that y.L = HJ(T), which yields (3.23).
We may suppose that k E HJ(RJDl) for some R> 1 From (3.21) we have
L.(Fk) =
4~~(Fk) = 4~ (OF k) = 4~(Gk), OZ Oz oz Oz OZ
and Green's Theorem in the form
h
(uvn - unv) =
I1
(u L. v - (L.u)v),
45
3.4. Other corona probleDls
with u = Fk and v = log~, now yields using with 6v = 80 and k(O) = 0,
l
k
o
F(eiO)k(e iO )_ • 21r
IJl ~ Jl
-
21r
Vn
= 1 and v = 0 on 'll', together
6(F(z)k(z)) log -I 1 Idxdy Z
ITJ)
k'(z)G(z) log
2J rk(z) 8z (z)
+; I
8G
JITJ)
1~ldXdY 1 log ~dxdy
+ II.
We estimate integral I using the factorization k = Ig with I, 9 E H2(]!))) and 1 (Corollary C.8 (p. 187) in Appendix C), along with the CauchySchwarz inequality as follows
11/1I2,lIg1l2 ::;
III
< 0 < 0
J
llUg'
+ g/')(z)IIG(z)1 (1 -
Izl2)dxdy
(J ll/(z)12IG(z)12 IzI2)dXdy) (J llg'(z)1 2 Iz I2)dXdY) (1 -
1
"2
1
X
(1 -
"2
+ a similar term corresponding to 9 /' , < 0 IItLIIICar(H2(ITJ»)) 1I/I1211g112 ::; 0(0, IIfIlHOO(ITJ»))' by (3.16) and the case m = 1 of (3.18). To estimate I I we proceed as follows:
IIII
0,
Z
E]J).
j=l The Gelfand theory reduces the proof to constructing {gj};=l in HOO(I!») n D(]J) satisfying J
Lh(z)gJ(z) = 1,
z E]J).
j=l
Let W 1 ,2(]J) as in Definition B 3 (p. 174) in Appendix B be the Hilbert space of functions f in the disk with locally square integrable weak derivatives normed by
IIfll~,
2(1f)
=
1IV'
f(z)12 dxdy.
See Theorem BA (p. 174) in Appendix B for the completeness of W 1,2(]J). As in the proof of Carleson's Corona Theorem for HOO(]J), it suffices to solve the d-bar equation (3.12), namely {}b m(z) = G(z), Z E]J), but this time with bE C 1 (]J)nC(D)nWl,2(]J), and most importantly, with control of both the L OO (,][,) and W 1 ,2(]J) norms of b in terms of the HOO(]J) and V(]J) norms of the corona data f = {fJ};= 1· Recall that G (z) = 'Pi (Z) 'P j (z) for any fixed i, J where 'Pi(Z) =
Ih(z)12~lfJ(Z)12
{}
tz
and
1
J
m'P)(z) = (E:=1Ifk(Z)1 2)2
~fk(Z){fk(Z)f;(Z) -
f;'(z)h(z)},
for 1 ~ i,j ~ J. In the present situation, unlike in the proof of Carleson's Corona Theorem, the measure dJ.L(z) = IG(z)ldxdy is actually an H 2 -Carleson measure:
J.L(T(I»
=
r
IG(z)1 dxdy
iT(J)
< C
(r
iT(!)
~C
r
If'(z)1 dxdy
iT(J)
If'(z)1 2 dXdY )
~ IT(I)I! = C IIfliv III·
47
3.4. Other corona problems
Thus from the previous subsection (or Remark 3.7 (p. 42», we see that with
F(z) = -1. 27rt
l
G«()
-
- - d ( Ad(
D (- Z
as in (3.20), then tzF = G and inf
hEA(D)
IIF + hll V", (T)
:::;
C IIf!!Hoo .
Thus there is a solution F + h with IIF + hIlLoo(T) :::; C(l\fll v + IIfllclO). We also have a good estimate for the W 1,2(lJ) norm of F using that tzF = G and {) 1 [G«() -
in «( _z)2d( A d( =
{)z F = 27ri imply
BG(z)
tz F is the Beurling transform BG of G = tzF given by
11-
BG(z) = -2. 7rZ
ID
K«(,z)G«()d( Ad(,
where the kernel K«(,z) = «'!Z)2 is a singular integral on C the theory of singular integrals in [46] we then have
IIFII~Tl 2(ID)
=
klV
= ]R2.
Indeed, from
F(z)12 dxdy
ll!F(Z)r dxdy+ l\:zF(Z)r dxdy
10 IG(z)12 dxdy + 10 IBG(z}12 dxdy and
10 IBG(z)1 2 dxdy
< C
k IG(z}12 dxdy
< C(lIflloo)
L
Jf'(z)1 2 dxdy
=
C(l!fll oo ) IIfll~,
by the boundedness of the Beurling transform B on L2(lJ)}. Remark 3.8 The kernel K«(, z) = the standard kernel estimates
«.!Z)2
of the Beurling transform B satisfies
IK«(, z)1
I, and define x E X.
Tnf(x),
Then
and
~ ("f I~TII) n T n+ f(x) 1
Th(x)
"f
IITII ~("f ItTl1 )nTn f(x)
"f IITII (h(x) - f(x))
and IIS(v)lIoo ::::; C !!vl!car S C !!/lllv-car where d/l(z) = !G(Z)!2 dxdy is a V-Carleson measure.
53
3.4. Other corona problema 1
We must now also show that S(v) dp,s(v)
==
Xi (T), i e. that IVS(v)(z)1 2 dxdy ITE
is also a 'D-Carleson measure. Of course /zS(v) = G in Jl} and we obtain immediately that! tz-S(v)(z)!2 dxdy = dp,(z) is a D-Carleson measure. Rather than attempting to use the singular Beurling transform B to estimate -IzS(v), we instead use the following clever device of Xiao introduced above (but used first by Xiao here) that greatly ameliorateb the singularity. We introduce the function S(v)(z) =
Jl
K(a,z,()dv«)
where K(a,z,()== 2i 7r
I-~122exp{J
11 - (zl
{
Jlwl~Io(T)
= distL"o(Cp, BOO) ~
Ilf
=
anznll
n=l
IJP-cpIlBMO(T)
(4.1)
BMO(T)
where P_ is orthogonal projection from L2 to H:'
See the next subsection for the definition of BMO and Fefi"erman's Duality Theorem 4.4 (p. 59). The bounded function cp in Theorems 4.1 and 4.2 is called the symbol of the Toeplitz operator B and Hankel operator A respectively. The symbol
56
4. Toeplltz and Hankel Operators
of a Toeplitz operator is uniquely determined (since all of its Fourier coefficients are prescribed), while that of a Hankel operator is only determined up to a bounded holomorphic function (since only its negative Fourier coefficients are prescribed). Note that with P+ the orthogonal projection from L2 to H2 and P_ the orthogonal projection from L2 to H'3..., we have
Indeed, for k
~
0 and j
(p_(cpzk),zj) = (
~
1, we have
L
ij3(n)zn+k,zj) = ij3(-k - j) = ak+j = (Azk,zj), n+k II (Re Hl)*
= C IIAbll(HJ)* .
Similarly the linear functional ~ defined by
hjh i lh, = -
III
Iq>al ::; 1Iq>lIl1allReHl ::; C 1Iq>1I.
Taking the supremum in 1 shows that
~ f == q>1 =
to
f
E £2,
alii
62
4. Toeplitz and Hankel Operators
is bounded on ReHl with norm that
1I~ll
=
1Ib(w) given by the Poisson integral. Then it is enough to prove that
But this reduces to the case w = 0 after changing variables with the automorphism that interchanges wand 0, and this case is just IT
b E L2 with IT b =
o.
Ib(eit ) 12 dt
= IT
Ib(eit ) 12 dt
for
See Corollary 2.5 in Chapter VI of [20} for details.
Remark 4.6 Elementary functional analysis yields the characterization (ReH1(11'»* ~ LOO('Jl')
+ Loo('Jl').
Indeed, the embedding I: ReH1('Jl') -+ £1('Jl') EB L1('Jl') defined by If = (/,1) is continuous with closed range, and so Proposition AAO in Appendix A shows that
(L1('Jl') EB L1(11'»* j(IReHI ('Jl'».L (L 00 ('Jl') EB L 00 ('Jl') ) j {(h, h) : h E L 00 } LOO('Jl') + Loo('Jl'), since
IT lh = - IT Iii implies (IRe HI ('Jl'».L = ((g,h) E LOO('Jl')
+ LOO('Jl') :
1
= {(g, h) E LOO(T) + £ClO(11') : 9 -
Ug + lh) =O,f E ReHI('Jl')}
h = OJ.
64
4. Toeplitz and Hankel Operators
4.2 Compact Hankel operators
We begin with Kronecker's characterization of finite rank Hankel operators. Theorem 4.7 Let Hip be a Hankel operator with symbol cp. Then the following conditions are equivalent· 1. Hip is of finite rank, 2. ker Hip = b1 H2 for a finite Blaschke product b1 , 3. ~cp E HOO for a finite Blaschke product ~, 4. P _ cp is a rational function. Moreover, we have the following equalities: rank(Hip) = #{zeroes of b1 } = #{poles of P_cp}. Proof: First we note that for f E H2 and g E H'3..., (P-cpf,g)
(Hipf,g)
=
(cpf,g)
=
(f,<j5g)
=
(f,P+<j5g)
(I, zP_zcpg) = (I, zHipzg) = (zHipzg,/) Thus f E ker Hip if and only if 1 1. range(T) where Tg = zHipzg. Now zg ranges over H2 as 9 ranges over H'3..., and since multiplication by z is an isometry, we see that rank(Hip) = dim range(Hip) = dim range(T) = dim(ker Hip)-L. Now ker Hip is invariant under multiplication by z (see (4.27) below) and so by Beurling's Theorem C.16 (p. 193) in Appendix C, ker Hip = bH 2 for an inner function b. Thus rank(Hip) = dim(bH2)-L.
We now claim that dim(bH 2 )-L is finite if and only if b is a finite Blaschke product Indeed, if b(z)
=
N
N
n (l~;Z)mn = n bAn (z)m,. is a finite Blaschke product, n=l n=l
n
N_ bm" H2 where then bH 2 = n n-l An
(b~n H2)-L
{J E H2 : (b'!:"ng, f) =
v'!1n-l{bt k } = J=O
An An
= O,g E H2}
Vm,,-l { J=O
(410)
(An - z)j } (1 _ AnZ)J+l '
and kA,,(z) = l-~nz is the reproducing kernel for H2 at An ElI}. This is obvious if An = 0, and the general case follows by composing with the automorphism CPA..{z)
= bAn (z) since bAn 0
CPA..{z)
=
N
n b~n we have
z. Thus for b =
n=l
(4.11) and if ker Hip
= bH 2, then N
rank(Hip) = dim(bH 2 )-L =
L
mn,
n=l
the number of zeroes of b counting multiplicity in lI}. Conversely, if dim(bH 2 )-L is finite, then (bH2)-L is spanned by the rootvectors of S*, since by Beurling's Theorem C.16, the orthogonal complements (bH2)-L are the S· -invariant subspaces of H2. Recall that if S* on (bH2)-L is represented by
4.2. COlnpact Hankel operators
65
a complex m x m matrix A in Jordan canonical form, the rootspaces of A are the subspaces ker(A - AnI)mn, 1 :::; n :::; N, where An are the eigenvalues of A and mn is the size of the largest Jordan block with diagonal element An. The elements of a rootspace are called rootvectors. However, S*J =).J with J E H2 if and only if [en) = Ai.(z) = l!Xz is the unique (up to a constant multiple) corresponding eigenvector. More generally,
°
ker(S* - "XI)m = V,?="(/ {
zi.} ,
(4.12)
(1 - AZ)J+l
follows using the identities ker( S* - "XI) m
H2
(S - AI)mH2
e (S -
.\I)mH2,
{J E H2 . J(j)(A)
= 0,0:::; j :::; m
-I},
(d~Y (f, k>.) (J, (d;Y 1 ~ ).z)
J(j) (A)
=
l' (/, (1 _
~z)i+l)'
O:::;j:::;m-l.
It follows that the Jordan canonical form of the restriction of S* to the invariant subspace (bH2)i. has, for each 1 :::; n :::; N, a single mn x mn Jordan block with diagonal elements An, and has rootspace V,?='O-l{ (l-::Z)Hl} corresponding to the eigenvalue An. Thus dim(bH2)i. = L:~=l mn and N k (S* (bH 2)i. -- Vn=l er -
~I)m" An
- VN
n=l vmnJ=o
-
1 {
zi
(1 _ AnZ)J+l
}.
(4.13)
However, we have equality of the linear spans in (4.10) and (4.12) when mn = m and An = A· m-l {
Vi=o
(A - Z)j}
(1 _ AZ)i+l
m-l {
= Vj=o
zj
(1- Az)i+l
Indeed, this is easily established by induction on musing
}
IAI < 1
Combining (4.11)
N
and (4.13), it follows that b =
IT
n=l
b';n is a finite Blaschke product. T~
Thus rank(H'f» < 00 if and only if there is a finite Blaschke product b such that ker Hcp = bH2 , i e. H'f>b = 0, which is in turn equivalent to cpb E Hoo or cp
= bJ
tfH
for some / E Hoo. Now if b =
= Q(z)
+ h(z) where hE Hoo
N
IT b';",... n=l
is a finite Blaschke product, and if
and (4.14)
is the principal part of the meromorphic function {" then
66
4. Toeplitz and Hankel Operators
since
2:':=1
I-tz
Z-\n
E H:'('Jr). Thus P-'P = Q is a rational function with mn = rank(Hcp) poles in the disk. Conversely, if P-'P is a rational func-
tion with principal part Q as in (4.14), then P -'P
=
Q and so 'P
N
Il
br;!:nn E H OO .
n=I
Definition 4.8 Given T . HI ---+ Hz a bounded linear operator between Hilbert spaces HI and H 2, we define the essential norm of T by
IITIless
== inf{IIT - KII : K is compact from HI to Hz}.
Sarason's Theorem will allow us to compute the essential norm of a Hankel operator. Theorem 4.9 The set HOO('Jr)
+ G('Jr)
is a closed subalgebra
01 the
Banach
algebra LOOer). Proof: The natural embedding j . G('Jr)jA('Jr) ---+ LOO('Jr)jHOO('Jr) is an isometry since LOO('Jr)jH=(T) can be identified with the second dual of G(T)jA(T). Indeed, (G('Jr)jA('Jr» * = HJ and (HJ)* = Loo(T)jH=(T). Thus the image j (G ('Jr) j A('Jr») of G ('Jr) j A('Jr) in the quotient space L = ('Jr) j H= ('Jr) is closed. If 7r : Loo('Jr) ---+ Loo('Jr)jHOO('Jr) is the quotient projection, then Hoo(T) + G('Jr) = 7r-I{)(G('Jr)jA('Jr))} is closed as well. Finally, the equality
H=('Jr)
+ G('Jr) =
closureLOO(T)(U~=oznH=('Jr)
shows that Hoo('Jr) + G('Jr) is an algebra. Here is a direct proof that the map J above is an isometry. We use the Poisson kernel f'/(re iO ) = Pr * 1(0). Since Ir ---+ I in L= for IE G(T) and Il/rlloo S 11/11= for I E H= ('Jr) we conclude that
distu,,"(j, Hoo('Jr)) = distLoo(j,A(T»,
IE G('Jr)
(4.15)
Indeed, S is trivial and for 9 E H= ('Jr) we have III - gll=
> :~ 1I(j - g)rll= > limsup{111 - grll= - III - Irlloo} r--->l
lim sup III - grll= ~ distu",(j,A('Jr). r--+l
From (4.15) we see that the natural embedding j : G('Jr)jA('Jr) is an isometry
---+
LOO('Jr)jHoo(T)
Here is the computation of the essential norm of a Hankel operator. Theorem 4.10 Let'P E L=(T). Then
IIHcplless
=
dist£,X'(T)('P, H=('Jr)
+ G('Jr».
Proof: If p is a trigonometric polynomial, then Hp is of finite rank by Kronecker's characterization, hence compact. Since the trigonometric polynomials are dense in G (T), (4.1) shows that if lEG ('Jr), then H f is a norm limit of the compact operators HPn where Pn ---+ I in G('Jr), and thus H f is compact. We then have from Nehari's Theorem,
dist Loo(1r)('P, H=('Jr)
+ G('Jr))
inf dist LOO (T) ('P -
fEC(T)
inf
fEC(T)
I, HOC> ('Jr»
IIH
o.
XEM-L,lIxlI:O:;l
Let Xk E M.L, IIxkll ~ 1 be such that lim (TXk' TXk)
k-+oo
= k-+oo lim (T*Txk, Xk) = A.
By compactness of T we may choose a subsequence, which we continue to denote by {Xk}~l such that TXk --+ Yo in norm, and by the Banach-Alaoglu Theorem for a separable Banach space (see Corollary A.45 (p. 167) in Appendix A), we may also assume that Xk --+ Xo weak star. But then TXk --+ Txo weak star implies Yo = Txo, and we have
I(Txo, Txo} - (TXk' TXk} I < I(Txo, Txo - Tx",} I + I(Txo = (II Txoil + IITxklD IIYo - TXkll
TXk, TXk} I
4. Toepiitz and Hankel Operators
72
tends to zero as k
-+ 00.
Thus Xo
M.l. with lIxol/
E
(T*Txo,xo) We claim that T*Txo
E
=
1 and
= A = I(T*Txo,xo)l·
sufficient for uniqueness is the following condition in terms of Hankel operators. The resulting form of the extemal will prove very useful. The proof uses the above duality argument together with the inner-outer factorization of H2 functions. Proposition 4.23 Let
0 such that c IIflb ~ IITI"fll2 ~ II cpf II 2
,
f
E
H2.
Consequently,
c II:zn fll2 = c IIfll2 ~ IIcpfll 2 = IIcpZ" fll2 , and since the closure of u;::"=o:zn H2 = L2, we obtain c IIfll2 :::; IIMl"f1l2, and thus that 1. E L oo . Finally, let h be an outer function satisfying Ihi = Icpr!. Since I"
79
4.3. Best appl"OxiDlation
o
H:' given by
= MeAPK ,
which clearly satisfies IIA* II = li A". We first note that AT = T A is equivalent to PKzAPK = APKz, which by (5.3) is in turn equivalent to
A*z
-
MeAPKz = MePKzAPK
=
Me(P+ - ep+8)zAPK = (Me - P+Me)zAPK (1 - P+)zMeAPK
=
which is the commutation characterization that A. is a Hankel operator. It then follows from Nehari's Theorem 4.2 (p. 55) that A. = H f for some f E Loo where "Hfli = distLoo(f, Hoo) For every h E H2 we have P K (8h) = 0 and so also
P-(f8h)
= Hf(8h) = MeAPK(eh) = O.
Thus Moreover, for 9 E K we have
MeAg
= P-MeAg =
A*g
= Hfg = P-fg
and so by (5.3) again,
Ag
=
8P_(8cpg) = 8(1 - P+)(8cpg)
(5.4)
(P+ - ep+8)(cpg) = PK(CPg) = cp(T)g. Clearly we can replace the function cp E Hoo in (5.4) by any function in the coset cp + 8Hoo , and in particular we can choose a sequence {CPn}~=l in cp + eHoo such that
3 • .1. ".l"ne
commutant;
We may also assume by Corollary A.45 (p. 167) in Appendix A that {.z
(f, t/J(>')k>.
>
H2
= f(>.)'I/J(>') =
(ft/J, k>.) H2
=
(I, P+1j)k>.) £2 '
which implies
P+1j)k>. = t/J(>')k>.. Suppose that ( is in the right side of (5.5) Now with 8(>')k>., and then with t/J =
.
P+(~){k>. - 8P+8k>.}
(ep(T)* - (I)Pe k>. =
(ep(>.) - ()k>. - 8(>')P+(
..
:;=
92
5. Hilbert Function Spaces and Nevanlinna-Plck Kernels
From p+ek>. = e('\')k>. again, we have
IIPek>.lI~
=
Ijk>.l1~ -lIep+ek>.lI~
=
IIk>.lI~ _le(.\.)j2I1k>.II~,
and so we obtain inf 11(.lI~
>'ED
inf l'EJ[)
(I + 18(.\.)111'1' -
VI _18(.\.)12
(lice
= O.
This shows that also an interpolating set for MH Indeed. if Z is interpolating for H, then {k zj }~1 if> a Riesz basis, and consequently satisfies the unconditional basic sequence condition: if laj J~ Jbj J, then 00
~ C lI{aj}~1Ile2(l'z) ~ C lI{bJ}~11Ie2(l'z) ~ C Lbjkz3 j=l
Now we seek to solve the interpolation
cp(zJ) = {j,
1~J
< 00,
with cP E MH of norm at most one whenever I){~j}~dl ~ 8, with (j > 0 sufficiently small But for 8 ~ we have l~jAJ ~ and the unconditional basic sequence condition implies
I %1,
b
2
2 00
=
L
J,m=l
(1 - {j{m)kzj (Zm)AJAm.
94
5. Hilbert; Function Spaces and NevanUnna-Plck Kernels
The Nevanlinna-Pick property now yields the desired solution
f2(J-tz) is bounded and onto, if and only zf Z is interpolating for M H, i. e R· H ----> fOO (Z) is onto Now we apply Sarason's Theorem 5.2 to prove that the Hardy space H2 has the Nevanlinna-Pick property. We should emphasize that we do not need the full force of Theorem 5 2 for this application, but only the special case where the inner function 8 is a finite Blaschke product, a case which is relatively easy. For a direct elementary proof of the Nevanlinna-Pick property for H2 see Theorem 2.2 in Garnett's book [20] For convenience we restate the Nevanlinna-Pick property for the Hardy space the unit disk JI)). Definition 5.7 The Hardy space H2 on the disk has the Nevanlmna-Pick property if for every pair of finite sets {Zn}~=l and {Wn}~=l of points in JI)), there exists ip of norm at most one in the multiplier algebra HOO of H2 that interpolates the data, 1
~ n ~
N,
(5.10)
if and only if the matrix
(511) is positive semidefinite. Theorem 5.8 The Hardy space H2 has the Nevanlmna-pzck property Proof. Let 8 be the Blaschke product corresponding to Z == {Zn}~=l' let K be the corresponding model space invariant under S*, and let T be the compression of the forward shift operator S to K. N
8(z) K T
=
II Zn -=-:
Iznl, n=l 1 - Zn Z Zn
Z
E JI)),
H2 e8H 2, PKSIK.
Note that T* = S* IK. Now K is N-dimensional and all the eigenvalues of the finite dimensional operator T on K are simple Indeed, 8H 2 is the space of functions in H2 that vanish on Z, which is clearly the orthogonal complement of the span V{kz"}~=l of the reproducing kernels kz,,«() = l-~n.(" Thus
K = V{kz,J~=l.
95
5.1. The commutant
Now for cp E Hoc = MH2 we have that the adjoint M~ of the multiplication operator Mcp on H2 has eigenfunctions kz with eigenvalues cp(z) since
(!,M~kz)
= (M",J,kz ) = cp(z)J(z) = (J,CP(z)kz ).
Then the compression PKM~ IK of with eigenvalues cp(zn)·
M~
to K will also have eigenfunctions k zn
P K M~ IK kZn
= PKcp(zn)k zn = cp(zn)k zn · (5 12) In particular T* = S* \K= PKS* IK is the compression of the backward shift S* and satisfies T*kzn = znkzn, 1 ~ n ~ N Thus we see that T* has N distinct eigenvalues {Zn};;'=l with corresponding eigenvectors {kz" };;'=1. Now the map A : K ~ K defined by Ak zn = wnkzn commutes with T* (see the remark below) Then A* commutes with T and by Sarason's Theorem,
= cp(T) = cp(PKS IK) = PKM", IK, H oo with IIA*II = Ilcplloo Then A = PKM~ IK and using (5.12) we A*
for some cp E see that (510) has a solution "All ~ 1 if and only if
cp E Hoo
with
\lcp'' ""
~ 1 if and only if
IIAII
~ L But
(5.13) for all sequence& {~n}~=l. The left side of (5 13) is
and the right side of (5 13) is
(t,en k." ,t,'n k.. )~ ,t/ (k."k,)e, ~ Jt,', j
1-
~z,'"
so that (5.13) is equivalent to N
L
1
_
~j~e(1 -
WjWp)
j,£=l
1 -
~ 0,
zJz£
which is the positivity of the matrix (5.11)
Remark 5.9 An operator A : K ~ K commutes with T* if and only if k zn is an eigenvector of A for all 1 ~ n ~ N. Indeed, if T* A = AT* then T*(AkzJ = A(T* k zn ) = znAkzn shows that Ak zn is an eigenvector of T* with eigenvalue Zn, hence Ak zn is a multiple of kzn · Conversely if Akz" = Ankzn' then T* Ak zn = Anznkzn = AT* kzn ·
96
5. Hilbert Function Spaces and NevanUnna-Pick Kernels
5.1.3 Generalized Blaschke products in Hilbert spaces with the NP property. The purpose of this subsubsection is to obtain a purely Hilbert space
proof of Carleson's Interpolation Theorem 2.4 (p. 15) - see Conclusion 5.17 (p. 102) - and a generalization. This will lead us to interpolation in Hilbert function spaces with the complete Nevanlinna-Pick property, and also to generalized Blaschke products in such spaces - a possibly new topic motivated by work of Shapiro and Shields on extremal problems [44]. Let H be a Hilbert space of analytic functions with the Nevanlinna-Pick property Let Z = {zo, Zl, . ,zN-d be an N-point set in the extremal problem (58) above, which we now quickly review. Let 10 be the unique solution to Re/o(zo) = sup{Re/(zo) I(zd = 0 and II/IIH :::; I}. In terms of the data
2
N-l
0:::;
L
=
~o
N-l
-!>.0/0(zo)!2 =
>'Jkzj
j=O
L
(1 - ~J~m)kzj (zm)>'J>'m.
(514)
Jm=O
H
Since H has the Nevanlinna-Pick property, there is '0, ,>'N-I, i e. such that the matrix above is positive semidefinite. Now all of the principle submatrices of this matrix, not containing the upper left entry, have nonnegative determinant Thus M solves
o=
det B - M2 det A
O.l.. Tile
commutant
where B
A
and so we obtain kZ(l (zo) det A - KZoco[Al' Kzo
M
detA
(5.16)
-,
1 _ Kzoco[A]' Kzo kzo (zo) det A
where Kzo = [kzo(Zl) .. kzo(ZN-l)] and colA]' denotes the transposed cofactor matrix of the square matrix [AJ. Next we note that if Z ib a set of points and Hz = {h E H . h(z) = 0, Z E Z} is the closed subspace of functions vanishing on Z, then V{kZ}ZEZ = H~.
Indeed, (h, kz) projection
= h(z) = 0 for
all
Z
E
Z if and only if h E H It follows that the
PV{k'}ZEZI
= PHtl
takes the same values on Z as I, and moreover PH1-f minimizes the norm over those z functions in H taking the same values on Z as I. Thus 10 E V{kz}zEZ and we see that there are numbers AO,· ., AN-l such that lo(z) = Aokzo(z)+' +)..N-1k zN _1 (z). 'vVe mUbt have 1
=
(which follows automatically from the calculations below) and
Thub M M = d et1 B CO [)' B Mel = --co[BJ'el =-r det B det B '
5. Hilbert Function Spaces and Nevanllnna_Plck Kernels
98
where
r
~o
= [
] is the first column in the transposed cofactor matrix co[B]',
IN-l
and we obtain (517)
fo(z)
Since fo(z) =
II kzo(lf k ZQ
l -
Z'm
}(zm)2
== rn>l inf -
~
1- c. and f.tz
= 2:;1 IIkzj
r 28
zJ
II d(zmozn)2 > 0 n¥l
At this point we collect what can be said about interpolating sequences for a Hilbert space H with the NP property Theorem 5.13 Suppose H ttl a Hilbert space of anal1ltic functions with a complete Nevanlinna-Pick reproducing kernel k(x, y) so that H = 1tk Suppose Z = {zJ}~l is a sequence and let f.1z = 2:;:1 IIkZJ W-.2 8 zj be its associated positwe measure. Denote by R the restnction map Rf = {f(Z)}~lo and let Bi\{zm} be the associated generalized Blaschke products Then the relations
1 {=} (2
+ 3) and if Z is l,eparated 3 {=} 4
hold among the following four conditwns 1. 2. 3. 4.
R maps MH onto COO(Z), R maps H onto C2 (f.tz), R maps H into C2 (f.tz), inf7n~l B~\{zm}(Z7n)2 == 8z > O.
0 • .1.
Tlle cOmmutant
Proof. Conditions 2+3 are equivalent to condition 1 by Theorem 5.6, and condition 3 is equivalent to condition 4 by Proposition 5.12
at
Remark 5.14 Let kz" and let
= II~:~ II
be the normalized reproducing kernel with pole
Zn,
B(n)(z) = B~\{Zn}(z) be the generalized Blas('hke product associated to the set Zn = Z \ {zn} with pole at Zn In order to prove that condition 4 implies condition 2, it suffices to show tha.t the Grammian [(9m,971»);;','.n=1 is bounded on £2 where n2':l.
Indeed, =
B(n)(zn)-l (kzm' MB(n)k z,,)
=
B(n)(zn)-l (M'i:J(n)k z"", k zn )
=
B(n)(z17£) ( - - ) 17£ B(n)(zn) kZm' k zn = bn ·
If the Gram matrix [(917£.9n)];;'.n=1 were bounded on ~
=
£2,
we would then have for
{~j}~1 E £2,
sup
If
111111,2=1 n=1
en77nl =
f bUP_1 (f: bUP_1
11111112-1
111111,2-1
< where C*
==
11'1
n=1
m=1
If
sup 1111ll t 2=1 rn,n=1
b:e 77 17£
n
l
(kzm,9n)em77nl ern k Zm ,
f:
n=l
77n9n )
I
C'II~, (mk,·l '
sUPIITIII,2=1 IIE:'=11Jn9nIlH coincides with the norm of the Grammian
[(gm,gn)l~,n=l
on £2 The inequa.lity 1I~lIt2 ::; C* !IL:=1 (17£kzml!H implies condition 2 by standard functional analysih - Corollary A.39 (p 164) in Appendix A Note that using bz ::; B(n)(zn) ::; 1 and PropObition 5 12, we at least have that the functions 9n are uniformly bounded in H'
5. Hilbert Function Spaces and Nevanlinna-Pick KernelS
Problem 5.15 With notation as in the above remark, when is it true that 1IL.:~=l17ngnIlH ~ CIl17I1£2, or equivalently that the Grammian 00
[I MB(rn)C MB(n)k:;,)] B(m)(zm) 'B(n)(zn) m,n=l 00
[(gm,gn)]m,n=l =
\
is bounded on £2? Later in Subsubsection 5.4 (p. 112), we will give Boo's proof [12] that condition 3 implies condition 2, and hence is equivalent to condition 1, for Hilbert spaces H as above having the additional property that whenever a Gram matrix with entries ( k Zi , k zj ) is bounded on £2, then the corresponding Gram matrix with entries / (kzi' kZj ) / is also bounded on £2
Remark 5.16 It is not true that 2 implies 3 in general. The classical Dirichlet space satisfies the complete NP property, and yet condition 2 holds there for a much wider class of ~equences Z than does 3 (see e g. [10], [11) and [7)). Nevertheless, for the classical Hardy space H2, condition 2 is equivalent to condition 3 by Theorem 2.4, and this can in fact be seen directly using Hilbert space techniques since H2 has the Nevanlinna-Pick property. Indeed, the third identity in (6.19) below easily shows that the generalized Blaschke function 'I/J~~ introduced above is given by the usual single product Blaschke function,
'l/JZO(z) = Zl
Z1 -
Z
( ...!l...::EL) 1-Z"lzo
1-Z1 z/z1- z o/ 1-z1 Z o
that vanishes at Zl and is positive at Zoo The open mapping theorem, together with the fact that we can factor out the generalized Blaschke product B~(z) (which is the usual Blaschke product here) corresponding to the zeroes of an H2 function without increasing the H2 norm of the function, shows that 2 implies 4, which is equivalent to 3. For the converse implication 3 implies 2, we give a purely Hilbert space proof in Remark 5.34 in Subsubsection 5.4 below
Conclusion 5.17 Thus for the classical Hardy space H2 we have, given Remark 5.34, a purely Hilbert space proof (in the presence of separation) of the equivalence of the four conditions in Theorem 5.13 (these are conditions 1, 2, ::I and 5 in Theorem 2.4) Problem 5.18 For which Hilbert spaces with a complete Nevanlinna-Pick reproducing kernel are the four conditions in Theorem 5.13 equivalent? What about just the three conditions 1, 3 and 4 (it is widely suspected that these three conditions are always equivalent)? 5.2 Higher dimensions We now consider the characterization of interpolating sequences for the Hilbert spaces B2" (lB n ) , 0 ::; (1 < ~, as well as the corona theorem for the corresponding multiplier algebras MB~(Bn)' As the techniques used are largely based on the theory of Hilbert spaces, we will also investigate two other problems related to Hilbert space theory; namely von Neumann's inequality and the characterization of multiplier algebras. These problems all involve the notion of Carleson measures, to which we
10.i
5.2. Higher dbneIlJllo....
now turn. The material on Carleson measures presented here is taken largely from [4], [5J and [6].
5.2.1 Carleson Measures. By a Carleson measure for B 2(18 n ) we mean a positive measure defined on 18n such that the following Carleson embedding holds for f E B 2(18 n ),
fen If(z)1 2 dp, ~ CI-' Iifli~~(Bn) .
(5.22)
The set of all such is denoted CM(B2(18 n » and we define the Carleson measure norm 1iP,ll carleson to be the infimum of the possible values of C;/2. In [5] we described the Carleson measures for B2 (18 n ) for a = O. Here we consider a ~ O. We show that, mutatis mutandis, the results for a = 0 extend to the range 0 ~ a < 1/2. FUndamental to this extension is the fact that for o ~ a < 1/2 the real part of the reproducing kernel for B 2(18 n ) is comparable to its modulus. However, even though the reproducing kernel for B;/2(18 n ) has positive real part, its real part is not comparable to its modulus. For that space a new type of analysis must be added - see [6]. For 1/2 < a < n/2 the real part of the kernel is not positive, our methods don't apply, and that range remains mysterious. For a ~ n/2 we are in the realm of the classical Hardy and Bergman spaces and the description of the Carleson measures is well established [39], [53J. We will need to extend the dyadic tree T in the disk to the ball in higher dimensions as in [5].
5.2.2 Construction of the Bergman tree. Let (3 be the Bergman metric on the unit ball:IB n in en. Note that the set
Sr
= oB{3(O,r) = {z
E 18n
:
(3(O,z)
=
r}
is a Euclidean sphere (with different radius) centered at the origin for each r In fact, by (lAO) in [53] we have (3(0, z) = tanh-Ilzl, and so
1 - Izl2 = 1 - tanh2 (3(O, z) 4 - e 2 {J(o.z) + 2 + e- 2 {3(O,z) ~
> O.
(5.23)
4e- 2{3(o.z)
for (3(O, z) large. We recall the following elementary abstract construction from [5] (Lemma 7 on page 18)
Lemma 5.19 Let (X, d) be a separable metric space and)" > O. There is a denumerable set of points E = {xJ }~r J and a corresponding set of Borel subsets Qj of X satisfying X --
OO or JQ u j=l j,
Qi n Qj = 00
where Ne
= 5Up card{fJ E T· /3 > a and
d({3) = dCd')
+ f},
nET
along with a similar definition for the lower dimension '!leT) using lim infr_oc in place of lim sUPe_oo If the upper and lower dimensions coincide, we denote their common value, called the dimenston of T, by neT) Note that if T i:, a homogeneous tree with branching number N, then N = (Ne) ~ for all £ :::=: 1. The choice of base 2 for the logarithm then yields the r elatiOllship N = 2n , consistent with the familiar interpretation that the dyadic tree ha:, dimension 1 and the linear tree has dimension The proof of the following lemma is in [5J
°
Lemma 5.21 The tree Tn, con~tructed above with positive parameters A and 0, and the unit balllffin satisfy the followmg properties 1 The balllffi" is a pairwise diSjoint union of the kubes Ko:, a E 7". and there are posztive constants C 1 and C 2 dependmg on A and 0 such that
B{3(co" C 1 ) C Ko. c B{3(co., C 2 ),
a E
2 U{3~o. K(3 is "comparable" to the Carles on tent co., where Vz = {w E lffin . /1 - 'ill. Pzl ::; 1 -
Tn, n :::=: 1. v;." associated Izl},
to the point
105
5.2. Higher dimensions
and pz denotes radial projection of z onto the sphere 8lffi n . More precisely "comparable" means there are positive constants aI, a2 such that al Vee. C U,6~a K{:J C a2Vca where rVz == Vz(r) and z(r) is the positive multiple of z satisfying I-iz(r)i = r I-izi
3 The invariant volume of Ka is bounded between positive constants depending on ).. and 0, but independent of 0: E Tn. 4 The dimension n(7;,) of the tree 7;, is 1~82 n S For any R > 0, the balls B(:J(c n . R) satisfy the finite overlap condition
L
ABl(r" R)(Z) S;
CR ,
z E lffi n .
"ETn
5.2.3 Characterization of Carleson measures on the ball. Now we return briefly to an investigation of Carleson measures for B2(lffin)' As the proofs are rather lengthy, we postpone them to the following section, and content ourselves here to careful statements of the results for use in the applications deo,cribed in this section. Let Tn denote the Bergman tree constructed above. We show in the next section (Theorem 6 9 (p. 141)) that the tree condition,
L
[20"d(6) 1* p(8)]2 S;
C 1* p( ex)
-n - 1 Now the atomic decomposition of Besov spaces, Theorem 6 6 in [53], implies in particular that the indllbions of the Besov spaces Bp(lRn ) are determined by those of the £P spaces. Thus Bg(lR n ) c Bgr(lR n ). r> 1 110re generally, we use that the proof of Corollary 6 5 of [53] ShOWb that a variation on the operator of radial differentiation of order n+~+o: - a, namely KY' n+~+",_o defined in (6 23) below, is a bounded invertible operator from B2 onto the weighted Bergman bpace A~, provided that neit.her n+7 nor n+7+ n+~+a -a is a negative integer Then the proof of Theorem 66 in [53] with B 2(lR n ) in place of Bg(lRn ) yieldb Thus we have
since qJ... qA > 1 Allio.
r:
\(,pj)(k)(O)! ::;
c
r:
t",(k-J)(O)f(i)(o)i 1
k=O )=0
k=O
,; c (~ 1,,K"
IIJ(SE IK)II K --+ K :::; IIf(S*)/l H2--+H2 where S* is the backward shift on the Hardy space H 2([j). Next from S = Mz and STL = Mzn we have for P( z) = ~;:=o anz n , N
N
n P(S) = ~ L-t anS = ~ ~ anMzn = M"v
LJ'II=(}
n=O
If we set P(z) =
a n z"
= Mp
n=O
fez), then we conclude that IIf(S*)II H2-->H2
= /lP(S)II H2--+H2 = IIMpllH2--+H2
Finally, Lemma 2.9 (p 21) shows that IIMpIlH2-+H2 = IIPIIHoo(D) = IIfIIHOC(lD) , and altogether we have IIf(T)IIH-->H :::; Ilf(S*)II H2-+H2 = /lfIlHoo(l[)) when IITI/ H-+ H < 1 A simple limiting argument establishes the general case Now we turn to Drury's generalization [19J Let A = (AI, . ,An) be an ncontraction on a complex Hilbert space 1t, i.e an n-tuple of linear operatoIs on 1t satisfying n
AjAk = AkAj for all 1 ~ ], k :::; n, and
L
IIAjhll2 :::; IIhll 2 for all hE 1t
j=l Equivalently, the AJ commute and the row operator A = (AI, .. , An) is bounded
with norm one from to 1i: 11~;=1 AJhJ!12 :::; in [19J that if I is a complex polynomial on en, then
E9;=11t
II/(A)11 :::; 1I/I1M.qa,,) ,
~;=l/lhJI/2.
Drury showed (535)
5.3. Applications of Carleson measures
for all n-contractions A on 1t where II/(A)1I is the operator norm of leA) on 1t, and II/IIM denotes the multiplier norm of the polynomial I on Drury'b Hardy "(Ill,,) space of holomorphic functiollb
K(Bn) =
{~akzk, z E Bn: ~ lakl I~~! < oo}, 2
denoted by H~ in Arveson [9] (who also proves (5.35) in Theorem 8 1). Moreover, equality holds in (535) when A is the n-tuple (Si,. ,S~) where SJ = M zj • The proof consists in finding the appropriate analogue of the isometry V in the proof of von Neumann's inequality above See [19] or [9] for details. Chen [16] has identified the Drury-Arvebon Hardy space K(Bn) = H2n as the 1 llebov-Sobolev space Bi (Bn) cOIlbisting of those holomorphic functions Lk akzk in the ball with coefficients ak batisfying "
2
L..-Iakl
Ikl"-1 (n - l)!kl (n _ 1 + Ikl)l < 00.
k 1
Indeed, the coefficient multiplier!> in the definitioIlb of K(Bn) and B1 (Bn) are easily seen to be comparable It now follows that the multiplier norms are equivalent:
II/I1Mqp u) ~ 1I/11l\! ~ /)2
(!I,,)
\Ye note in passing that a number of important operator-theoretic properties of the Hilbert space H~ are developed by Arveson in [9] that ebtablish its central position in multivariable operator theory. Recall from the previous 8ubbubbection Theorem 5 28 of Ortega and Fabrega I that bhows I is a pointwibe multiplier on Bi (Bn) if and only if I is a bounded holomorphic function and the measure
dJlj(Z)
=
IR (
"+1)
-2
I(z) 12 (l-Izl 2)dz I
is a Carlebon meabure for the Drury-Arvcbon Hardy space Bi (Bn) In fact, we can replace dJl f by any of the meru:.ures
dJlj(z)
=
i/(m)(z)i 2 (1 _lzl.2)2m- n dz.
Tn
>
n;
1
U"ing tllli. we obtain the following estimate Theorem 5.30 For any
Tn
. II/(A)II
sup
A an n-("ontrachon
> ";-1. ~ 11/1100 + sup V2d(a)I*Jlj(a)
(5.36)
OIET"
+ sup
aET.,
for all polynomials
I on en
The right side of (5.36) can of course be transported onto the ball using that u#?o:K/J is an appropriate nonisotropic tent in B n , and that 2- d (a) ~ (1 - Iz12) for
z
E
Ka
112
5. Hilbert Function Spaces and Nevanlinna-Pick Kernels
5.4 Interpolating sequences for certain spaces with NP kernel
Given a, 0 :-:; a < 1/2 and a discrete set Z = {Zi}f=1 C Bn we define the associated measure P,z = 2:;:1 (1 - IZj 12)20' 8ZJ We say that Z is an interpolating sequence for B2"(Bn) if the restriction map R defined by (Rf)(Zi) = f(zi) for Zi E Z maps B2"(Bn) into and onto £2(Z, P,z). We say that Z is an interpolating sequence for MB2(B nl if R maps MB~(Bn) into and onto £OO(Z,P,z). Using results of B. BoE' [11], J. Agler and J. E MCCarthy [1], D Marbhall and C. Sundberg [24], C Bishop [10], along with the above Carleson measure characterization for Bg(Bn) we now characterize those sequences. Denote the Bergman metric on the complex ball Bn by {3. TheoreIll 5.31 Suppose (J. Z. and ILZ are as de.scrtbed, then Z is au interpolating sequence for B2"(Bn) if and only if Z is an interpolating sequence for the multiplier algebra MBf(B,,) if and only if Z satisfies the separation condition infi#J {3(Zi, z)) > 0 and p'z is a B'{(Bn)-Carleson measure. equivalently. it satisfies the tree condition (5 25) Proof The case a = 0 was proved in [24] when n = 1 and in [5] when n > 1 If 0< a < 1/2, then Corollary 1 12 of [1] &hows that the reproducing kernel k(z, w) = (l-k J20' has the complete Nevanlinna-Pick property Indeed, the corollary states that k has the completE' Nevanlinna-Pkk property if and only if fm any finite set {ZI' Z2," , zm}, the matrix Hm of reciprocals of inner plOducts of reproducing kernels kz, for Zi, i.e.
H
- [
m -
1 k )
/k \
Z"
]m
,
i )=1
ZJ
has exactly one positive eigenvalue counting multiplicities (kZi' -1 by the binomial theorem as
kzJ
\Ve may expand
oc
(1- zJ . Zi)2 nonnegative semidefmite since m
L
(i(z) Zi)(i = 1((lZl . . , (m z m)12 ~ 0
i 7=1
Thru; by Schur's Theorem so i& [(z) . ZYli.~=l for every f ~ 1, and hence, also. so is the sum with positive coefficients. Thub the positive part of the matrix Hm i& [In:i=l which has rank 1, and henc,e the sole positive eigenvalue of Hm is m Once we know thib it then followb from Theorem 56 (see aL~o Theorem 919 of Marshall and Sundberg [24]) that the interpolating sequences for MB2< Bnl are the bame as those for B2"(Bn). Thus we need only consider the case of B2"(Bn). We remark that the standard reproducing kernel for the Dirichlet space B~(Bn) also has the complete Nevanlinna-Pick property (see e g [1]). We now invoke a theorem of B Boe [11] (see Theorem 532 below) which says that for certain Hilbert spaces with replOducing kernel, in the presence of the separation condition (which is necessary for an interpolating sequence, see Ch. 9 of
5.4. Interpolating sequences for certain spaces with NP kernel
113
[21) a necessary and sufficient condition for a sequence to be interpolating is that the Grammian matrix associated with Z is bounded. That matrix is built from normalized reproducing kernels; it i~ (537) ThE' space~ to which Boe's Theorem applies ale those where the kernel has the completE' Nevanlinna-Pick property, which we have already noted holds in our case, and which have the following additional technical property. Whenever we have a the dCbirf'd property (It is this step that precludes including a = 1/2 ) Finally, as also pointed out in [11], the boundedness on £2 of the Grammian matrix io.; E'quivalent to flz = 2:;1 Ilkz ) 2bz , = 2:~1 (1 - IZi12)20" 8zj being a Carleson measure Thub the obvious generalization to higher dimensions, which we give in Theorem 5 32 below, compktes the proof (m)e presents his work for the dimension n = 1. but. 8b he notes, it ('xtend~ directly to general n.) In Older to state Boe's Theorem, we briefly recall the theory of Hilbert spaces with a complete Ne,anlinna-Pick kernel k(x, y) in Agler and MCCarthy [1] (see Subsubsection 7.3.1 in Appendix A below), keeping in mind the classical model of the Szego kernel k(:r. y) = 1 l lY on the unit disk]]) Let X be an infinite set and k(x. y) be a positive definite kernel function on X, i.e. for all finite subsets {:rdf~lOf X.
11-
m
L
aiaJk(xi
Xj)
~ 0 with equality
¢:}
all ai
= O.
ij==l
Denote by 'Hk the Hilbert bpace obtained by completing the &pace of finite linear combinations of kx, 'b, where k, (y) = k(x. y), with rebpect to the inner product
The kernel k is called a complete Nevanlinna-Pick kernel if the solvability of the matrix-valued Nevanlinna-Pick problem is characterized by the contractivity of a certain family of adjoint operatoTh R""A (we refm to [1] for an explanation of this generalization of the classical Pick condition - see al&o Definition 5 35 below for a le&s informative discussion) Theorem 5.32 Suppose H is a Hilbert space of analytic functions wzth a complete Nevanlinna-Pick reproducing kernel k(x,y). so that H = 'Hk. Suppose also that the Grammian property mentioned above holds whenever {zJ }~1 is a sequence for which the matrix (5.37) is bounded on 12 then the matrix with abbolute values
114
5. HUbert Function Spaces and Nevanlinna-Pick Kernels
is also bounded on £2. Then a sequence Z = {Zj }i=1 is interpolating for H if and only if Z is separated and J.Lz = Z=~l
Ilk 11- 2 8 Zj
zj
is a Carleson measure for H
Remark 5.33 The Grammian matrix (5.37) is bounded on £2 if and only if -+ £2 be the normalized restriction map Tf = {If~::il }i=I' Then J.Lz is a Carleson measure for H if and only
J.Lz is a Carleson measure for H. To see this let T : H
if T is bounded. But
T*{~J}i=1 = Z=':l J
e ~llkz and so the matrix representation Ilkzj II j
of TT* relative to the standard basis {ej }i=1 of £2 is the Grammian:
[ ( T(
II~:: II)' ej )]:=1
[( ~;k:~i?) l:~, ~ [( 11::11' 1/::;1/)l:~,· Now use that T is bounded if and only if TT* if'> bounded. Proof of Theorem 5.32: If Z is interpolating for H, f'>tandard arguments show that Z is separated and that J.Lz is a Carleson meabure for H. Conversely, Remark 5.33 shows that the Grammian matrix (5.37) is bounded on £2. To show that Z is interpolating for H it suffices by Bari's Theorem A 35 (p
161) to show that {k z Ji=1 is a lliesz basis, where kZi = ":::,, is the normalized reproducing kernel for H Let {h }~1 be the biorthogonal functions defined as the unique minimal norm solutions of
~:~I? =
(fn, k zm ) =
8~
If P denotes projection onto the dosed linear span V'i=1 kZJ of the kz ;, then
and so fn
= Pfn
E VJ=lkz1'
By Bari's Theorem A 35 again, {kzJ~J is a Riesz
basis if and only if both [( kz" , k zm ) ] : n=l and [(fn. 1m) l~,n=1 are bounded matrices on £2 We already know that [( k zn , k Zm ) ] 00
_
m,n-l
is bounded, so it remains to
show that [(fn, 1m)1~,n=1 is also. For A c Z = {Zj}J=1 let HA = {J E H . f(a) = 0 for a E A} If k~(z) is the reproducing kernel for H A , then
IIk:'lI.;! =
k:'(w) and
k~(w) = sup{11(w)1 f E HA with 11111 = Ilk~Ii}· It follows that with Zn = Z \ {ZT'}' we have n;:::l.
Note in particular that
II n -
Ilf
Ilk zn II and
Ilk!.:-ll
k;""(zm) _ fn(zm) _ 8~ [! Ilkz"J - IIkzm II II In II - IIfnll'
Ilk:""
5.4. Interpolating sequences for certain spaces with NP kernel
115
We now compute the entries (In, 1m} in the biorthogonal Grammian [(In, 1m) l~,n=l in terms of the corresponding entries kZn' kz",) in the Grammian
despite the nonnegativity of Re kw However, if Bw (z)
=
n
1~"in~ ~ denotes the
wEW
Blaschke product with zeroes W c JD>, then it is known that the extremal functions IT! and (z) d m() BZm n (z) B Z'Il. (z n )-\\k II an 'Pn Z = B Zn Zm. n (z)' m
so that
and
Altogether we then have
_ IIln\!2 'P:'(Zn) = 1 - zmzn IZn\ zm 1 . 1 - znzm Zn IZml B z "" (zm)Bzn (zn)
15. Hilbert Function Spaces and Nevanllnna-Pick Kernel..
118
Now we use
to obtain that
(fn'!m)
-lIlnll 2 'P~(zn) (k zm , kzn )
=
Cznl ;;" (zn») It is now clear that
Czml;;: (zm») (k
zn •
k
zm ) .
[(fn, 1m) l~,n=l is bounded on £2 if [(k zn , kzm )] oc
_ is bounded
m,n-l
on £2 since the factors
,z" IB~"" (z,,)
satisfy
Note that this argument provides the last link in a purely Hilbert space proof (see Subsubsection 5.1 3 (p. 96) for the other arguments) of the equivalence of the four conditions 1, 2, 3 and 5 in Carleson's Theorem 2.4. 5.5 The corona problem for multiplier spaces in C n
Here we show that if X is a Hilbert function space of holomorphic functions in an open set n in C n with a complete Nevanlinna-Pick kernel (see below for the definition), then the corona problem for the multiplier algebra Mx is equivalent to the so-called baby corona problem for X: given 'PI' .. ,'PN E Mx satisfying
l'Pl(Z)1 2
+ .. + I'PN(Z)1 2 ?: c> 0,
zEn,
(5.44)
is there a constant 6 > 0 such that for each hEX there are II, ... , IN E X satisfying
IIIIII~ + .. + IIfNII~
$
JIIhll~,
+ . + 'PN(z)IN(Z)
=
h(z),
'Pl(Z)!r(Z)
(5.45)
zEn?
More succinctly, (5.45) is equivalent to the operator lower bound
M",M; -
§Ix ~
(5.46)
0,
where 'P == (CPl, ",'PN), Mop: $NX ----+ X by M",j = 'L:=lCPOtlotl and M;h = (M;" f)~=l' To see this equivalence note first that (5.46) is equivalent to (5.47) From Corollary A.39 (p. 164) in Appendix A below, we obtain that the bounded map M", . (!}N X -> X is onto. If N = ker M"" then.M:, N..L -> X is invertible Now (5.47) implies that
M:,*: X
By duality we then have satisfying M",I = h and
----+N..L
is invertible and that
IlcM:,)-ll! ::; 7.5.
2 X 1l/11EeN
IlcM",*)-l!1 $-36.
Thus given hEX, there is f
-1 h 112 = 11 CM",) Ee N
2 ' x ::; -;s1 IIhllx
E
N..L
5.5. The corona problem for multiplier spaces In
en
119
which is (5.45). Conversely, using (5.45) we compute that
IIM~hll6>NX
lNXI
sup IIgll(j)N x 9
> i(MNX =
1 0 if and only if there are II, ... J N E Mx such that
IIfll Mult (x e>tv X) 0 :"uch that condition (5.49) holds. 5.5.1 Calculus of kernel functions and proof of the Toeplitz Corona Theorem. A crucial theme for the proof of the TOE'plitz Corona Theorem i& that operator bounds for Hilbert function :"paces, &uch as M"X =
L 0I,{3=1
or
N
.J
L L
J
L
C~~k(x~,x?),
8B
i.1=1
C~f[{8~ - fo:(x't)f{3(x~)}k(x7' x?)] ~ O.
0:,{3=1 i,1=1
If we view f(') E B(C, CN) we can rewrite this last expresbion as
(554)
{leN - J«()f(>')*}k«(,>') t: 0,
which i& the required matrix-valued kernel equivalence of the multiplier bound in (5.49) Now we turn to the proof of Theorem 536. To see that (549) implies (5.45) just mUltiply M«,f = v'8 by to get Mrp$ = h whE're
-!h
II ~II:NX
~(llhhlli + .. + IlfNhlli) 1
2
2
2
1
2
< 8(lIftll Mx + .. + IIfNII Mx ) IIhllx ::; 8l1hllx'
122
5. Hilbert Function Spac.... and Nevanlinna-Pick Kernels
The computation (5.48) above then shows that (5.46) holds for the same 0 > O. However, we can give another short proof, but using the language of positive semidefinite kernel functions to characterize operator boundedness. This will afford us our first opportunity to use the "calculus" of positive semidefinite forms. If (5.49) holds then
'P«()* f«() = Mcpf =
VJ,
and (5.54) holds.
{lcN - f«()f(A)*}k«(,A)
~ 0
These two relations imply the positivity of the kernel function,
{('P«(),'P{A»CN - 6}k{(,A) {('P«(),'P(A»CN - VJVJ}k«(,A) =
{('P«(), 'P(A»CN - 'P«()* f«()f(A)*'P(A)}k«(, A)
=
'P«()*[{lcN - f«()f(A)*}k«(, A)]'P(A) ~ 0
By (5.50) this is equivalent to (5.47), and hence to (5.46). Conversely, normalize k at a fixed point Ao En so that k>.o == 1. Since k is an irreducible complete Nevanlinna-Pick kernel, we can find a Hilbert space J( and a map b : n -+ J( with b(Ao) = 0 and such that
1
(555)
k«(, A) = 1 _ (b«(), b(A»),.;:
This theorem has a long history and is not easy to prove (Theorem 7.31 in [1]). In fact, one can take J( to be the Drury-Arveson Hardy space H! for some cardinal number m (Theorem 8.2 in [1]), but we will not need this. From (5.46) we now obtain (5.50):
K{(, A) == (('P«() , 'P(A»CN - 6}k«(. A) ~ O. By a kernel-valued version of the Lax-Milgram Theorem, we can factor the left hand side K«(, A) as (G{(), G{A»1t where G : n -+ 1t for some auxiliary space 1t. Indeed, define F: -+ X K by F«() = K( so that
n
K«(,A) = (K>.,KdxK = (F(A),F«(»xK' Now fix an orthonormal basis {eo:},,, for X K and define a conjugate linear operator by
r
Then G =
r
0
F satisfies
K«(, A) = (F(A), F{()hK = (r 0 F«(), r
0
F(A» XK
=
(G«(), G(A» XK '
with 1t = X K as required. Hence
(c,o«(), c,o(A»CN - 6 = [1 - (b«(), b(A»,d (G«(), G(A»1t, or equivalently,
('P«(), 'P{A»CN
+ (b«(), b(A)h: (G«(), G(A»1t =
0+ (G«(), G(A»1t.
(5.56)
5.5. The corona problem for multiplier spaces in
en
123
Now we rewrite (556) in terms of inner products of direct sums of Hilbert spaces,
(cp«(), CP(>'»eN = (vg, 1>0
+ (b«() 0
G«(), b(>.) 0 G(>'»K:®1-l
vg) e + (G«(), G(>'»1-l'
that it can be interpreted as saying that the map that sends the element
(cp(>.)u, b(>.) 0 G(>')u} E C N EB (!C 0Ji)
(5.57)
with u E C to the element
(vgu,G(>')u) E CEBJi
(5.58)
is an isometry! Here the space!> Nl and N2 are given by
Nl
Span { (
b(>..r~>'b(>.)
)
Nz.
Span { (
~)
EC, >. En} c C EB Ji.
) u.u
U'
u
EC, >. En} Cc NEB (!C 0Ji),
Thus using (546) we have obtained (5.56) that defines a linear isometry V' from the linear span Nl of the elements cp(>.)u EB (b(>.) 0 G(>'»u in the direct sum C N EB(!C0Ji) onto a bubspace N2 of the dire')
(5.61)
f . n ---+ B(C, CN) (which is of course isomorphic to CN) by f(>.)* = A + B{b(>.) 0 (I - DEb(A»-lC},
where Eb is the map Eb : H
---+
!C 0 H given by
EbV = b0v,
v E H.
(5.62)
124
5. Hilbert function Spaces and Nevanllnna-Plck Kernels
Note that this formula for f(>•.}* is obtained by solving the second line in (5.61) for G(A) = (I - DEb()..»-lC.)(! - DEb(>.)}-lC +C*(! - Eb«)D*)-1 Eb«)D*C +C*(I - Eb«)D*)-l Eb«) (D* D - I)E/)()..)(/ - DEb(>.»-lC C*(I - E*b«,) D*)-l x{(I - Eb«)D*)(I - DEb(>.» + (/ - Eb«()D*)DEb(>') +Eb«()D*(I - DEb(>.» - Eb«,)Eb(>') + Eb(()D* DEb()..)} xCI - DEb()..»-IC C*(/ - E;«,)D*)-l(I - Eb«;)Eb()"»(I - DEb()..»-lC
(1- (b«(),b(A»dC*(1 - Eb«)D*}-l(J - DEb()..»-lC, where the last line follows from (563). Thus using (555) the left side of (564), which is an N x N matix-valued kernel function, has its complex conjugate equal
5.5. The COrona problem for multiplier spaces in
en
125
to C*(1 - Eb((;)D*)-l(1 - DEb(A»-1C
= «I -
DEb (A»-1C, (I - DEb «»-1C)'H'
which is an N x N matix-valued Grammian, hence a positive kernel as required. 5.5.2 Two generator case of the baby corona theorem. Ortega and Fabrega have obtained a partial rebult toward the baby corona problem in [33] for the Besov-Sobolev ::;paces B2(~Tt) on the ball ~n with 0 :::; a < i e. from the Diric,hlet space B~(~n) up to but not including the Drury-Arveson Hardy space
4,
H~
1
= Bi (~n)' Their partial result i::; that (5.46) holds when N = 2 for the::;e Besov-
Sobolev ::;pace::; We remind the reader that only the case of N = 2 generators in Carleson's Corona Theorem is needed to prove the spectral mapping theorem 5.4 for the HOC functional calculus. Now for 0 :::; a :::; the spaces B2(~n) are reproducing kernel Hilbert spaces with the complete Nevanlinna-Pick property, and so the Toeplitz Corona Theorem 536 bhows that the corona theorem for BH~n) (i e. the Banach algebra MB~(Bn) has no corona), 0 :::; a :::; would follow from (5.46) for all N 2: 2.
!,
4,
Remark 5.38 The corona theorem for the multiplier algebra MH2 of the Drury-Arveson Hardy space H~ (as well as for the multiplier algebra MB~(Bn) when 0 < a :::; h&.,> recently been obtained by Costea, the author and Wick [18] by using the Koszul complex in the next subsection to establish the baby corona problem for Bl(~n)' and then invoking the Toeplitz Corona Theorem 5.36.
4)
a
1'. Also 9 E MB~(Bn) if and only if 9 is bounded and JL'; is a B1 (lffin )-Carleson measure. We now give a very rough sketch of the proof that for 9 E MB;(B,,) and M large enough, T j is bounded on B2(lffin)' From (6.19) below we have
hence
11 - wzl 2 - (1 - IwI2)(1 - IZI2) = 11 - wzI21 0 is such that Bj3(O, r) = B(O, t) (note from (6.20) that Bergman metric balls centered at the origin rue Euclidean balls), then the .a-balb are the ellipsoids ([39], page 29)
Bj3(a, r) = { z E lBln
IPaz - cal;! 2 2 t Pa
z2
l + -IQa 2- < t p"
1
}
,
where (1 - t 2 )a Ca = 1 _
t 2 1a1 2 ' Pa
1 - lal 2 = 1 _ t 2 1al 2
We have the reproducing formula of Bergman ([39]. Theorem 3.1.3), fez) -
n!
-
Jrn
r (1 -
il8"
few) dw W z)n+l '
n R(lBln ) ,
(6.21)
Res> -1,
(622)
f E Ll(dAn)
and the following variants ([39], Theorem 7 1.2) fez) = n! ( n Jrn
+s
n
)
r
iBn
(l-=:-/w/2)S f(w)dw, W z)s+n+l
(1 -
valid for all f E R(lBl n ) for which the integrand ib in Ll We now recall the invertible "radial" operatOI& R'Y t in [53J by
Rr,t fez) = provided neither n
+7
-7
R(lBl,,) given
~ r(n + 1 + 1')r(n + 1 + k + 7 + t) /. (z) ~r(n+1+7+t)r(n+1+k+7) I. ,
(623)
nor n
+7 +t
.
R(lBl n )
is a negative integer, and where fez) Note that
=
E~ofk(z) i.& the homogeneous expansion of f
r(n + 1 + 7)r(n + 1 + k + l' + t) ~ (1 rCn + 1 + l' + t)r(n + 1 + k + 1')
+ k)t
If the inverse of R'Y t is denoted R-y,t, then Proposition 1 14 of [53J yieldb
(1 -
W
(1 -
W
1 z)n+1+I'+t'
(6.24)
1 z)n+l+'Y'
for all w E lB n . Thus for any 7, R'Y,t is approximately differentiation of order t. From Theorem 6.1 and Theorem 64 of [53J we have that the derivatives R'Y,m fez) are "£2 norm equivalent" to E;;'':OI lV'k f(o)1 + V'm fez) for m large enough. Proposition 6.5 (Theorem 61 and Theorem 6.4 of [53J) Suppose that n + 7 E R(lBn ). Then the following four conditions are
is not a negative integer, and f
139
6.2. Invariant metrics, m ......ures and derivatives
equivalent: n
(1 - IzI2)mV'm fez)
E
(1-lzI2)mV'mf(z)
E
L2(dAn) for some m > 2' mEN, n L2(dAn) for all m > 2,m E N,
(1 - IzI2)7Tt R'Y,m fez)
E
L2(dAn) for some m > ~, m
(1 _lzI 2)mR'Y,m fez)
E
L2(dAn}for all m > ~, m
+ n + 'Y 't
+ n + 'Y 't
-N,
-N.
Moreover. with a(z) = 1 - Izl 2 . we have, C- 1 1Iaml R'Y·7Tt lfll L 2 (d).. .. )
s;
(6.25)
,%1\'i'/(O)\ + (hn \(1-lzI2)m2vm2 l(z)\2 dAn(Z») ~
s; C lIaml R'Y Tltl IIIL2(d)",,) lor all ml,m2 > ~, ml + n + 'Y 't -N, m2 EN, and where the constant C depends only on mI. m2, n, and 'Y Proposition 65 persists with the obviolIb modifications when (1 - Iz12)7Tt is replaced by (1 - IzI2)m+ - l.
Definition 6.1 For 0: > -1, we use (-, )'" to define an inner product on B2 (which we also denote by (', )"') as follows.
(Rc;.tl±"f,R~tl±ag) a 2
=
2
~HI! {(l-lzl
JSf Rc;.tIt"f(z)R(~+-l±"g(z)dvo,(Z) 2
Il
2
2 ntlt
)-z-Rc:.tItf(z)} 2
With K;;;(z) the reproducing kernel fOl A~, we haye that the kernel k~(z)
=
(R~)-I(R';';tJt,. )-1 K~(z) 2
(627)
2
satibfies the following reproducing formula for B2
few)
= (I, k;:Ja = f
1Bn
R';';tI±n f(z)Rc;.tlt 2
k~(z)dva(z),
(628)
2
Thus we have the following theorem.
Theorem 6.8 Let 0: > -1. Then B2 is a Hilbert function space with the inner product ( , )"" and the reproducing kernel k~ for this space is given by {6.27}. From (627) and (6.24) we have
(6.29)
R
(X-
1l+~+Ck
n+~+o
«(1
- -'w .
z )-(n+1+ a ))
)_ »tIt,. _ ( 1-w z 2.
Using this formula one can show that the B2 norm of the reproducing kernel k~ is comparable to (1 + log 1_~11I12)4. Finally we note that there are similar resultf:> for the reproducing kernel Hilbert spaces B~ (Ja" ), a ~ 0
6.3 Carleson measures on the ball Jan Recall that given a positive measure f.t on the ball, we denote by ji the associated measure on the Bergman tree Tn given by ji(o:) = df.t for 0: E Tn. We will often write f.t(o:) for ji(o:) when no conftL.'lion should arise Let a ~ O. We say that It is a B~ -Carle50n measure on Jan if there L'l a positive constant C such that
IK"
(6.30) for all f E B~. In this section we show (Theorem 6.9) that f.t is a measure on lffin if ji is a B~(Tn)-Carleson measure, i.e if it satisfies
B~-Carleson
f~O,
(6.31)
6.3. Carl......n meaaurea on the ball Bn
which is (6.5) with w(o)
(L
=
141
1-'(0) and v(o)
[2 ud(a) 1*91-'(0)]2) 1/2
~C
=
(L
2-2ud(a). The dual of (6.31) is 9(0)21-'(0») 1/2,
92:0.
(6.32)
"'ETn
aETn
Theorem 6 1 shows that (6.31) is equivalent to the tree condition:
L[2UdU:l)1*I-'({:JW ~ CI*I-'(O)
8f(·)
=
:1 -Carleson measure if and only if the linear
Ix
Relx( ) l(x)dJL(x)
is bounded on L2(X,JL). Proof T is bounded if and only ifthe adjoint T* is bounded from L 2(X,JL) to
:1, i.e. IIT* III~ = (T* I, T* f}.7 :S C 1I/1I~2(J')' We have for x E X,
1 E L2(JL).
(6.35)
T* I(x) = (T* I, jx).7 = (j, Tlxh2(J')
=
J J
l(w)]x(w)dJL(w)
jw(x)/(w)dJL(w).
=
and thus we obtain
IIT* III~ = (T* I, T* f}.7
(J JJ = JJ =
jwl(w)dJL(w),
J
jW I(W')dJL(W'»).7 1
(iw,jw / }.7 l(w)dJL(w)/(w')dJL(w')
=
jw(w'}/(w)dJL(w)/(w')dJL(w').
Having (6.35) for general I is equivalent to having it for real 1 and we now suppose is real. In that case we continue with
I
IIT* III~
=
JJ
Relw(w')/(w)/(w')dJL(w)dJL(w')
(81,/)£2(1'-).
=
The last quantity satisfies the required estimates exactly if 8 is bounded; the proof is complete. In the case of current interest Lemma 6.10 gives that JL is a measure exactly if we have estimates for
(T* I, T* I) B;.(8.,) for
I
~
=
B~(Bn)-Carleson
JJ _!.
w,)2a l(w)dJL(w)/(w')dJL(w')
1 )2U l-w w'
II-w.w' \2U
Re( 1
o.
Now we use that
Re
(
~
1
(6.36)
6. Carleson Measures for the Hardy-Sobolev Spaces
144
for 0
0,
'Y:::;,xAa'
and so the left side above is approximately
L
L
22crdhl f(a)J-t(a)f(a')J-t(a')
=
L
22Jdh ) I* fC'Y)2.
°
Thus for < 0" < 1/2, J-t is B 2 (lffi n )-Carlel:>on if and only if (6.32) holdl:> where ranges over all unitary rotationb of a fixed Bergman tree. By Theorem 61, this is equivalent to the tree condition (6.33) where Tn ranges over all unitary rotations of a fixed Bergman tree. However, we need only consider a fixed Bergman tree Tn since if J-t is a positive measure on the ball whol:>e discretized measure J-tTn on Tn satisfiel:> the tree condition, then its dil:>cretization J-tUTn to any unitary rotation UTn also satisfies the tree condition (with a possibly larger, but controlled constant). Indeed, Theorem 6 1 showl:> that J-tTn is B2('T,:,)-Carlewn, and hence so is the fattened measure defined by
Tn
J-tt(a)=
L
J-tr,,(!3),
aET"
d(o that the tree condition (6.33) il:> l:>ufficient for J-t to be a B 2 (lffi n )-Carleson measure. This completel:> the proof of Theorem 6.9
APPENDIX A
Functional Analysis Here we give a brief introduction to some of the functional analysis that arises in function theory We begin with an example taken from Schechter's book ([42]). The pair of function1> {cos x, sin x} is a fundamental solution set on the real line li for the homogeneous second order equation
yf/(x)
+ y(x)
=
0,
x E li,
and the general solution is given by
Yhom(X) = Yhom(O) cos X + Y~om(O) sinx,
x Eli.
(A.I)
We now wish to solve the more general equation
y"(x) where
(7
+ y(x)
= (7(x)y(x),
is a continuous function on li. First we solve the inhomogenoeous equation
y"(x)
+ y(x) = I(x)
by writing it as a system in y = [ ;, ]:
y'
[ ;:, ] =
[~l ~] [ ;, ] + [ ~ ]
Ay+f. Then the Wronskian matrix
W(x) = [ cosx
sinx]
cos'x sin' x
=
[cosx - sin x
sinx ] cos x
satisfies
Thus
W-1y' + (W-1),y W-ly' _ W- 1 Ay = W-1f implies
A. Functional Analysis
146
and so a particular solution Ypart(X) is derived from (A.2)
foX [~~XX ~::] [~~:: ~:~~t] [ f~t)
r [**
sin X cost - cosxsint ] [
10
*
]dt
] dt
0
J(t)
[ fox sin(x ~ t)J(t)dt ] Now we see from (A.l) and (A.2) that the solution to the initial value problem
{
Y" + Y yeO) y'(O) =
uy 1 0
satisfies the integral equation
vex)
+ fox sin(x - t)u(t)y(t)dt,
= cos x
x
E
JR,
and vice versa If we write u(x) = cos x and
Lh(x)
=
fox sin(x - t)u(t)h(t)dt,
we can rewrite this equation as
y=u+Ly,
(A 3)
an example of a Volterra integral equation A.O.l Volterra Equations. To solve the Volterra equation (A.3) for x E [-N, N], we start with a guess Yo = yo(x) where Yo is any continuous function on [-N, N], and plug it into the right side of (A.3), defining
Yl(X)
YI
u(x) + Lyo(x)
cos x
+ fox sin(x -
= Yo
(highly unlikelyl) we are done. Otherwise set Y2
=
If it happens that YI and inductively
=
Yn
=
x E [-N,N].
t)u(t)yo(t)dt,
u + LYn-1 on [-N, N],
n
=
1,2,3, ..
= U+LY1 (A.4)
We hope that this sequence of functions {Yn}~=l converges in some sense. Since uniform convergence yields a continuous limit, we define
!/hl/ = and hope that IIVm - Ynll convergence) .
~
max Ih(x)J
Ixl: m,
IIYm - Ynll
= IIL mu + ... + L n- 1u + Lnyo - LfflYolI ~ II LfflUl1 + ... + !!Ln-1ull + IILnYol1 + II Lmyo II ,
and in particular this will tend to zero as m, n convergence of orbit series":
-+ 00
(A.6)
provided we have the "absolute
00
L
IIL nvll
pace
°
There are versions of these definitions and thof:>e below when the scalar field is the real field ~ instead of the complex field C Normally there is little difference in the interaction of the concepts, and we will usually use the complex scalar field C - but will explicitly mention the scalar field ~ when it matters. We will denote by C(K) the Banach space of continuous functions on a compoct topological space K, equipped with the supremum norm Ilflloo = sUPxEK If(x)1 Now we examine the properties used of the map L from the Banach space C([-N, N]) into itself First, in (A.5) we used thdt L is linear, and then in (A.9) we used that L takes bounded sets in C([-N, N]) to bounded sets. This motivates the following definition of a bounded linear operator between normed linear spaces. Definition A.2 A map L from one normed linear space X to anothm Y is linear if L(AX + y) = ALx + Ly for all x, y E X and A E C, and bounded if there is a nonnegative constant C such that IILxlly ::; C IIxllx for all x EX
The proof of the next result is easy and is left to the reader. Lemma A.3 Let LX ...... Y be linear where X. Yare normed linear spaces Then L is bounded ~ L is wntinuous on X ~ L is continuous at
°
Remark AA If Y is the scalar field, then in addition we have that L i5 continuous ~ the null space of L is closed. Indeed, if the null space N of L is closed and Xo ¢ N, then there is a ball B(xo, r) c X \ N Now if L(B(O, r)) is unbounded, then it must be all of the scalar field C, and so L(B(xo, r)) = C as well, contradicting the fact that L(B(xo, r)) C L(X \ N), where the latter set doesn't include Thus L(B(O, r)) is bounded as required. However, this equivalence of continuity and closed null space fails for general Y = X as evidenced by the space X of polynomial., on [0, 1J with the supremum norm, and L : X ...... X by LP = P' for P EX
°
Our arguments above prove the following general theorem in a Banach space. Theorem A.5 Let X be a Banach space and L be a bounded linear operator from X to itself If 2:.':=0 IILnxl1 < 00 for all x E X, then the equation
x = y+Lx has a unique solution x
E
X for every y EX.
= Y + LXI and X2 = Y + LX2. Then LXI - LX2 = Lx = L2X = .. = Lnx
Proof: To see uniqueness, let Xl
x =
Xl -
X2
=
149
A.2. HUbert spaces
for all n ~ 1 implies that IIxI - x211 = IILnxll ~ 0 as n ~ 00, which implies Xl = X2· The existence is proved as above using the approximating bequence {Xn}~=o defined inductively by Xn = Y + LXn-l, Xo arbitrary in X The contraction mapping theorem is a special case. Theorem A.6 Let X be a Banach space and L be a bounded linear operator from X to itself. If L is a contraction. i. e. there is a constant 0 ::; I < 1 such that
\I Lx II ::; I
\lx\I ,
XEX,
then L has a unique fixed point x, i. e. x = Lx
=
Proof. We can apply the previous theorem with y L~=() Itt \lx\l = l~-Y \lx\l < 00.
0 since L~=o \lLnx\I ::;
A.2 Hilbert spaces
There is a class of bpecial Banach spaces that enjoy many of the properties of the familiar Euclidean spaceb 1R1t and en, namely the Hilbert spaces, whose norms ari5e from an inner product We follow the presentation in Rudin ([38]).
Definition A.7 A complex vector space H is an inner product space if there is a map ( , ) from H x H to e satisfying for all x, y E H and ). E e, (x,y)
=
(y,x),
().x, y)
=
). (x, y) ,
(x,x)
>
o and
(x, y)
(x+z,y)
+ (z, y) , (x, x)
= 0 {=}
X
= O.
vi
Then \lxll = (x, x) defines a norm on H (see below) and if this makes H into a Banach space, i e the mE-tric d(x, y) = IIx - yll is complete, then we say H is a Hilbert space
A simple example of a Hilbert bPace ib real or complex Euclidean space ]Rn or en with the usual inner product More generally, the space £2 (N) of square summable sequences a = {an };::"=1 with inner product (a, b) = L~=l anbn is a Hilbert space Both of these examples are included as special cases of the Hilbert space L2(f.,t) where f.,t is a pObitive measure on a measure space X and the inner product is (f, g) = Jx fgdf.,t Note that an inner product ( , .) on an inner product space H can always be recovered from its norm 11·11 by polarization: 4 Re (x, y)
=
\Ix + yll2 -
IIx - yl12 ,
x, Y E H.
Lemma A.8 Let H be an inner product space and define xEH Then II II is a norm onH andforallx,YEH,
Ilx\lllyll, \lyll ::; lI).x + yll Jor all ). E e IIx + y\l2 + Ilx - yl12 = 211xll2 + 211yll2 l(x,y)1
Ilxll =
.j(x,x) for
::;
iff (x, y)
= 0,
Proof. For X,y E H and), E C,
o ::; lI).x + yll2 = 1).!2\1x\l2 + 2Re()' (x, y» + lIyl12.
(A. 10)
150
A. Functional Analyau.
Thus (x, y) = 0 implies /lYIl : : : IIAx + yll for all A E C. Conversely, if minimize the right side of (A.lO) with A = -11;ff~ to get
IIAx + Yll2
x i=
0 we
= _I(x, y~12
+ IIyll2 . Ilxll This shows that Ilyll :::::: IIAX + yll fails for some A if (x, y) -=1= 0, and also proves the Cauchy-Schwarz inequality l(x,y}l:::::: /lx/illyli. With A = 1 in (A lO) we now have 0< -
IIx + y/l2
0 so that x - E ~ B x (0, r) Then we conclude 1
1
A(Bx(O,r» c Ax - AE c By(O, 2) - By(O, 2) c By(O, 1), which implies IIAII : : : ~ for all A E r A.4.1.1 Nonconvergence of Fourier series of continuous functions Recall that U = {eint}nEZ is an orthonormal set in L2(1I'), i e ( eimt , eint )
r eimteint dt = {O Jo 27r
=
211"
if m # n if m = n
1
The Stone-Weierstrass Theorem shows that U is dense in H = L2(1I'), and thUh by Theorem A.15 the map F L2(1I') ~ £2(Z) given by Ff(n) = f(n)
(j,e int ) =
=
127r f(t)e- int ::'
nEZ,
is a Hilbert space isomorphi5m of L2(1I') onto [2(Z) Now consider the symmetric partial sums Snf of the FOUIier series of f E L2(1I').
L n
f(k)e ikx =
k=-n
r ~)
27r
127r o
f(t) {
t
L Inr
k=-n
d
17r
n
0
f(t)e-ikt-.!..eikx 211"
eik(X-t)} dt
k=-n dt f(t)Vn(x - t)- = f 211"
211"
* Vn(x),
where n
L
k=-n ei(n+~)9
=
_
e-i(n+~)9
sin(n + ~)O sin ~2
157
A.4. COD1pleten...... theore.....
satisfies
> 2
r !sin(n + ~)Ol I!!
Jo
21(n+~)".
11"
0
dJ)
211" d(}
Isin (}I -()
1 lk7r -11"2 L k Isin (}I k=l 11" (k-l)7r 11.
>
4
n
d()
1
1I"2Lf' k=1 and so tends to 00 ab n --4 00 From the Hilbert space theory above, we obtain that 8 n l converges to £2(T) for all IE £2(T).
118.J - 1112
L \in
For I E G(T) we ask if we have pointwise convergence of 811.1 to I on T However, the property bUPn::::l IIVnIILl(1r) = 00 of the Dirichlet kernel V n , when combined with the uniform boundedness principle, implies that there are continuous functions I E ikx fail to converge at some points in T. In G(T) whobe Fourier series L:~=-oo fact there is a dense Go subset E of G(T) (a set is a Go subset of X ifit is a countable intersection of open subsets of X) such that {x E T : 811.1 (x) fails to converge at x} contains a dense Go subset of T for every lEE 7r l(t)Vn(t)g! Then An E G(T)* and To see this, set Ani = 811.1(0) = IIAnll* = fo21l" IVn(t)1 g! / 0 0 as n --4 00 By the uniform boundedness principle we cannot have (A.I8) sup IAnl1 = sup 18nl(0)1 < 00
i1 IAnl1 is a lower semicontinuous function of I, we albo have that Eo is a GIj subset. Now choose {Xi}i=1 dense in T = [0,211"), and by applying the above argument with Xi in place of 0, choose Ei to be a dense Go bubset of G(T) such that sup 18n l(Xi)1
11.::::1
= 00,
IE E i , i ~ l.
By Baire's Theorem, E = n~IEi is also a dense GIj subset of G(T). Thus for every lEE we have sUPn::::118n l(xdl = 00 for all t ~ 1 Now we note that sUPn>118n l(x)1 is a lower semicontinuous function of x (since it is a supremum of continuous functions), and thus the set { X E
l' : sup 18n l(x) I = n~1
Do}
158
A. Functional Analysis
is a G6 subset of l' for every I E C(1'). Combining these observations yields that there is a dense G6 subset E of C(1') such that for every lEE, the set of x where the Fourier series of I fails to converge contains a dense G6 subset of 1'. Remark A.27 In a complete metric space X without isolated points, every dense G6 subset is uncountable. Indeed, if E = {Xk}k::l = n~=1 Vn, Vn open, is a countable dense G 6 subset of X, then W n = Vn '" {Xk }k=l is still a dense open subset of X, but n~=1 Wn = be given and choose P(x) = Ilf - PII£l('f) < c Since Pen) = for Inl > N, we have
2:::=-N cneinx
such
l en)1 = if-P(n) I5 Ilf - PII£l('f) < c for Inl > N Thus F: £1(1[') --+ t'o(Z) with norm 1 where fo(£:.) is the closed subspace of t'OC(Z) consisting of those sequences with limit zero at ±oo The following application of the open mapping theorem bhows that not every such sequence arises a.. 0 buch that (A 21)
But (A.21) fails if we take f
=
V .. for n large, since
1I~leil"(z) = lix{ -n.l-11, while IIVnIlLl(T)
n-l n}
iifg'(Z) =
1
/00
A.4.3 The closed graph theorem. If X is any topological spa<e and Y is a Hausdorff space, then every continuous map f . X ~ Y hM> a dosed graph (exercise' prove this) A statement that givef> conditions under which the converse holds is referred to as a "dosed graph theorem" Here is an elementary example Suppose that X and Y are metric f>pace5 and Y i::, compact. If the graph of f if> closed in X x Y then f is continuous Indeed, for metric space::, it is enough to show that every sequence {x.J~=1 in X converging to a point a:. E X has a subsequence {x n /,,}k'=1 such that f(xnlJ ~ f(x) as k ~ 00. HoweveI, since Y is compact, {J(Xn)}~=l has a convergent subsequence. say f(x,./,) ~ y E K as k ~ 00. Thus (x, y) i~ a limit point of the gIaph G = {(x, f(x) . x E X}. and since G is assumed dosed, we have (x,y) E G, i e y = f(x) The next theorem gives the same conclusion for a linear map from OIle Banach bpace to another Note that
linearity is needed here since f . IR
~ IR by
f(x) =
{5
!~ ~ ~ ~
has a dosed
graph, but is not continuoUb at the origin. Theorem A.32 (closed graph theorem) Suppose that X and Yare Banach spaces and A : X ~ Y is linear If the graph G = {(x, A(x» x E X} is closed tn X x Y, then A is continuous Proof. The product X x Y is a Banach space with the norm II(x,y)11 = Since A is linear and the graph G of A is clobcd, G is also a Banach space Now the projection 7rl : X X Y ~ X by (x, y) ~ x is a continuous linear map from the Banach space G onto the Banach 5pace X, and the open mapping theorem thus implies that 7rl is an open map. However, 7rl i5 dearly one-to-one and so the inverse map 7r1"1 . X ~ G existf> and if> continuous. But then the composition 7r2 07r}1 X ~ Y is also continuous where 7r2 : X X Y ~ Y by (x, y) ~ y. We aIe done since 7r2 07r}1 = A
Ilxli x + Ilyll y
As a consequence of the closed graph theorem, we obtain the automatic continuity of symmetric linear operators on a HilbeIt f>pace. Theorem A.33 (Hellinger and Toeplitz) Suppose that T is a linear operator on a Hilbert space H satisfying (Tx, y) = (x, Ty) for all x, y E H. Then T is continuous. Proof: It is enough to show that T has a closed graph G So let (x, z) be a limit point of G Then there is a sequence {Xn}~=l C X such that Xn ~ x and TX n ~ z For every y E H the symmetry hypothesis now shows that
(T(x n - x),y)
=
(x n - x, Ty)
~
0
161
A.4. COInpieteness theoreIns
as n
---4
00.
But we also have
= (Txn,y) - (Tx,y) (z,y) - (Tx,y) (z - Tx, y) = 0 for all y E H and so z = Tx, which
(T(xn - x),y) as n ---4 00 Thus (x,z) E G.
---4
shows that
Finally we have the following characterization of Riesz bases in a Hilbert space. Definition A.34 A bet X = {x,,} aEA in a Hilbert space is a Riesz basis for H if there is a linear ibomorphism U . H ---4 K onto another Hilbert space K (i.e. U and its inverse U- i are linear and bounded, but do not necessarily preserve the inner productf» such that U X = {U xoJ aEA is an orthonormal set in K. The operator U is called an Olthogonalizer of X More generally, we say that X is a Riesz basis if it is a Riesz baf>is for its closed linear span VX If X = {XoJaEA is a Riesz babis, then every x E VX can be expanded in a Fourier series relativt' to X·
2:
2:
(Ux, Ux,,) UXrr. = (x, U*Uxa) xa· o<EA Now let H be a Hilbert space and let X = {Xo}aEA be a Riesz babis for H. Denote x = U- 1 Ux = U- 1
by X' = {x~} aEA the biorthogonal sYbtem defined by the relations (Xa, X~) = ntical to that above. A.5.l Separation, subspaces and quotients. Now we discuss some useful applications of the Hahn-Banach Theorem Suppose X is a normed linear space. Then X* beparates points on X Indeed, given Xo i= 0, let M = Span{xo} and define I on M by I(AXo) = Allxo!! The Hahn-Banach Theorem supplies a bounded linear functional A E X* of norm 1 buC'h that A IM= I. If Xl i= X2, then with Xo = Xl - X2 we have
AXI -
AX2
= A(XI -
X2)
= Axo = I(xo) = IIxoll =
!lXl -
X2!!
i= 0
Note moreover that Axo = I(xo) = IIxoll so that
!Ix!! = sup{\Axl
IIAII ~ I}.
(A.24)
The Hahn-Banach Theorem actually yields an even stronger separation result. Namely, if M is a closed subspace of X and Xo ~ M, then there is A E X* with Axo = 1 and A IM= O. For this take I(x + AXo) = A for x E M and A a scalar, and use dist(xo, M) = ~ > 0 to obtain
ll(x + Axo)1 = IA\
~ IAI!!A - I XO+ xoll
=
~ IIx + Axoll.
A. Functional Analysis
164
Now let A be a norm preserving extension of f to X. There is also a separation result for disjoint closed balanced convex sets A and B, one of which is compact - the case A is a singleton is especially useful. A set E is balanced if o.E c E for all 0. E C with 10.1 :::; l. Corollary A.38 Suppose that A and B are disjoint closed convex balanced subsets of a Banach space X If A is compact. there are A E X* and /3 < 'Y such that ReAx:::; tJ < 'Y:::; RE-Ay
for all x E A and y E B. In the bpecial case A = {a} is a singleton. there is A E X* such that xEB (A.25) IAxl:::; 1 < IAxol, Proof' It suffices to prove the case of real l>calars Let r > 0 be buch that Ar = A + B(O, r) and B are disjoint Let p be the Minkowski functional of r = Ar - B + Xc, i.e.
p(x)
= inf{t >
°
t-1x E
r}
Since r is a convex balanced neighboUI hood of the origin, straightforward computations show that p is a seminorm on X Fix ao E A. and bo E B and set Xo = bo - ao Then Xo rt- r since Ar n B = a linear functional A withAxo = 1 and IAxl :::; p(x) for all x E X Since r is a neighboUIhood of the origin, we have A E X* Now for a E A and bE B we have Aa - Ab + 1 = A(a - b + xo) :::; p(a - b + xo)
joint convex subsets of lR. with AAr to the left of AB. Now A. is open and A il> an open map, and thus AA,. is open If we set 'Y = sup AA,., we obtain ReAx
< 'Y:::; Ay,
xEA.,yEB
Now we observe that AA is a compact subset of the open set AAr to get such that
ReAx:::;
/3 < 'Y:::; Ay,
/3
et5 W2 form a local bas(' for the topology T p at Ao Since K C X* n P, we have WI n K = W2 n K and (I) is proved. To bee (2), suppose that fo is in the Tp closure of Kin P. Then we have that fo is linear Indeed, simply approximate fo by f E K at the points x, y, ax + f3y and note that the linearity of f yields f(ax + f3y) = af(x) + f3f(y). We also have that Ifo(x)/ ::; 1 for x E V by again approximating fo by f E K at x and then using If(x)1 ::; 1. Thus fa E K and K is Tp closed. Corollary A.45 If X is a 5eparable normed linear space, and K is the closed unit ball in X*, then K is metrizable and compact in the weak* topology. Proof: If {Xn}~=l is dense in X, then the functions W ~ W(x n ) are weak'" continuous on K and separate points on K. It follows that
d(A,4.» =
'f n=l
Tn IA(xn )
-
SUPWEK
4.>(x n )1 Iw(xn)1
defines a weak* continuous metric on K. Since the metric topology T generated by d is Hausdorff, and contained in the compact weak* topology , on K, it follows from the rigidity of compact Hausdorff spaces that the metric topology T coincides
168
A. Functional Analysis
with the weak* topology "1 on K (since T C "1, the identity map Id takes (K, "1) continuously one-to-one onto (K, T), so if G E "1, then Gc- is ('ompact in (K, "1), Id(GC-) is compact in (K, T), hence GC is closed in (K, T) and GET) A.6 Compact operators
A linear operator T mapping one Banach space X to anot heI Y is said to be compact if T B is preeompact in Y where B is the unit ball in X - precompact means the closure is compact Thus if {Xn}~=1 is any bounded sequence in X, the &equence {TXn};:C=l has a convergent subsequence in Y. Examples of compact operators include all bounderllinedI operators T . X ---+ Y into a finite dimensional space Y, as well d,l:; bounded lineal operators Twit h finite dimensional range RT Lemma A.46 If F is a finite dimensional subspace of a normed linear space Y, then F zs closed in Y, and the re8tnctWrt of the Y topology to F coincides with the topology mduced by any linear isomorphzsrn of Y with C' Proof: Let f en ---+ F be a linear isomorphism Since f (Z 1, , Z7l) = zd(eJ) + + znf(en ) and vector space opelations are continuous in Y, it follows that f is continuous Thus f(§n-l) is compact and disjoint from 0, and there is r > 0 such that B(O,r) nf(§TL-l) = ¢ where §?I-l = oIffi n and Iffin i:-, the unit ball in en Now 0 E E = f-l(B(O, r) n F) is convex, hence connected. and it follows that E C Iffin From this we obtain that each component of f- 1 . F ---+ e" is a bounded linear functional on F (with norm at most:), and bO f- 1 is continuous by Lemma A 3 This proves that the restriction of the Y topology to F coincides with the topology induced by the isomorphism f It remains to prove that F is closed in Y. Pick y E F Then y E 2~B(O,r) = tB(O, r) and so
y E F n tB(O, r) = t(F n B(O, r» since in Y
f
continuous dnd
tiffin
C
f(tlffin)
compact imply that
C
f(tlffin) =
f(m n )
f(m n ) i..oc Tnx11 converges, say to Yn, fOI each n ::::: 1. It is now easy to show that the sequence {yn}~=l convelges to y E Y, and that lime_co TX Jt = y. This proves T is compact In the case that both X and Y are Hilbert spaces, the converse can be proved with the aid of the Schmidt decomposition - sec Theolem 4.16, Proposition 417, and (4.23) We leave the details to the interebted reader. Finally we mention that it was not until 1973 that P. Enffo produced an example to show that not every compact operator OIl a separable Banach space can be approximated by finite rank operators.
A.6. Compact operators
169
Theorem A.48 (Fredholm alternative) Suppose that T: H ~ H is a compact operator on a Hilbert space H Then 1. either the equation (I - T)x = 0 has a nonzero solution x E H, 2 or the equation (I - T)x = y has a unique solution x E H for each y E H In this case, the inverse linear operator (I - T) -1 is bounded on H. First we re a conl>tant C l>uch that IIxll:::; CIISxll,
xEH
(A.26)
Indeed, if not then there il> a l>equence {X1t}~=l C H with IISx n ll = 1 and IIxnll /' 00 Then Zn = "::,, it, in the unit ball of H and IISznll \.. 0 Since T is compact, there il> a l>ubsequence {znJ. }k'~=l l>uch that TZnk converges in H, to say w. But then ZlI. = SZn. + TZnk convergeb to 0 + w = wand since Sw = limk->oo SZnk = 0, the assumption Ns = {O} yields w = O. This contradicts IIznA, II = 1 for all k, and completel> the proof of (A 26). Second, still assuming Ns = {OJ, we obtain from (A.26) and the boundedness of S that Ilxll S; C liSa-II :::; C f Ilxll for all x E H. This easily yields that Rs is closed, and moreovez that Stakes clObed sets to closed set&. Third, st.ill assuming N.s = {O}, we claim that Rs = H. Let Vk = Sk H Then VII ib dosed by induction using the previous step, and Vk+l C Vk for all k. We must have Vk = Vk+ 1 for somE' k bince otherwise there is YII E Vk '"Vk+ 1 with II Yk II = 1 and Uk .i Vk+l' But then we have for n > Tn,
by Lemma A 8 since SYm - SYn + y" E \!,.n+l and Ym .i Vm+ 1 Thus {TYn}~=l has no convergent l>ubsequence, contradicting, T compact. So VA. = Vk+ 1 for some k Then for Y E H we have Sky = Sk+1x for some :1:' E H Thus Sk(y - Sx) = 0 implicl> y = Sx upon iterating (A.26): lIy-Sxli
< CIIS(y-S.1)II:::;C2 \1S2(y-Sx)!1 < :::; C k IISk(y - Sx)1I = O.
This bhows that S ib onto, and then (A 26) bhows that S-l . H ~ H exists and i~ bounded. Fourth, we claim that Rs = H implies Ns = {O}. This time let Vk = S-k( {O}). Then Vk is closed by the continuity of S, and Vk C Vk+l for all k. Ar argument analogous to that above shows that there is Tn such that Vn = Vm fOJ all n 2: Tn Given y E v'n, an induction using Rs = H shows that Rs.n = H, ane so there is x E H such that y = smx Thus s2mx = STRy = 0 by the definitioI of y E Vm . So x E V2m = Vm implies that y = smx = 0 as well. Thus Vm = {O~ and hence SO also the smaller space VI = S-l ( {O} ). Thib completes t.he proof tha
Ns = {OJ
A. Functional Analysis
170
We will use the following lemma in the proof of the spectral theorem for a compact operator on a Banach space. Lenuna A.50 T : X
----+
Y is compact if and only if T* : Y*
----+
X* is compact
Proof' Suppose that T is compact, and let {Y~};:"=l be a sequence in the unit ball of Y* Then the sequence of functions {fn};:"=l given by fn(Y) = (y, y';.> if> equicontinuous on Y. Since T(B(O, 1)) has compact closure in Y, Ascoli's Theorem shows there is a subsequence {fnJ~l converging uniformly on T(B(O, 1)). Now
sup I/Tx,y~i-Y~j)l= sup !fn,(Tx)-fnj(Tx)!, II T*Y~, -T*Y;jll= 113"11:51 \ IIxll::O;l together with the completeness of X* shows that {T*Y~i }~1 converges Thus T* is compact The reverse implication can be proved by the same met.hod Theorem A.51 Suppose X is a Banach space and T ts a compact linear operator on X Then 1. If A of. 0, the following n1Lmbers are equal and finite'
a {3 a* (3*
dim ker(T - AI), dim X/range(T - AI), dim ker(T* - AI), dim X* /range(T* - AI)
2 If A of. 0 and A E aCT). then A is an eigenvalue ofT and T* 3. a(T) is compact, at most countable. and has at most one hmit point. namely
O. Note that by Theorem 31 and Remark 3.2 above, aCT) is compact ('ven when T is merely bounded Proof. From Proposition AAO we have dim(X/M)
= dim(X/M)* = dimM.L
(A.27)
whenever M is a closed subspace of X. We now claim M = range(T - AI) is dosed in X Indeed, the restriction of T to the closed subspace ker(T - AI) is compact, one-to-one and onto, hence open by Theorem A 29. But then the unit ball in ker(T - AI) is compact, hence ker(T - AI) is finite dimensional, i e a < 00. Now a finite dimensional subspace is easily seen to be complemented in X, i e there is a closed subspace N in X such that X = ker(T - AI) E9 N. Define S : N ----+ X by Sx = Tx - AX Then S is one-to-one and bounded on N Moreover, the range of S coincideb with M, the range of (T - AI) So it suffices to show that IIxli
s CIISxjj,
X
EN.
(A 28)
However, (A.28) can be proved in the same way (A.26) was proved above (that. proof was in the context of Hilbert spaces, but the argument carries over here) Thus M is dosed Also. M.L = ker(T* - AI), and so (A.27) implies {3 = dim(X/M) = dimM.L = a*.
Since T* if> compact by Lemma A 50, we obtain {3* = a as well. If we can prove that a S (3, then a* ::; (3* follows by the same proof since T* if> compact. We would then have a ::; {3
= a*
::; {3*
= a < 00,
which would complete the proof of assertion 1.
171
A.S. Compact operators
So we must show a ~ f3. Suppose in order to derive a contradiction, that a > f3. Since a < 00, both ker(T - AI) and Xjrange(T - AI) are finite dimensional, hence complemented in X. Thus there are closed subspaces E and F in X such that
= ker(T - AI) ffi E = range(T - AI) ffi F. (A.29) Since dim ker(T - >..I) = a > f3 = dim F, there are a linear map cP from ker(T - AI) X
onto F and a point Xc
E
ker(T - >..I) with cp(xo) = 0 and Xc 4.>(x) = Tx
+ CP1l(x),
X
E
i- O.
Define
X,
where 11 X --+ ker(T - AI) is projection relative to the first direct sum in (A.29). Clearly 4.> i& bounded. Since cp has finite range. cp is compact, hence so are cp7r and 4.> Now we observe the following &tatement~,. 4.> - >..I (4.> - AI)(E) (4.> - >..I)x range( 4.> - >../)
T - >../ +
cp7r,
range(T - AI), ..px, X E ker(T - >..I), range(T - AI) + F = X.
J
Then we have (4.> - >../)xo = cp(xo) = 0 so that ker(4.> - AI) i- is also compact. By an argument bimiliar to that used to prove Theorem A.49 above, we conclude that range(4.> - AI) i- X, the desired c,ontradiction We note that in the argument used to prove Theorem A.49 above, we can replace the Hilbert space Lemma A.8 with this elementary buhstitute if M is a nondense subspace of a normed space X, then for every r > 1 there ib x E X satibfying
Ilxll < r
and IIx -
yll
2 1 for all
y E Y.
(A.30)
Assertion 2 followb from assertion l' if A is not an eigenvalue of T, then a = 0 and assertion 1 implies f3 = 0, i.e range(T - AI) = X Thus T - >../ is invertible and A fj. aCT) Finally, to establish asbertion 3, it remains only to show that 0 is the only possible limit point of aCT) (we already know aCT) is compact from Theorem 31 and Remark 32 above) To see this suppose in order to derive a contradiction that {>..1t1~=1 ib a sequence of eigenvalues of T with IAnl > r > 0 for all n 2:: 1. Choose corresponding eigenvectors Vn and let Mn = Span{vk}k=l We have (A.31)
n
(T -- AnI)x =
L ak(Ak -
TI-1
An)Vk =
k=l
L
ak(Ak - An}Vk
E
M n- 1 •
k=l
Thus we have
T(Mn) (T - AnI)(Mn) Using (A.31) and (A.30) there are Xn
II x nll < r
C C E
Mn, n 2 1, M n- 1 , n 2:: 2.
Mn for n 2:: 2 such that
and I\xn - xii 2 1 for all x
E
Mn -
I.
(A 33
172
Fix 2 ~ m gives
A. Functional Analysis
< n and set z = TX m - (T - AnI)xn E Mn- 1 IITxn - TXmll
by (A.32). Then (A.33)
= IIAnXn - zll = IAnlllXn - A~lZII ~ IAnl > T.
Thus {Xn}~=l is bounded and {TXn}~=l has no convergent subsequence, contradicting T is compact.
APPENDIX B
Weak Derivatives and Sobolev Spaces We follow Gilbarg and Trudinger [21J, beginning with a discussion of weak derivatives. B.l Weak derivatives
Given f, 9 E LioAQ) and 1 :-::; J :-::; n, we say that 9 is the weak derivative of f in Q provided
f
f
gcpdx = -
10.
f(
10.
~Cp )dx,
jth
partial
for all cp E C~(Q).
UXj
Here C~m (Q) denotes the normed linear space of all continuously differentiable functions cp with compact support in Q, and norm given by
IICPlieI
(0.)
com
= sup [cp(x)J + sup ]Vcp(x)l. xEn
xEn
When 9 is the weak lh partial derivative of f in Q we write 9 = .!!L88 • This should Xj cause no confusion since it is easily verified that this weak definition is an extension of the classical definition of partial derivative for a continously differentiable function f (use integration by parts). We next develop a useful approximation property of weak derivatives, but first need a standard lemma on convolutions with a smooth approximate identity. Lemma B.l Ifu E Lioc(Q), then u*e
o as £ - t 0 for every compact K c Proof· Suppose first that
f
If 6 - f[
u in Lioc(Q), i.e. IK lu * e - u[-t
E C~om(Q). Then for b
111
1*
-t
Q.
> 0,
{f(x - y) - f(Y)}6(Y)dY\ dx
< sup sup If(x - y) - f(y)J x
lyl~6
tends to 0 as 5 ~ 0 by uniform continuity of f. Now given u E L;oc(Q), K compact in Q and £ > 0, choose K c Q' <E Q and use that COcom(Q') is dense in L1(Q') to find f E COcom(Q') such that Io,lu- fl 0-/1+ il(J-u)1
.1...)' which then converges to v in L[o("(n). The converse is easy. m B.2 The Sobolev space wI.2(n) Definition B.3 Let w1.2(n) ("onsist of those (complex-valued) functions
I
E
L2(n) with V IE L2(0) in the weak sense Define an inner product on WI 2(n) by (J,g) =
k/g + k dX
Vf Vgdx.
(B1)
Theorem B.4 W 1 ,2(n) is a Hilbert space with the inner product (B.1). Proof· We prove completeness If {Jd~1 is Cauchy in WI 2(0), then {fdk::l and {VlkHo=1 are Cauchy in L2(0) and @n£2(n) Thus there are l,gl . . ,9n E L2(n) such that fk ---; I and ~ ---; 9j in L2(0) for 1 ::; J ::; n We must now show ) that 9J
=
3!- in the weak sense. Letting k ---;
00
}
in the equation
{ Ik( 8.p )dx = _ ( 8fk ..pdx,
10.
8Xj
10. 8Xj
yields fn/(if)dx = - fngjcpdx for all cp E C(lom(O) as required. J Now we give an explicit example of a ("ompact opelator T on the Hilbert space 00 For this we need the thE' Sobolev Embedding Theolem which shows that functions I in W1.2(0) are bettel than square integrable, namely IE L2' (0) for 21, = ~ - ~, at least when n ?: 3 and 0 is a Lipschitz domain. The difficulties with the boundary disappear for the subspaC'e W6",2(0), defined to be the closure in Wl,.2(O) of the spaC'e C;om(O). Clearly WJ 2(0) is a Hilbert space with the inner product (B 1). We have the following embedding Wl,2(0) with dim RT =
Theorem B.5 (Sobolev embedding) Let 0 be an open subset o/]Rn WJ,2(0), then u E L 2' (0) where 21, = ~ - ~ when n ?: 3, and lor all 2* < n = 2 Moreover we have
Proof: For any inequalities
If u E 00
when
I E WJ,2(0), extended to vanish outside 0, that satisfies the 1 ::; j ::; n,
(B 2)
B.2. The Sobolev space W , ,2(n)
where
x) = (Xl,
[!(x)[""
175
.. ,Xj-l,tj,Xj+l, ... ,xn )
(for example
I
E
C';'m(O)), we have
"{llf-~ [1i;!(X;)[dt;}"" ~ II {(_~ 11i;!(X')ldt,}~;
Now integrate over
Xl
in lIt and use Holder's inequality n-l
!!h1 ,
.
,hn - 1 !!L1 :=;
II !!h}IiL',-l
(B.3)
j=l
to obtain that
1:-00 I/(x)1 "~l
dx l
g
1 Indeed, lull' is absolutely continuous in x) and the pointwise derivative OJ lull' batisfies 10j lul")'l = 'Y lul'Y- 1 !ojul a.e. since equality holds if either u =1= 0 or 'Y
= O. For this we note that the set {u = 0 and OjU =1= O} is an Fq set, hence measurable, with at most countably many points on each line parallel to the lh direction. Integration by parts shows that OJ !ul" is the lh weak partial derivative of lul1', that (B.2) holds and that !V!un = 'Y !U!"-l IVuj a.e. With 'Y = 2:=~ we have
OjU
as required. The case n
= 2 is similar and is left as an exercise.
176
B. Weak Derivative" and Sobolev Spaces
Finally, to extend this inequality to arbitrary U E W~,2(Q), choose a sequence C C~om(Q) that converges to U in W 1,2(Q) Then we obtain that
{Uk}
so that {Uk} converges in L2* (Q) to a function which must be u. Then
When q < 2* and Q is bounded, the Lebesgue space Lq(Q) is strictly larger than L2* (Q), and the natural embedding of W~ \Q) into U(Q) via the identity map turns out to be not only continuous, but also compact. Theorem B.6 (compact embedding) IfQ is a bounded open subset ofJRn, then W~,2(Q) embeds compactly in Lq(Q) for all q < 2* = 2n~2 Proof· Recall that a set E in a metric space is compact if and only if every sequence has a convergent subsequence. We will also use a btandard chaI acterization of compactness in a complete metIic bpace· E is compact if and only if E is closed and totally bounded (for every I': > 0, E C Uf=1 B) for some finite collection of balls {B) }f'=1 of radius 1':). Also, an interpolation inequality for Lebebgue spaces (which follows from Holder's inequality) together with the Sobolev Embedding Theorem above shows that
lIuIlLQ(n)
S
lIull~l(l1) lIull~;!(I1) S lIulI~l(l1) (Cn Ilull w,; l(I1)1-8,
1
{}
1-{}
q= 1 + ~
(BA) Thus it is enough to prove the compactneSb of the embedding of 2(Q) into £l(Q). Indeed, if {Uk}k=1 is a bounded bequence in WJ,2(Q) that converges in Ll(Q), then (BA) show& that {Ud~1 is Cauchy in U(n), hence convergent there as well So let A be a bounded set in W5,2(Q). We must show that A is compact in Ll(Q), and for this we may assume without loss of generality that A c C~om(Q) and lIullwg 2(11) S 1 for all u E A. Let ¢ be a smooth nonnegative function supported in the unit ball of JRn having integrall, and set ¢h(X) = h-fl¢(~) for h > 0 Then Uh = u * ¢h is smooth and the following elementary estimates hold.
W(;
IUh(X)1
0 the set Ah = {Uh : U E A} is a bounded equicontinuous subset of C(n). By Arzela's Theorem, Ah is precompact in C(n), and thus precompact in Ll(Q) since the embedding of C(n) into Ll(Q) is obviously continuous.
B.S. Maxln>al functions
177
Next we observe that writing
Ilu - uhll£l(n)
z = Izi w,
liU(X) - { n
<
0 (since it is precompact), (B.5) ~hOWb that A i~ totally bounded in L1 (n) as well, and thus precompact as required. \Ve dObe this section with a version of the chain rule for Sobolev spaces
f
0
Um
Lemma B.7 Let f E C1(JR) with l' E Loo(JR) and suppose u E W 1 ,2(n). Then u E w1.2(n) and '\l(l 0 u) = (I' 0 u)'\lu Proof By Lemma B.2 there is a sequence {u m } of smooth functions such that u and '\lum -+ '\lu in Lfor(n) Then for n' <s n,
-+
< 111'1100 { IUrn - ul -+ 0 in' { 11'(um )'\lurn - 1'(u)'\lul < 111'1100 { l'\lurn - '\lui in' in' ( If(urn) - f(u)1
in'
+ (
in'
11'(um )
-
as m
-+ 00;
1'(u)II'\lul
also tends to 0 llli m -+ 00 upon applying the dominated convergence theorem to the last integIal ll~ing the continuity of l' and assuming, as we may by passing to a bubbequence, that Urn --+ U 3..e in n'. This ~hows that f 0 Um --+ f 0 u and '\l(l 0 um) = (I' 0 um)'\lu m --+ (I' 0 u)'\lu in LioAO), and we're done.
B.3 Maximal functions Let V = {2 k (J + [0, l)n)}JEZn,kEZ be the grid of dyadic cubes in JRn, and define the dyadic maximal function Mdv f of a locally integrable function f on JRn by
Mdy f(x)
= sup IQll ( If(y)1 dy, XEQE'D
iQ
x E JRn .
Thus Mdy f(x) is the optimal upper bound for all the dyadic averages of fat x. In order to study the convergence of the dyadic averages of f we consider the dyadic upper oscillation r dy f of f:
r dy f(x)
= lim
~~ I~I 10 If(y)1 dy,
x
E
JRn,
where it is understood by the expre55ion Q -+ x that Q is a dyadic cube containing x whose side length is shrinking to zero in the limit Clearly we have
rdY(l - f(x))(x) = 0 implies f(x) =
4i~T I~!
k
f(y)dy.
(B.6)
178
B. Weak Derivatives and Sobolev Spaces
Of course we have
r dy I(x) $
(B.7)
Mdy I(x),
and the key properties of the maximal function Loo(JRn) and of weak type 1 - Ion L1(JRn ):
Mdy
are that it is bounded on
A>O
(B 8)
To see (B.8) define
nA
{x
E
CPA
{Q
E 1):
JRn . Mdy I(x) > A},
I~I
kII(y)1
dy > A},
and let {Qm}m be the set of maximal dyadic cubes in pairwise disjoint and we have
nA =
U
Q=UQm.
QE>.
m
CPA'
Then the cubes Qm are
Unravelling definitions yields
InAI =
L IQml < L ~ 1 If(y)1 dy $ m
m
Qm
~
in
If(y)1 dy
IR
As a consequence we obtain the maximal theorem.
Theorem B.8 For 1 < p $
00
we have
Proof' The following argument it, from Marcinkiewicz interpolation (page 14 1>.. = x{IJI>~}f so that MdY(j - fA) $ by the boundedness of Mdy on Loo(JRn) with norm 1. Consequently, by the subadditivity of Mdy we have
q
in [46]). Define
and thus
{x
E lR,n : Mdy I(x)
> A}
C
{x
E lR,n . Mdy f>..(x)
>
~}
(B.9)
B.3. Maxinuol functions
179
Fubini's theorem, (B.9) and then (B.8) applied to
hn
IMdYf(x)IP dx
=
Ln {foMdY!(Z} P fooo l{x
. (x) >
1 {~h" 11>.(x)1 1 {11 00
00
o
p
\
-
A
21fCZ)1
I/(x)1
~}\ )..p-1d)..
dx} )..p-1d)..
{zERn If(z)l> ~}
1 1 ~
1>. yield
I/(x)1 dx
AP- 2 dAdx
=
}
)..p-1d)..
_P_2 v- 1 p-l
0
1
If(xW dx.
~
The weak: type inequality (B.8) also yields the Lebesgue Differentiation Theorem.
Theorem B.9 For IE Ltoc(]Rn) we have
I(x) =
4~z I~I
k
a.e.x E ]Rn.
I(y)dy,
Proof· Since the conclusion of the theorem is local it suffices to consider Ll(]Rn) with compact support. Given c > 0 we can choose 9 E C(Rn) with J II - gl < c by the density of C(]Rn) in Ll(]Rn) However, rdY(g - g(x»(x) = 0 for every x E R n since 9 is continuous. It follows from the subadditivity of ~Y and (B 7) that
I
E
rdY(J - I(x»(x)
< rdY(J - I(x) - [g - g(x)])(x) + rdY(g - g(x»)(x) < rdY(J - g)(x) + rdY(J(x) - g(x)(x) < MdY(J - g)(x) + I(J - g)(x)l·
Now we have
{x E R n : rdY(f - f(x»(x) > )..}
c {x E R n : MdY(I - g)(x) >
~} U {x E Rn
:
1(1 - g)(x)! >
~}
and so
!{x
E
Rn
:
rdY(J - I(x»(x) > )..}I
::; I{x E Rn : MdY(J - g)(x) > +
< ~)..
l{
f
x
E]Rn :
II - gl
!(J - g)(x)1 >
f
~}l
~}
I
+ ~A If - gl < ic. A
Now let £ ----+ 0 to obtain I{x E Rn : rdY(J - f(x»(x) > )..}l = 0 for all ).. > O. This proves that rdY(I - f(x»(x) = 0 for a.e. x E JRn, and (B.6) now concludes the proof of Lebesgue's differentiation theorem.
B. Weak Derivatives and Sobolev Spaces
180
B.4 Bounded lllean oscillation and the .John-Nirenberg inequality
We say that a function I
E
Lloc(JRn) has bounded mean oscillation, if
1If1I BMO (Rn) = s~p I~/
10 I/(x) - IQI dx
0, set
s.
n). == {x E JR"
MdYh(x) > A}.
As in the previous section, we can write
m
where {Q~}m are the maximal dyadic cubes in the colleetion ~>. = {Q E V·
We daim that with 'Y = 1 +
2"{".
/n-y>.
and A ~
n Q~I S.
IhlQ > A}.
1I/i1*,
~ !Q;'./
Indeed, if Q~. is the dyadic parent of Q;'" then
and 1
[
IQ~I iQ~, /hl By maximaIity,
1
=
[
/Q~tl iQo Ihl
(B.IO)
for all m
Q;n
c Qo since otherwllie Q~
1
s. IQ~I"/II* IQol $
Ilfll*·
J
Qo
B.4. Bounded mean oscillation and the John-Nirenberg inequality
EOI'>" n Q~
XQA (h - hQ>. ), take 1 > 1 and x such that x E Qj>" C Q~ ~nd Now let 9
=
,).
..1
Then there is some i
-1-1 Igl + Ih>:: I: ; -1-1 Igl + ).. IQ7>"1 Q?>' Q IQtl Qi"
Ihl
" n Q~\ < \{Md'J g > b -
b
1»'}\ ::; b ~ 1».
JIgi
~ 1». k~ II - IQ~.1 ::; b ~ 1».11111* !Q~I 2n
b - 1».11111* \Q~\.
11f1l •. Now sum (B 10) in m to obtain \0>"+211111.\ = 101'>. 1 = L \0,>.. n Q~\ ::; L ( :n1». 11111* \Q~\ = ~ 10",1· (B.ll)
This establishes (B.IO) with,
= 1+
n
2 +
1
m
m
We now obtain with c = \{x E
21itl!.
'
that
Qo : MdYh > ).}\ ::; Ce-C.>"IQol
for all ). ~ 11111* by iterating (n 11) and using \0211111.\ ::; IQol For)' < II/I!. we simply use {x E Qo : Md"h > ).} c Qo and increase C to ecllili • if necessary to obtain that
Finally, Lebesgue's Differentiation Theorem B.9 shows that
MdY[XQ,,(I - IQo)](x) 2:
1(1 -
IQo)(x) I ,
a.e.x E
and this completes the proof of Theorem B.1O. Corollary B.ll For IE BMOdY(lRn ) we have
k
elf(x)ldx
0 and let Bn be the Blaschke product formed with the first n zeroes (listed according to multiplicity) of I. For each fixed n, Bn is smooth near 11' and so )B n (re i9 ) - t 1 uniformly on 11' as r
-t
1. Thus
f,;
satisfies
I! -i lip = 1If1lp
I
As n
---> 00,
If,; I
increases to Igl so that
0< r < 1, by the monotone convergence theorem. Letting r - t 1 we obtain IIgli p :-: ; hence equality. The case p = 0 b left as an exercise.
1If1l p'
We can now obtain the important factorization of an HI function as a product of H l. functions. Corollary C.B Suppose 0 < P < 00, I E HP, I '" 0, and let B be the Blaschke product associated with the zeroes 01 I Then there is a zero-free Junction h E H2 2 such that I = Bh"P In particular, every I E HI is a product gh with g, h E H2 and U/IIHI = IIgllH2 IIhllH2 Proof The zero-free function
= i.
such that e'fJ
= e~'fJ E
Then h
Bh~. If p = 1 put 9
-Ii satisfies I -Ii lip = l!fllpand there is
0 and P(z.eit)dt = 1 for z E JI)), we have 11/.11 1 :::; Illlll for 0 < r < 1, hence f E HI. But then f = 1P'[f*] by Theorem C 10, and the uniqueness of the Poibson reprebentation implies that dfJ,(t) = f*(t)dt. Uniqueness of the Poisson integral is proved ab follows' if 1P'[fJ,1 == 0 for a complex Borel measure fJ" and if IE G(T), then by the symmetry of thE' Poibson kernel,
h
(IP'[/]). (e i6 )dfJ,(0)
=
h
(1P'[fJ,])r( eio)f( ei8 )dO = 0
for all 0 < r < L Since (1P'[j]), -+ f uniformly on T as r IdfJ, = 0 for all I E G(T), hence Jl = 0
IT
-+
1, we conclude that
C.l Factorization theorems
We say that a holomorphic function M on the disk is an inner function if M E HOC and M* (e i9 ) = 1 a e on T By Theorems C.5 and C.6, BI8.bchke products are examples of inner functions, and M(z) = e::; is an example that is e-. d.nd "y.,tems an easy reading, Volume 2 110del operators and systems Translated from the French by Andreas Haltmann and revised by the author. Mathematical Surveys d.nd 1Ionographs 93, A 1\.1 S Providence. RI, 2002 [32] J 11 ORTEGA AJIOD.J FA-BREGA Pointwise multipliers and wrona type decomposition in BMOA, Ann Inst Fourier, Glenoble 46 (1996). 111-1'37 [33] J 11 ORTF,GA AND J F ABREGA Pointwise multipliers and decomposition theorems in analytic Besov spaces Math Z 235 (2000), 53-81 [34] V V PELLER A~D S V HRI'S( r::v. Hankel operators, best approximations, and stationary Gaussian processes. RUSf>icl.D Math Surveys 37 (1982), 61-144 [35] R ROCHBERG, Restoration of unimodular functions from their holomorphic proJections, 1999 [36] W ROS5 The classical Dirichlet space, Contemporary 1I.1ath [37] W RUDIN Real and Complex Analysis, 1IcGrd.w-IIill, 3rd edition. 1987 [38] W Rl'DIN. Functional Analysis, Internation~l Selies in Pure and Appl I\lath. 1\.IcGraw-Hill. 2nd edition, 1991 [39] W RllDiN. Function Theory in the unit ball oj cn, Springer-Verlag 1980 [40] D SARA,)ON, Function theory on the unit circle, Virginia Poly lnst and State Univ Dept of Math, Blacbburg, Va 1978 !41] E SAWYER AND R L WHCEDEN, Holder continuity of 'Subelliptic equation'> with rough I;o(>fficients, Memoirs of the A M S 847 [42] 1\.1 SCHECHTER, Principles of Fum tional Analysi." Academic Press. New York - London, 1971 [43] K SEIP. Interpolation and sampling in spaces of and.lytic function& University Lectur(> Series 33, American Md.thE'matical Societ} 2004 [44] H SHAPIRO AND A SH1ELD~, On the zeroes of juncfiom with finite Dirichlet inteqral and some related function spaces Math Z, 80 (1962) [45] D STEGEN(,A, Multipliers of the Dirichlet space, Illinois J I\I~th 24 (1980), 113-139 [46J E 11 S rF1N. Harmonic Analysis real-1Jariable methods orthogunality, and osdllatorv integrals, Princeton University Press, Princeton, N J 1993 [47] B SZOKErALVI-NAGY ANn (' FOIA!~, HMmonic analy.,is of operdtors on Hilbert &pace NOlth Holland, Amsterdam, 1970 [48] TCHOUNDJA, preprint [49] S TREIL .. ND B WICK, Anal'l/tic- proJections, corona problem and geometry of holomorphicvector bundles, preprint [50] V A TOLOKON~IKO\'. The corona theorem in algebr(U, of bounded analytic junctions, Amer Math Soc Trans 149 (1991).61-93 [51] N TH VAROPOULO'5 BMO junctions and the "& equation, Paociated measure, 112 Banach space. 148 Banach-Alaoglu theolem, 166 Bari's theorem, 161 Bergman metric, 137 B(,'SOv-Sobolev spac(> 1 best appIoximation 51,76.78,81 Beurling transform, 47 BeUlling''l thcorE'm, 193 Blaschke product. 40, 94, 117 finite, 4, 64 genE'zalized 99 in U 15 bounded me.m oscillation, 4, 59. 180 analytic, 59 Carlcson measurE' for B~ (Ill" ), 103 fox HP(I[]», 10 V-,53 for the Dirichlt spac.e V(I[]». 8 fox the Dirichlet bPace B~(J8n), 133 for thE' tre
