CID. On the other hand, Iw..]- ar(w..) > Iw,.I - (~c2(1 - Iw, I) = (1 + ac2)lw,.I- ac2 -+ 1 as n -+ ~. Thus, for ally 0 < 1' < 1, there exists N > 0 such that when n > N, Iw..
Integr Equat Oper Th Vol. 21 (1995)
HANKEL
0378-620X/95/040460-2451.50+0.20/0 (c) Birkhiiuser Verlag, Basel
OPERATORS
SPACES WITH
ON THE WEIGHTED
EXPONENTIAL
TYPE
BERGMAN
WEIGHTS
P E N G LIN AND R I C H A R D R O C H B E R G *
Let AL~(ID) denote the closed subspace of L 2 (D, e-2~~
consisting of analytic
functions in the unit disk D. For certain class of subharmonie ~ : D -+ IR, the Hankel operator Hb on AL2~(D) with symbol b E L2(ID) is studied. Criteria for boundedness and compactness of such kind of Hankel operators are presented.
1. I N T R O D U C T I O N
Let dA denote the area measure for tile unit disk D in the complex plane C. Let L2(D) denote L2(D, dA) and let L~176 denote L~(D, dA). Let ~ : ID -+ ~R be a subharmonic function. Let L~(ID) be the space of all measurable functions f on ]I) such that e-~f E L~(D) and let H~(ID) denote the subspace of L~(D) consisting of analytic functions. Let L~(D) be the Hilbert space of all measurable functions f on ][3)such that
]lf]]L~ ----: (,g If] 2e-2~ dA) 1/2
- 1 . In the present paper, we study the Hankel operator on the weighted space AL~(D) for certain class of subharmonic qo. The typical examples of our weight e -2~ are
(1 -Iz]Z) A, A
> 2 and (1 -IzlZ)Aexp{--B/(1
--Izl2)},
A > 0, 13 > 0. Characterizations of
the symbol functions b E L2(D) are given so that the Hankel operator
Hb is,
respectively,
bounded and compact from AL~(D) to L~,(D). Tile special case 9) = --~ log(1 --Izl 2) (a > 2), gives the corresponding results for the standard weighted Bergman spaces A 2'~ for a > 2. On the standard (weighted) Bergman space, the reproducing kernel plays a crucial role in all kinds of estimates. But on the space AL~(D), generally we can not write out its reproducing kernel. This makes things harder. To overcome this, instead of using the reproducing kernel of AL~(D), we construct a function ~:w(z) E AL~(ll)) which locally has a similar lower bound estimate as the normalized reproducing kernel does in the standard (weighted) Bergman space. Tile paper is arranged as follows. In Section 2 we discuss the Carleson measures on AL~(D).
In the same section, We also construct the so-called extremal function in
ALL(D ) . Such kind of functions are involved in all the necessity part proofs of the characterization theorems about the Carleson measures and the Hankel operators. In Section 3, we consider the boundedness of the Hankel operator Hb on ALL(D ) with b E Lz(D). We use HSrmander's estimates [HI] for the 0 equation on L~(D) and the results in Section 2 to prove the theorem characterizing the boundedness of Hb from AL~(D) to L~(D) for certain class of subharmonic ~o. Section 4 contains the compactness results. In the last section we discuss the corresponding results for the Hankel operator Hb on ALL(C), the weighted space of entire functions in C. The special case qo =
Iz12/2 gives the
corresponding results
for the Fock space. Throughout this paper, we will use the letter C to denote constants and they may change from line to line.
462
Lin and Rochberg
2. CARLESON MEASURES ON
AL~(I[})
Let Iz be a locally finite nonnegative Borel measure on tile unit disk D, dA be the area measure on D and ~ : ID -+ R be subharmonic function. Let L2 ~,.(D) be the space of all m e a s u r a b l e functions f on D such that
IIflILL, ' =
(/;,
fl2e-U~ d#
)1,,
< oc.
Let L2~(D) denote L~,dA(D ) and AL~(D) be the closed sul)space of L~(D) ,:onsisting of analytic functions. 2.1. # is called a Carleson measure on AL2~(D) i[ the inclusion
DEFINITION
m a p from A L 2 (D) to L~,,,,(D) is a bounded linear map. 2.2. # is called a vanishing Carleson measure oil A L 2 (D) i[ tile
DEFINITION
inclusion m a p h'om A L 2 ( D ) to L2~,,~(D) is a compact linear map. 2.3. For real valued function ~ E C2(D) with A ~ > O, let
DEFINITION
7(z) = (A99(z)) -1/2. We say that 99 E 73 if the following conditions are satMied. (1) There exists a constant C1 > 0 such that IT(z) - w(~)l 0 such that 7(z) 0 such that 7(z) 2. T h e corresponding weight r
is the s t a n d a r d
weight (1 - I z l 2 ) A for A > 2. (it) ~2(z) = 8 9
- Izl 2) + B / ( 1 - Iz12)), A > 0, B > 0. Tile corresponding weight
e - 2 ~ is the exponential weight (1 - I z l 2 ) A e x p { - - B / ( 1 --Izt2)}, A _> 0, B > 0. (iii) 991 + h, a n d qo2 + h, where ~1 and 992 are as in (i) and (it) respectively, and h C C2(iD) can be any harmonic function on tD. For 99 E 73, we have the following theorem a b o u t the Carleson measure on
A L 2 (D).
Lin and Rochberg
463
THEOREM
2.4. Let qo E 29. Then # is a Carleson measure on A L 2 (D) if and
only if there exists a constant a E (0, rain(C11, C~-t)/4) such that sup
1
#{~c e D : I~c - z I < a t ( z ) ) < oo.
zED ~
The sufficiency of Theorem 2.4 was proved by Oleinik [O] under the condition (1) and (2) of Definition 2.3 and for any a E (0, min(C~ -1, C~-1)/4). To prove the necessity, we need some lemmas. The following lemma of HSrmander [H1] plays an important role in this paper. LEMMA
2.5(Lemma
4.4.1 o f [H1]). Let T be a rea/ valued function in
C2(D) such that A T > O. Then for every f satisfying the condition
s Ifl~e-~/zxTdA < O0 there exists u E L~(D) such that Ou = f and ./D [u12e-2~~ dA < ,/D IfI2 e-2~~/ A T dA. LEMMA
2.6. Let T C 29. Let/3 = min(C~ -1, C2~)/2 where C1 and C2 are the
constants of T in Definition 2.3. For any fixed w C D, let p(w) = (~r(w) and q~ be a function analytic in [z - w[ 1 and
10'1 _< 3.
Let us consider the function
where the function g is to be chosen in such a m a n n e r t h a t C~zJC,,, - 0 on D. For 9 we o b t a i n the c5 equation
Og(z)
-
(z
w)-l~
-
'N~) =--h,(z).
(2.1)
In (2.1), h, is a s m o o t h function with s u p p o r t in the ring p ( w ) / 2 _< [z - w[ _< p(w). Let 50 be as in L e m m a 2.7, a n d let pl(z)
1
= ~o(z) - ~ l o g ( p ( w ) :
+ , ' ~ S g l ~ - *ol~)-
T h e n by L e m m a 2.7 there exists constant C > 0 such t h a t C A ~ l ( z ) _> p(w) 2 + lz _ wl 2 > 0.
(2.2)
Now a p p l y i n g L e m m a 2.5 for ~1, we obtain a solution g of (2.1) such t h a t
9L ['q(z)t2e-2~l(z)dA(z) C where the last inequality is by Lemma 2.6, u - ~ _> 0. This completes the proof of Lemma 2.8. Now we can prove Theorem 2.4. PROOF
OF THEOREM
2.4. For the sufficiency, see [O]. In the following
we will prove the necessity. Suppose that tt is a Carleson measure on AL~(D), i.e. the imbedding AL~(D) C L~.~(ID) is bounded. For any w E ]D, let .F,,,(z) be the function constructed in Lemma 2.8. Then by (2) of Lemma 2.8, we have
117~,,11~, _> f ,"
,][~-wl c~{~ e D: I~ - < -< ~o~(w)}.
468
Lin and Rochberg
Since p is a Carleson measure on AL~(D), we have
I1.,%11~, 0 and a sequence w., C D satisfying ]wn[ --+ 1 as n ~ oo such that 1 v(w..)2#{4 e ]D: l( - w,,j < a r ( w . ) } > 6 for all n.
(2,6)
Since ~ E 7?, by condition (2) of Definition 2.3 we have c~r(w.,) < aC2(1 - ]w,~]) < 88 -[w.,f). Tans {~ e l i ) : 14 - w - I < ar(w..)} CID. On the other hand, Iw..]- ar(w..) > Iw,.I - (~c2(1 - Iw, I) = (1 + a c 2 ) l w , . I - ac2 -+ 1 as n -+ ~ . there exists N > 0 such that when n > N, {4 E D :
Iw..I
-
Thus, for ally 0 < 1' < 1,
~r(w.) > r. Hence, when n > N,
I ~ - w,J _< ar(w,.)} C {4 e l D : r < I~I < 1}. Then, when n > N we have If,-. (~)l 2e-%'(e) dp(~) -> 'f( [ 9
le-~(Z)L,,(z)l- -2,T(w)
andiz-w[
(~)1/2
-
= 89 ()
1( C -
7
for ',~ > NI(~)and I~-~'1 < ~o/3~-(w).
On the other hand, since I I f ~ ; ) - L,,IIL5 --+ o N2(w) > 0 such that when n > N2(w) we have
]if!, 7`) [[L~ --~ [[f,,~iiL~ + 1 _< C Let N = m a x ( N l ( w ) , N2(w)).
~1/2
~-(~,)~'
as ~ -+
,~,
there exists another constant
(by (1) of L e m m a 2.10).
Now for the to which was fixed at the beginning of the
proof, we define h:w(z) by
k,,,(z) = /.~/v+i)(z). Then h:~, 9 H~~
and satisfies the condition (1) and (2) of L e m m a 3.3.
The second l e m m a is a covering lemma.
472
Lin and Rochberg
L E M M A 3.4. Let ~ e 7) and let a E (0, min(C~-t, C~-l)/4). Then there exists
a sequence of points {zj} C D, such that the following conditions are satisfied. (1) zj 6 D(aT(zk)), j # k. (2) u j D ( ~ ( z j ) )
= D.
(3) D(aT(zj)) C D(3aT(zj)), where b(a~-(zj)) -- U=eD(~(z~))D(az(z)),
(4) {D(3~T(zj))} PROOF.
j = 1, 2 , . . . .
is a co~ering o~D of finite m,,itipli~ity N.
See Lemma of Coverings in [O].
Now we can prove Theorem 3.1. PROOF
OF T H E O R E M
3.1. First we prove (1) ~
(2)
By (1), Hb is
bounded in the L~ norm. For any w E D, let k~,(z) E H ~ ( D ) be tile analytic function in Lemma 3.3. Then [##:,, - P(bL,,)IIL~ < cIIL,,IIL~ _< c. Writing this out as an integral gives
,fD [b(z)~:w(z) - P(bkw)(z)12e -2~(z) dA(z) N, IH6(L'")(z)12e-2~(z) dA(z) >_ ,/n(~,(,,,))tH6(~:,,,,)(z)lUe -2~(~) dA(z)
,s
> CF,,(w,.) 2 > C5 2.
(4.3)
T h e last inequality is by (4.2). But (4.3) contradicts to (4.1). Thus we must have
Now we prove (2) :v (3). By (2) we know that F~,o(w) ~ 0 as ]w] -+ 1 for some c~0 E (0, min(C~ -~, C f ~ ) / 4 ) . Then, by checking the definition of the function Fo(w), it is easy to see that
Fo(w) -+ 0 as
lu,] ~ 1
for all ct E (0, c~0).
(4.4)
A careful reading of the proof of (2) ~ (3) in Theorem 3.1, especially of (3.6), reveals that the function bl produced there actually satisfies
1
ID(~r(z))l.
/D(~.(~)) Ibm12dA < Csup{Fo(w)2:
w E n(aar(z))}.
(4.5)
478
Since 9~ 9
Lin and Rochberg
we have T(z) --4 0 as Izl --+ 1. Thus from (4.5) and (4.4), we obtain
/~
1
Ibll 2 dA -+ 0
as
Izl -+ 1.
The proof of (2) ~ (3) in Theorem 3.1, especially (3.3), also gives
II,.j(z)
-
z,.(~)l -< C s u p { F , , ( w )
: w
e D(3aT(z))}.
whenever D(~-(zj)) ~ D(3w(zk)) # O. This combining with (3.7) and (4.4) implies
0t (z) (n99(z)) 1/2
--+0
as
]zl-~l.
Finally we prove (3) ~ (1). By Theorem 2.9, the condition G,~(z) --+ 0 as [z[ --+ 1 in (3) implies that tile multiplication operator Mb~ is compact from AL~(D) to L2(D) and therefore Hb~ = (I - P)Mh, is also compact. It remains to show that Hh2 is compact. In tile proof of (3) ~ (1) in Theorem 3.1, we already proved 062
IIH flt B _> q(Iz?),
for Izl < R.
Finally, let a = max(a1, a2), then (5.4) is true for this a. Hence (5.3) is true. It then follows from (5.2) and (5.3) that the condition (2) is satisfied. In order to prove the corresponding theorems for the Hankel operator Hb on
AL~(C), we need the corresponding lemmas for C. L e m m a 2.5 is also true for the entire plane C (see Theorem 4.4.1 of [H2]). For ~o E C, the covering lemma, L e m m a 3.4, is also true for C (see [O]). In lOP], Oleinik and Perel'man proved that under the condition ~0 E C Theorem 2.4 ami Theorem 2.9 are also true for C. In their proofs they also constructed a function ~-,,,(z) E AL~(C) which satisfies the same condition as in L e m m a 2.8. Finally, using the similar method of tile proofs of Lemma 2.9 and L e m m a 3.3, we can prove that for F E C L e m m a 3.3 is also true for C. Then by following tile same procedure of Section 3 and Section 4. we can prove tile following theorems for tile Hankel operator Hb on AL~(C). Let A denote the Lehesgue measure on C. For the houndedness we have THEOREM
5.3. Let 99 E C and suppose that H~ is dense in AL2(C). oo
b E L2(C) and let H~ be defined on H~~ by H~f = bf - P(bf).
equivalent. (1) H~, is bounded in tile L~ norm.
Let
Then the following m'e
482
Lin and Rochberg
(2) The function F,~(z) defined by
F,~(z) 2
=
9 t L (--(=)) I ~ - t,,I= d~ : h, ~ J y t i ~ in D(~---(~))} lnf{iD(aTr(z))[,
is bounded for some a E (0, rain(l, C{-1)/4).
(3) b admits a decomposition b = bl + b2 where b2 E C 1(C) and satisfies Oh2 (A+)I/-------~ e L + ( C ) , while 51 satisfies the following condition: the fimction G~,( z ) defined by G.(z)2
_
/'D
1
is bounded for some a e (0, rain(l,
]t)112~/~
C{-1)/4).
For the compactness we have
THEOREM
5.4. Let ~ E C and suppose that H ~ is (lense in AL~(C). Let b e L2(C) azM let Hb be defined on H ~ by H h f = b / - P ( b f ) . Then the following are equivalent.
(1) Hb is (extends to) a compact operator from AL~(C) to L~(C) . (2) The function F~(z) defined by 9 I Y,,(z) 2 = 'nf{]D(a~.(z))l
.Z(.~(z))
]b - hi 2 dA: h, anMytic in D ( o J ( z ) ) }
tends to zero as z --+ ~ for some a E (0, min(1, C{-1)/4).
(3) b admits a decomposition b = bl + b2 with b2 C CI(C) so that (A~(z)) 1/2 -+0
as z --~ oo.
and for some c~ C (0, rain(l, C l l ) / 4 ) , G . ( z ) -4 0 as z -4 ~c, where the function Go(z) is defined by
G~(z)2 _
1 /D Ibl 12 d.~. ID(,~-(~))I. (~.-(z))
Lin and R o c h b e r g
483
ACKNOWLEDGMENT W e w o u l d like t o t h a n k t h e r e f e r e e for s o m e h e l p f u l s u g g e s t i o n s .
REFERENCES
[AFP] [B] [H1] [He] [.Jpa] IN] [KM] [LR] [Lu] [O] [OP] [Sl [St] [Trl
J. Arazy, S. Fisher and J. Peetre, Hankel operators on weighted Bergman spaces, Amer. 3. Math. 110 (1988), 989-1054. B. Berndtsson. Weighted estimates for 0 in domains in C 1, Duke Math. J. 66 (1992), 239-255. L. HSrmander, An Introduction to Complex Analysis in Several Variables, 3rd. ed. rev., North Holland, Amsterdam 1990. L. HSrmander, An Introduction to Complex Analysis in Several Variables, New York: Van Nortrand Reinhold, 1966. S. Janson. J. Peetre and R. Rochberg, Hankel forms and the Foek space, Rev. Mat. Iberoamericana 3 (1987), 61-138. S. G- Krantz, Function Theory of Several Complex Variables. 2nd. ed. Wadsworth, Belmont, 1992. T. L. Kriete III and B. D. MaeCluer, Composition operators on large weighted Bergman spaces: Indiana Univ. Math. J. 41 (1992), 755-788. Pang Lin and R. Roehberg, The essential norm of Hankel operator on the Bergman space, Integral Equations and Operator Theory. 17 (1993), 361-372. D. Luecking, Characterizations of certain classes of Hankel operators on the Bergman spaces of the unit disk. J. Functional Analysis 110 (1992), 247-271. V. L. Oleinik. Embedding theorems for weighted classes of harmonic and analytic functions, J. Soviet Math. 9 (1978), 228-243. V. L. Oleinik and G. S. Perel'man, Carleson's imbedding theorem for a weighted Bergman space, Mathematical Notes 47 (1990), 577-581. E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press. 1970. K- Stroethoff, Hankel and Toeptitz operators on the Fock space, Michigan Math. J. 39 (1992), 3-16. T. Trent. A m.easure inequality, preprint.
D e p a r t m e n t of M a t h e m a t i c s C a m p u s B o x 1146 Washington University St. Louis, M o 63130 USA
Submitted;
April
5, 1994
