Hankel Operators on the Weighted L P -Bergman Spaces with Exponential Type Weights

Abstract

We characterize the boundedness and compactness of the Hankel operator with conjugate analytic symbols on the weighted [superscript] L P [/superscript] -Bergman spaces with exponential type weights.

(ProQuest: ... denotes non-US-ASCII text omitted.)

Academic Editor:Abdelghani Bellouquid

Department of Mathematics, Pusan National University, Busan 609-735, Republic of Korea

Received 5 November 2013; Accepted 20 December 2013; 12 February 2014

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Let ... be the unit disc in the complex plane ... and d A ( z ) the area measure on ... , and denote by H ( ... ) the space of all analytic functions in ... . Let [straight phi] ∈ ^{C 2} ( ... ) with Δ [straight phi] > 0 . For 0 < p ...4; ∞ , the weighted Bergman space ^{A [straight phi] p} is the space of functions f ∈ H ( ... ) such that [figure omitted; refer to PDF] Note that ^{A [straight phi] p} is the closed subspace of ^{L [straight phi] p} : = ^{L p} ( ... , ^{e - p [straight phi]} d A ) consisting of analytic functions. Since the space ^{A [straight phi] 2} is a reproducing kernel Hilbert space, for each z ∈ ... , there are functions _{K z} ∈ ^{A [straight phi] 2} with f ( z ) = Y9; f , _{K z} YA; , where Y9; · , · YA; is the usual inner product in ^{L [straight phi] 2} . The orthogonal projection from ^{L [straight phi] 2} to ^{A [straight phi] 2} is given by [figure omitted; refer to PDF] where K ( w , z ) = _{K z} ( w ) ¯ .

Given σ ∈ ^{C 1} ( ... ) so that there exists a dense subset ...9F; of ^{A [straight phi] 2} with σ f ∈ ^{L [straight phi] 2} for f ∈ ...9F; , the big Hankel operator _{H σ} with symbol σ is densely defined by [figure omitted; refer to PDF] where P is the orthogonal projection of ^{L [straight phi] 2} onto ^{A [straight phi] 2} .

We write ∂ ¯ = ∂ / ∂ z ¯ . Then the ∂ ¯ -equation can be written by [figure omitted; refer to PDF] For f ∈ ^{A [straight phi] 2} , we look for a solution v ∈ ^{L [straight phi] 2} of minimal ^{L [straight phi] 2} -norm. Notice that the solution of minimal norm is the one that is orthogonal to the kernel of ∂ ¯ on ^{L [straight phi] 2} ; that is, v [perpendicular] ^{A [straight phi] 2} . Then, if u ∈ ^{L [straight phi] 2} solves (4), we get [figure omitted; refer to PDF] The linear operator N : ^{A [straight phi] 2} [arrow right] ^{L [straight phi] 2} given by [figure omitted; refer to PDF] is called the canonical solution operator to ∂ ¯ on ^{A [straight phi] 2} .

For any f ∈ ^{A [straight phi] 2} , obviously ∂ ¯ ( z ¯ f ) = f and [figure omitted; refer to PDF] That is, the canonical solution operator coincides with the big Hankel operator acting on ^{A [straight phi] 2} with symbol z ¯ . Motivated by this fact, we now consider Hankel operators with conjugate analytic symbols on ^{A [straight phi] 2} . For f , g ∈ ^{A [straight phi] 2} , we do not necessarily have g ¯ f ∈ ^{L [straight phi] 2} . Let ...9F; : = Span ... { _{K z} : z ∈ ... } . Then ...9F; is dense in ^{A [straight phi] 2} . For symbol g ∈ ^{A [straight phi] 2} such that [figure omitted; refer to PDF] we consider the densely defined big Hankel operator on ^{A [straight phi] 2} given by [figure omitted; refer to PDF]

A positive function τ on ... is said to belong to the class ... if it satisfies the following three properties.

(a) There exists a constant _{C 1} > 0 such that [figure omitted; refer to PDF]

(b) There exists a constant _{C 2} > 0 such that [figure omitted; refer to PDF]

(c) For each m ...5; 1 , there are constants _{b m} > 0 and 0 < _{t m} < 1 / m such that [figure omitted; refer to PDF]

In this paper, we characterize the boundedness and compactness of the Hankel operator with conjugate analytic symbols on the weighted ^{L p} -Bergman spaces with exponential type weights as follows.

Theorem 1.

Let 1 ...4; p < ∞ and g ∈ ^{A [straight phi] p} . Let [straight phi] ∈ ^{C 2} ( ... ) with Δ [straight phi] > 0 , and the function τ ( z ) = ( Δ [straight phi] ( z ) ^{) - 1 / 2} is in the class ... . Then _{H g ¯} extends to a bounded linear operator on ^{A [straight phi] p} if and only if [figure omitted; refer to PDF]

Theorem 2.

Let 1 ...4; p < ∞ and g ∈ ^{A [straight phi] p} . Let [straight phi] ∈ ^{C 2} ( ... ) with Δ [straight phi] > 0 , and the function τ ( z ) = ( Δ [straight phi] ( z ) ^{) - 1 / 2} is in the class ... . Then _{H g ¯} extends to a compact linear operator on ^{A [straight phi] p} if and only if [figure omitted; refer to PDF]

In [1], Luecking firstly proved the same results in the context of the ordinary ^{L 2} -Bergman spaces. For ^{L 2} -Bergman spaces with exponential type weights, the same results were proved in [2-4]. Moreover, Schatten-class Hankel operators are also indicated in their papers.

The expression f [<, ~] g means that there is a constant C independent of the relevant variables such that f ...4; C g , and f [approximate] g means that f [<, ~] g and g [<, ~] f .

2. Preliminaries

From now on we assume that [straight phi] ∈ ^{C 2} ( ... ) , Δ [straight phi] > 0 , and the function τ ( z ) = ( Δ [straight phi] ( z ) ^{) - 1 / 2} is in the class ... . The following notations will be frequently used: [figure omitted; refer to PDF] where _{C 1} and _{C 2} are the constants in the conditions (a) and (b) in Section 1 and [figure omitted; refer to PDF]

Lemma 3 (see [5]).

For each M ...5; 1 , there exists a constant C > 0 (depending on M ) such that for z , w ∈ ... , one has [figure omitted; refer to PDF]

By using the upper estimates for K ( w , z ) in Lemma 3, Arroussi and Pau [5] proved that the orthogonal projection P projects ^{L [straight phi] p} boundedly onto ^{A [straight phi] p} for 1 ...4; p ...4; ∞ .

Lemma 4 (see [6]).

Let 0 < ρ ...4; _{m τ} and w ∈ ... . Then, [figure omitted; refer to PDF]

By using the third Green formula, we get the following two approximation results.

Lemma 5 (see [7]).

For small K > 0 , there exists A = A ( K ) > 0 such that [figure omitted; refer to PDF] where _{h z} is a harmonic function in ^{D K} ( z ) with _{h z} ( z ) = 0 .

Lemma 6 (see [7]).

For small K > 0 , one has the estimate [figure omitted; refer to PDF] where _{h z} is defined in Lemma 5.

The following is a certain submean value property of | f ( z ) | ^{e - [straight phi] ( z )} . We follow the proof of ([8], Lemma 19).

Lemma 7.

Let 0 < p < ∞ . For any small r > 0 , there exists C = C ( r ) > 0 such that for any f ∈ ^{A [straight phi] p} and z ∈ ...

(a) | f ( z ) | ^{e - [straight phi] ( z )} ...4; C ( ( 1 / τ ( z ^{) 2} ) _{∫ ^{D r} ( z )} ... | f ^{| p e - p [straight phi]} d A ^{) 1 / p} ;

(b) | ∇ ( | f | ^{e - [straight phi]} ) ( z ) | ...4; C ( 1 / τ ( z ) ) ( ( 1 / τ ( z ^{) 2} ) _{∫ ^{D r} ( z )} ... | f ^{| p e - p [straight phi]} d A ^{) 1 / p} , provided f ( z ) ...0; 0 .

Proof.

(a) By Lemma 5, there exists some constant A = A ( K ) > 0 such that [figure omitted; refer to PDF] for z ∈ ... . Since _{h z} is harmonic, there is an analytic function _{Φ z} on ^{D K} ( z ) such that _{Φ z} ( z ) = 0 and Re _{Φ z} = _{h z} on ^{D K} ( z ) . Thus we have | ^{e _{Φ z}} | = ^{e _{h z}} . Hence by the submean value property together with Lemma 5, we get [figure omitted; refer to PDF]

(b) We begin as follows: [figure omitted; refer to PDF] Since _{h z} is harmonic, there is an analytic function _{Φ z} ∈ H ( ^{D K} ( z ) ) such that [figure omitted; refer to PDF] Note that [figure omitted; refer to PDF] On the other hand, [figure omitted; refer to PDF] For | z - ζ | = τ ( z ) , we have [figure omitted; refer to PDF] Hence we have [figure omitted; refer to PDF] Thus [figure omitted; refer to PDF]

Despite that the next result was proved in [4], we give the proof of different method by using Lemma 7.

Proposition 8.

There is an r > 0 independent of z such that [figure omitted; refer to PDF]

Proof.

Let r > 0 . By (b) of Lemma 7, we have [figure omitted; refer to PDF] Hence it follows that [figure omitted; refer to PDF]

Note that by Lemma 3 and (a) of Lemma 7, we have [figure omitted; refer to PDF] Thus we have [figure omitted; refer to PDF] if we choose small r > 0 .

3. Hankel Operators on ^{A [straight phi] p}

For the proof of boundedness of Hankel operator on ^{L 2} -Bergman spaces with exponential type weights in [2-4], they used Hörmander's ^{L 2} -estimates for ∂ ¯ . However, for ^{L p} -Bergman spaces, we need the following ^{L p} -estimates for ∂ ¯ .

Theorem 9 (see [6]).

Let [straight phi] ∈ ^{C 2} ( ... ) with Δ [straight phi] > 0 . Suppose that the function τ ( z ) = ( Δ [straight phi] ( z ) ^{) - 1 / 2} satisfies conditions (a) and (b) in Section 1. Let 1 ...4; p ...4; ∞ . Then there is a solution u to the equation ∂ ¯ u = f such that [figure omitted; refer to PDF] provided the right hand side integral is finite.

Let g be an analytic function in ... satisfying the condition (8). Let [figure omitted; refer to PDF] By the reproducing formula in ^{A [straight phi] 2} we get [figure omitted; refer to PDF]

Lemma 10.

Let 0 < p < ∞ . Then [figure omitted; refer to PDF]

Proof.

Consider [figure omitted; refer to PDF] First, [figure omitted; refer to PDF] Now, [figure omitted; refer to PDF] We take a large constant M > 1 so that p M > 2 . Then [figure omitted; refer to PDF]

Thus we get the result.

Theorem 11.

Let 1 ...4; p < ∞ . Let g ∈ ^{A [straight phi] p} . Then _{H g ¯} extends to a bounded linear operator on ^{A [straight phi] p} if and only if [figure omitted; refer to PDF]

Proof.

Assume first that [figure omitted; refer to PDF] By Theorem 9, there is a solution u of the equation ∂ ¯ u = f ∂ ¯ g ¯ such that [figure omitted; refer to PDF] Since _{H g ¯} f is the minimal ^{L [straight phi] 2} -norm solution of the ∂ ¯ -equation, we have _{H g ¯} f = ( I - P ) u . In [5], Arroussi and Pau proved that the orthogonal projection P projects ^{L [straight phi] p} boundedly onto ^{A [straight phi] p} for 1 ...4; p ...4; ∞ . Thus we have [figure omitted; refer to PDF] By (45) and (46), we have [figure omitted; refer to PDF] which shows that _{H g ¯} can be extended to a bounded linear operator on ^{A [straight phi] p} .

Conversely, assume that _{H g ¯} is bounded on ^{A [straight phi] p} . Then we have [figure omitted; refer to PDF] Using Proposition 8 and Lemma 3, there exists r > 0 such that [figure omitted; refer to PDF] Hence we have [figure omitted; refer to PDF] Since g is an analytic function in ... , by the Cauchy estimates applied to _{g ζ} ( z ) : = g ( z ) - g ( ζ ) , we can now conclude [figure omitted; refer to PDF] Thus we get the result.

Lemma 12.

Let 0 < p < ∞ . Then [figure omitted; refer to PDF]

Proof.

Let K be a compact subset of ... . We choose a large constant M so that M - ( 2 / p ) - 1 > 0 . Then we have for w ∈ K [figure omitted; refer to PDF] where dist ... ( K , z ) = min ... { | z - w | : w ∈ K } .

Theorem 13.

Let 1 ...4; p < ∞ . Let g ∈ ^{A [straight phi] p} . Then _{H g ¯} extends to a compact linear operator on ^{A [straight phi] p} if and only if [figure omitted; refer to PDF]

Proof.

Suppose now that _{H g ¯} is compact on ^{A [straight phi] p} . Then by Riesz-Tamarkin compactness theorem, we have [figure omitted; refer to PDF] uniformly in ζ ∈ ... . Now, by Lemma 12, [figure omitted; refer to PDF] as | ζ | [arrow right] ^{1 -} . Thus we have [figure omitted; refer to PDF]

We choose ρ > 0 so that [figure omitted; refer to PDF]

Then [figure omitted; refer to PDF] This implies that [figure omitted; refer to PDF]

For | ζ | > r + ρ , the inclusion ^{D ρ} ( ζ ) ⊂ { r < | z | < 1 } holds, and [figure omitted; refer to PDF] This implies that [figure omitted; refer to PDF]

Assume now that [figure omitted; refer to PDF] It is enough to show that for any sequence { _{f n} } that is bounded in norm and converges uniformly to zero on compact subsets, we have _{|| H g ¯ f n || p , [straight phi]} [arrow right] 0 as n [arrow right] ∞ . As in relation (46), we have [figure omitted; refer to PDF]

Now [figure omitted; refer to PDF] Since { _{f n} } converges uniformly to zero on compact subsets, [figure omitted; refer to PDF] Now [figure omitted; refer to PDF] Hence _{H g ¯} is compact.

Acknowledgment

This is supported by the National Research Foundation of Korea (NRF) Grant funded by the Korea government (MEST) (NRF-2011-0013740).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

References

[1] D. H. Luecking, "Characterizations of certain classes of Hankel operators on the Bergman spaces of the unit disk," Journal of Functional Analysis , vol. 110, no. 2, pp. 247-271, 1992.

[2] P. Galanopoulos, J. Pau, "Hankel operators on large weighted Bergman spaces," Annales Academiæ Scientiarum Fennicæ Mathematica , vol. 37, no. 2, pp. 635-648, 2012.

[3] P. Lin, R. Rochberg, "Hankel operators on the weighted Bergman spaces with exponential type weights," Integral Equations and Operator Theory , vol. 21, no. 4, pp. 460-483, 1995.

[4] P. Lin, R. Rochberg, "Trace ideal criteria for Toeplitz and Hankel operators on the weighted Bergman spaces with exponential type weights," Pacific Journal of Mathematics , vol. 173, no. 1, pp. 127-146, 1996.

[5] H. Arroussi, J. Pau, "Reproducing kernel estimates, bounded projections and duality on large weighted Bergman spaces," http://arxiv.org/abs/1309.6072

[6] H. R. Cho, S. K. Han, "Exponentially weighted ^{L P} -estimates for ∂ ... on the unit disc," Journal of Mathematical Analysis and Applications , vol. 404, no. 1, pp. 129-134, 2013.

[7] H. R. Cho, S. K. Han, I. Park, "Peak functions for Bergman spaces with exponential type weight on the unit disc," preprint

[8] N. Marco, X. Massaneda, J. Ortega-Cerdà, "Interpolating and sampling sequences for entire functions," Geometric and Functional Analysis , vol. 13, no. 4, pp. 862-914, 2003.