Compact Operators on the Bergman Spaces with Variable Exponents on the Unit Disc of C

We study the compactness of some classes of bounded operators on the Bergman space with variable exponent. We show that via extrapolation, some results on boundedness of the Toeplitz operators with general L1 symbols and compactness of bounded operators on the Bergman spaces with constant exponents can readily be extended to the variable exponent setting. In particular, if S is a finite sum of finite products of Toeplitz operators with symbols from class BT, then S is compact if and only if the Berezin transform of S vanishes on the boundary of the unit disc.


Introduction and Statement of Results
Variable Lebesgue spaces are a generalization of the Lebesgue spaces that allow the exponents to be a measurable function and thus the exponent may vary.These spaces have many properties similar to the normal Lebesgue spaces, but they also differ in surprising and subtle ways.For this reason, the variable Lebesgue spaces have an intrinsic interest, but they are also very important in applications to partial differential equations and variational integrals with nonstandard growth conditions.See [1] for more details on the variable Lebesgue spaces.
Let Δ denote the unit disc in C and  the normalized Lebesgue measure on Δ.For 1 ≤  < ∞, the Bergman space   =   (Δ, ) is the space of all analytic functions, , on Δ such that Let  be the Bergman projection from  2 onto  2 .Then  is an integral operator given by for each  ∈ Δ and  ∈  2 .Here, the function (, ) =   () = 1/(1 − ) 2 is the reproducing kernel for  2 .For  ∈  ∞ , the Toeplitz operator with symbol  is defined on   by    =  () ,  ∈   .
Toeplitz operators are amongst the most widely studied classes of concrete operators and have attracted a lot of interest in recent years.The behaviour of these operators on the Hardy spaces, Bergman spaces, and Fock spaces has been studied widely and a lot of results are available in the literature.The characterization of compactness has been studied in [2][3][4][5][6][7][8] just to cite a few.
Given Ω ⊂ R  , a measurable function  : Ω → [1, ∞) will be called a variable exponent.If  is a variable exponent then we denote Let P(Ω) denote the set of all variable exponents for which  + < ∞.
It is known (e.g., see chapter 2 of [1]) that the dual of  (⋅) is    (⋅) , where 1/(⋅) + 1/  (⋅) = 1.A straightforward computation shows that For simplicity, we will omit one set of parenthesis and write the left-hand side of each equality as   (⋅) + and   (⋅) − .Throughout this work, we shall use   (⋅) as the conjugate exponent of (⋅) and if  is a constant in (1, ∞) we shall use   as the conjugate exponent of .In other words, to study these spaces, some regularity conditions are imposed on the exponents.A function  : Ω → C is said to be log-Hölder continuous on Ω if there exists a positive constant  log such that for all ,  ∈ Ω with | − | < .We denote by P log (Ω) the exponents in P(Ω) that are log-Hölder continuous on Ω.For (⋅) ∈ P log (Ω) and a given measurable function, , define Theorem 2.34 of [1] shows that there exist constants  1 and  2 , depending on (⋅), such that The next result which establishes a relationship between the Lebesgue spaces with exponents  − ,  + , and (⋅) will be very useful in the rest of the work.It is Corollary 2.50 of [1].
Lemma 1. Suppose (⋅) ∈ P  (Ω) and |Ω| < ∞.Then there exist constants  1 and  2 such that The study of variable exponent Bergman space,  (⋅) , which is the space of analytic functions in  (⋅) , has been introduced in [9].There it was shown, amongst other things, that the Bergman projector  is bounded from  (⋅) onto  (⋅) .Also in [10], the authors studied Carleson measures in such spaces.
In this paper, we will extend the results in [3,7] on boundedness and compactness of operators for the Bergman spaces with constant exponents to the Bergman spaces with variable exponents.
For  ∈ Δ, let   be the analytic map of Δ onto Δ given by   () = ( − )/(1 − ).We define the operator   on  2 by Then   is a unitary operator on  2 .We shall show later that   is bounded on  (⋅) .For , a bounded operator on  (⋅) , we define   by   =     .
If  is a bounded operator on  (⋅) , then the Berezin transform of  is the function S on Δ defined by where   () = (1−|| 2 )  is the normalized Bergman kernel which also belongs to  (⋅) and ⟨ , ⟩ is the inner product of  2 .We let and set Our first result gives some conditions for the boundedness of Toeplitz operators with  1 symbols on the variable Bergman spaces.
We note here that this result was proved in [3] in the Bergman spaces   , where  is a constant.We also have the following result on compactness.Theorem 3. Suppose (⋅) ∈ P  (Δ), 1 <  0 ≤  − ≤  + < ∞, and  1 = min( 0 ,   0 ).If  is a bounded operator on  (⋅) such that for some  > ( 1 +1)/( 1 −1), then the following are equivalent: (1)  is compact on  (⋅) , (2) S() → 0 as  → Δ, This theorem is well known in the Bergman spaces with constant exponents; for example, see [3,7].However, the techniques here are different from those used in either of the papers for both the proof of boundedness and compactness.This is because their proofs depend on the use of Schur's test which does not hold in the variable Lebesgue space.However, using the Muckenhoupt weights we were able to develop some Schur-like tests from where we obtain the theory that builds upon the Rubio de Francia theory of extrapolation from the theory of weighted norm inequalities.The advantage of this approach is that it quickly yields to sufficient conditions for these operators to be bounded on variable Lebesgue spaces.Through such techniques, we are also able to obtain some norm estimates for bounded operators on the space  (⋅) .
Similar to the work of Miao and Zheng [7], we consider the case of the algebra of Toeplitz operators generated by symbols in the class .To be precise, we have the following.Theorem 4. Suppose (⋅) ∈ P  (Δ) and  is a finite sum of finite products of Toeplitz operators with symbols in the class .Then  is compact on  (⋅) if and only if S() → 0 as  → Δ.
This paper is organized as follows: in Section 2, we will study some basic concepts on the Muckenhoupt weights.Section 3 deals with the variable Bergman spaces and the proof of Theorem 2. In Section 4, we study some norm estimates on these spaces and in Section 5 we give the proof of the compactness results.
Let  be a locally integrable function in Δ.Then the Hardy-Littlewood maximal function relative to the pseudodistance  is given by where the supremum is taken over all pseudoballs containing .Suppose 0 < () < ∞ almost everywhere on Δ.Then we say that  is in the Muckenhoupt weight  1 if There are two equivalent definitions which are useful in practice.First,  ∈  1 , if for almost every  ∈ Δ, It follows that if  ∈  1 then and thus Alternatively,  ∈  1 if for every pseudoball  we have that For more details on the Muckenhoupt weights, see chapter 9 of [13] or chapter 4 of [1].
We will need some results on extrapolation.The following proposition is Theorem 5.24 of [1].
International Journal of Mathematics and Mathematical Sciences Proposition 6.Let Ω ∈ R  and suppose there is some  0 ≥ 1 and the family F such that for all  ∈  1 , where  (⋅) =  0 and  is some positive constant depending on the dimension of Ω.
The following is Theorem 3.

Variable Exponent Bergman Spaces
Given (⋅) ∈ P log (Δ), we define the variable exponent Bergman space  (⋅) as the space of all analytic functions on Δ that belong to the variable exponent Lebesgue space  (⋅) with respect to the area measure  on the unit disc.With this definition  (⋅) is a closed subspace of  (⋅) .By Theorem 4.4 of [9], the Bergman projection, , given by ( 2) is bounded from  (⋅) onto  (⋅) for any (⋅) ∈ P log (Δ).It is, thus, necessary to study the behaviour of Toeplitz operators on such spaces.Similar to the definition of Toeplitz operators on the Bergman spaces with constant exponent, we define the Toeplitz operator with symbol  ∈  ∞ on  (⋅) by Lemma 8.The operator   is bounded on  (⋅) for (⋅) > 1.
since () ≤ () for almost every  ∈ Δ.Now, for 0 <  < (Δ) there is a pseudoball  containing  such that It follows that where the last inequality comes from (27).This shows that It follows that the family {(|  |, ||) :  ∈   0 } satisfies inequality (29).Also, by Proposition 7 the maximal function Mf belongs to  ((⋅)/ 0 )  .Thus by Proposition 6   is bounded on  (⋅) .Remark 9. We just want to give an alternative argument to obtain the estimate (35), and this argument has different effects and may be useful in applications.
We recall that if  is locally integrable in Δ, then lim The proof of this statement can easily be adapted from that of Theorem 1.3 of [14].We use this statement as follows: Let  ∈  1 and  0 > 1.Then for any  > 0, we can find  > 0 such that for all  ∈ (0, ).Now, if we fix such  then for 0 <  <  we have for all  ∈  (⋅) .
In the application of Lemma 10, we may assume that ess inf ∈Δ () ≥ 1 for  ∈  1 , as the following lemma shows.Lemma 12. Let  ∈  1 be such that ess inf ∈Δ () < 1 and  =  for any  > 1 such that ess inf ∈Δ () ≥ 1.If the hypothesis of Lemma 10 holds for the weight , then the conclusion of Lemma 10 holds for the weight .
Proof.By (27), we have that 1 ≤ ([]  1 /(Δ))() for almost every  ∈ Δ.Now by Proposition 9.1.5 of [13], we have that for almost every  ∈ Δ.Thus we have that the constants  1 = []  1 and  2 are independent of  and, hence, independent of .Now since the hypothesis of Lemma 10 holds for the weight , we have that which gives the result.
The next lemma will be used frequently and is well known; see, for example, Lemma 3.10 of [15] for the proof.Lemma 13.Suppose  < 1 and  +  < 2. Then Proof of Theorem 2. Let  ∈  (⋅) .Then Then using the identity we have where the last equality is from the change of variable  =   .By Hölder's inequality, we have that (2) follows from the fact that ‖ ∘   ‖  = ‖‖  and ‖ ∘  ‖ (⋅) ≤ ‖ ∘   ‖  , which is given by assertion (1).
We also have the following estimate for operators in the Toeplitz algebra.To be precise, we have the following.
This completes the proof of the lemma.
Proof.Fix  ∈ Δ.Then where the second equality comes from the definition of   and the third equality from the definition of   .Thus, where . By the choice of , we have that sup ∈Δ ‖  ‖   < ∞ and (69) holds.
To prove (70), replace  by  * in (69), interchange  and  in (69), and then use the equation Finally, we use the same argument as in the proof of Theorem 2 to obtain that there is a pseudoball  containing  such that () ≤ () ≤ (Δ)(|| −1 + 1) and thus A similar argument as the one used to obtain the estimate (69) will give us (70).
and  1 = min( 0 ,   0 ) and suppose that  is a bounded operator on  (⋅) .If International Journal of Mathematics and Mathematical Sciences for some  > ( 1 + 1)/( 1 − 1), then there is a constant  such that Proof.For  ∈  (⋅) and  ∈ Δ, we have where the last equation follows from (69).Given that  > 0 that satisfies (53), we have by (69) that In a similar manner, we use ( 73) and (70) to get that for all  ∈  1 where the constant  depends on []  1 and not on .We now apply Proposition 11 to get the required result.
We will need the power series formula for the Berezin transform of the bounded operator  on  2 .From the definition of the reproducing kernel, we get that for ,  ∈ Δ.To compute S() = ⟨  ,   ⟩, we first compute   by applying  to both sides of (86) and then take the inner product with   , again using (86), to obtain as  ∈ Δ.Suppose S() → 0 as  → Δ.Fix  ∈ [1, ).We will show that ‖  1‖  → 0 as  ∈ Δ.
For  ∈ Δ, ,  = 0, 1, . .., an easy computation shows that Since we have that for some constant   .For ,  ∈ Δ, we have where the second equality comes from (87).Also, note that the power series in (100) converges uniformly for each  ∈ Δ.
We will first show that the operator  for all  > 0. The conclusion then follows from Theorem 3.