Derivative and Lipschitz Type Characterizations of Variable Exponent Bergman Spaces

The Bergman spaces were introduced in [1]. Since then, the theory of Bergman spaces has grown quickly, due to its connection with harmonic analysis, approximation theory, hyperbolic geometry, potential theory, and partial differential equations; see [2–5]. In particular, they can be characterized by derivatives and Lipschitz type conditions. Indeed, Zhu in [5] gave the derivatives characterizations of Bergman spaces. Wulan and Zhu in [6] gave Lipschitz type characterizations for Bergman spaces.We remark here that Lipschitz type characterizations for Sobolev spaces were considered in [7–11]. Recently, in [12], Chacón and Rafeiro introduced variable exponent Bergman spaces on the open unit ball of the plane and obtained that the Bergman projection and the Berezin transform are bounded and polynomials are dense in these spaces. Then in [13], Chacón, Rafeiro, and Vallejo gave a characterization of Carleson measures for variable exponent Bergman spaces.These results are generalizations of constant exponent Bergman spaces. The theory of variable function spaces has attractedmany authors’ attention for four decades. Since there are huge literatures, we only recommend [14– 16]. Motivated by those papers, in this paper, we shall extend the derivatives characterizations in [5] and Lipschitz type characterizations in [6] to variable exponent Bergman spaces on the open unit ball of C for any integer n. To state our results, we firstly recall some definitions. We denote the Euclidean norm on C by | ⋅ |.Then we let B = {z ∈ C : |z| < 1}, the open unit ball in Cn. Let d] be the normalized volume measure on Bn. For any α > −1, let d]α(z) = Cα(1 − |z|2)αd](z), where Cα is a positive constant such that ]α(Bn) = 1. In this paper, we only consider the case of α = 0. For a measurable function p : B 󳨀→ [1,∞), we call it a variable exponent and denote p+ fl ess supz∈Bnp(z), p− fl ess infz∈Bnp(z). Denote by P(Bn) the set of all variable exponents with p+ < ∞. Let p ∈ P(Bn). For a complexvalued measurable function f on B, we define the modular of f by ρp(⋅) (f) fl ∫ B 󵄨󵄨󵄨󵄨f (z)󵄨󵄨󵄨󵄨p(z) d] (z) , (1)


Introduction
The Bergman spaces were introduced in [1].Since then, the theory of Bergman spaces has grown quickly, due to its connection with harmonic analysis, approximation theory, hyperbolic geometry, potential theory, and partial differential equations; see [2][3][4][5].In particular, they can be characterized by derivatives and Lipschitz type conditions.Indeed, Zhu in [5] gave the derivatives characterizations of Bergman spaces.Wulan and Zhu in [6] gave Lipschitz type characterizations for Bergman spaces.We remark here that Lipschitz type characterizations for Sobolev spaces were considered in [7][8][9][10][11].
Recently, in [12], Chacón and Rafeiro introduced variable exponent Bergman spaces on the open unit ball of the plane and obtained that the Bergman projection and the Berezin transform are bounded and polynomials are dense in these spaces.Then in [13], Chacón, Rafeiro, and Vallejo gave a characterization of Carleson measures for variable exponent Bergman spaces.These results are generalizations of constant exponent Bergman spaces.The theory of variable function spaces has attracted many authors' attention for four decades.Since there are huge literatures, we only recommend [14][15][16].Motivated by those papers, in this paper, we shall extend the derivatives characterizations in [5] and Lipschitz type characterizations in [6] to variable exponent Bergman spaces on the open unit ball of C  for any integer .To state our results, we firstly recall some definitions.For a measurable function  : B  → [1, ∞), we call it a variable exponent and denote  + fl ess sup ∈B  (),  − fl ess inf ∈B  ().Denote by P(B  ) the set of all variable exponents with  + < ∞.Let  ∈ P(B  ).For a complexvalued measurable function  on B  , we define the modular of  by and the Luxemburg-Nakano norm by The variable Lebesgue space  (⋅) (B  , d]) is the set of all complex-valued measurable functions  on B  such that  (⋅) () < ∞.It is a Banach space equipped with the Luxemburg-Nakano norm.
If  ∈ P(B  ), then the variable exponent Bergman space  (⋅) (B  ) is the class of all holomorphic functions on B  which belong to the variable exponent Lebesgue space  (⋅) (B  , d]).It is easy to show that  (⋅) (B  ) is a closed subspace of  (⋅) (B  , d]).When  is a constant, these spaces are called weighted Bergman space with standard weights; see [3,4] for details.As usual, we denote by (B  ) the space of holomorphic functions on B  .
Given  ∈ (B  ), the radial derivative of  at  is defined by The complex gradient of  at  is defined by And the invariant complex gradient of  at  is given by where   is the automorphism of B  mapping 0 to .
For any  ∈ B  , let   be a biholomorphic map on B  such that   (0) =  and  −1  =   .The explicit formulas are available for   (see [5]).
Let  be the In particular, if  is fixed, then the volume of (, ) is comparable to For any  > 0 and  ∈ B  , we let (, ) fl { ∈ B  : (, ) < }, the hyperbolic ball centered at  with radius .If  ∈ (0, 1), then (, ) = (, ) with Consequently, if  is fixed, then the volume of (, ) is also comparable to (1 − || for all ,  ∈ B  such that | − | < 1/2.We will denote by P log (B  ) the set of all log-Hölder continuous functions on B  .Now, the main result of the paper is the following.
In Section 2, we shall collect some results which we shall need in the paper.The proof of Theorem 2 will be given in Section 3. Finally, we claim that the notation  ≲  means there exists a constant  > 0 such that  ≤ , and  ≈  means  ≲  and  ≲ .

Preliminaries
In this section, we recall some preliminary results that we shall need in our paper.Lemma 3 ([5], Lemma 2.20).Let  be a positive number.Then there exists a positive constant   such that for all ,  ∈ B  with (, ) < .Moreover, if  is bounded above, then we may choose   to be independent of .
The following Jensen type inequality was proved in [17] in the context of spaces of homogeneous type (SHT).
Let F denote a family of pairs of nonnegative measurable function and  1 denote the Muckenhoupt  1 weight.

Lemma 8 ([14], Theorem 5.24). Suppose that for some 𝑝
Given (⋅) ∈ P(B  ), if  0 ≤  − ≤  + < ∞ and the maximal operator is bounded on  ((⋅)/ 0 )  (B  ), then there is a positive constant  independent of (, ) such that For  ∈ C  , we define the following radial test function: where  > 0 is the normalizing constant in the sense where B  stands for the complex ball with radius .
For the second part, fix  > 0 and choose a function  with compact support on B  , such that ‖ − ‖  (⋅) (B  ) <  (see [ Thus, ‖  − ‖  ∞ (B  ) ≤ 1 and consequently Therefore, we have reduced the convergence to the case of a constant exponent.
In this case, from [3] we know that radial dilations (⋅) converge in  − -norm to .Consequently, if we define the translation operator   () fl (−), then by Minkowski's inequality and the result follows from the continuity of   on the space   − (B  ).
From the above lemma, we have the following lemma.
The Bergman projection operator  is defined for functions  on B  by To proceed, we need the class   for  ∈ (1, ∞).Let  be a positive measurable function, and  is called belonging to the class   if there exists a constant  such that, for every pseudo-ball  ⊂ B  , where () = ∫ B  ()d]().
In order to prove that  is surjective, we use the fact that  =  for every  ∈  2 (B  ).In particular, this equality holds for any polynomial.Thus, if  ∈  (⋅) (B  ), we use Lemma 11 to find a sequence of polynomials   converging in  (⋅) (B  ) to .But since   =   →  in  (⋅) (B  ) then we have that  = .Lemma 17 ([5], Lemma 2.24).Suppose  > 0,  > 0, and  > −1.Then there exists a constant  > 0 such that for all  ∈ (B  ) and all  ∈ B  .
The last result deals with the limits of (, ) and (, ) as  tends  in the radial direction.(38)

Proof of Theorem 2
Proof of Theorem 2 .We shall divide the proof into 8 steps.
In Step 1, by Lemma 13 we obtain that (b) implies (c) and (c) implies (d).
In Step 2, we prove (a) implies (b).We firstly consider that  ∈  (⋅) (B  ) such that ‖‖  (⋅) (B  ) = 1.We follow the idea of the proof of Theorem 2.16 in [5] and make some crucial modifications where needed.Fix  ∈ (0, 1).It follows from Lemma 2.4 in [5] that for fixed  >  = 0 there exists a constant  > 0 such that for all holomorphic  in B  .Now, for each  ∈ B  , let   be the biholomorphic mapping of B  which interchanges 0 to , and let  =  ∘   .Then by making an obvious change of variables according to Lemma 12, Integrating both sides of the above inequality over B  with respect to d]() and using Fubini's theorem, we have that By Lemma 14 we have that Thus using Lemma 3 and the above result we obtain that Therefore ∇ ∈  (⋅) (B  , d]).
In Step 3, to prove (d) implies (a), we assume that  is a holomorphic function in B  such that the function Let  be a sufficiently large positive constant.Then by the proof of Theorem 2.16 in [5] we have that where  is the Bergman projection operator.By Lemma 16 we know that (⋅)−(0) ∈  (⋅) (B  , V).Therefore  ∈  (⋅) (B  ).