# Derivative and Lipschitz Type Characterizations of Variable Exponent Bergman Spaces

Authors: Rumeng Ma; Jingshi Xu
Journal: Journal of Function Spaces
 (2018)

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## Abstract

We give derivative and Lipschitz type characterizations of Bergman spaces with log-Hölder continuous variable exponent.


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## Body

## 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 [2–5]. 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 attracted many 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 Cn for any integer n. To state our results, we firstly recall some definitions.We denote the Euclidean norm onCn by ·. Then we let Bn={z∈Cn:z<1}, the open unit ball in Cn. Let dν be the normalized volume measure on Bn. For any α>-1, let dνα(z)=Cα(1-z2)αdν(z), where Cα is a positive constant such that να(Bn)=1. In this paper, we only consider the case of α=0.For a measurable functionp:Bn→[1,∞), we call it a variable exponent and denote p+≔esssupz∈Bnp(z),p-≔essinfz∈Bnp(z). Denote by P(Bn) the set of all variable exponents with p+<∞. Let p∈P(Bn). For a complex-valued measurable function f on Bn, we define the modular of f by(1)ρp·f≔∫Bnfzpzdνz, and the Luxemburg-Nakano norm by(2)fLp·Bn≔infλ>0:ρp·fλ≤1. The variable Lebesgue space Lp(·)(Bn,dν) is the set of all complex-valued measurable functions f on Bn such that ρp(·)(f)<∞. It is a Banach space equipped with the Luxemburg-Nakano norm.Ifp∈P(Bn), then the variable exponent Bergman space Ap(·)(Bn) is the class of all holomorphic functions on Bn which belong to the variable exponent Lebesgue space Lp(·)(Bn,dν). It is easy to show that Ap(·)(Bn) is a closed subspace of Lp(·)(Bn,dν). When p is a constant, these spaces are called weighted Bergman space with standard weights; see [3, 4] for details. As usual, we denote by H(Bn) the space of holomorphic functions on Bn.Givenf∈H(Bn), the radial derivative of f at z is defined by(3)Rfz≔∑k=1nzk∂f∂zkz. The complex gradient of f at z is defined by (4)∇fz≔∑k=1n∂f∂zkz21/2. And the invariant complex gradient of f at z is given by (5)∇~fz≔∇f∘φz0, where φz is the automorphism of Bn mapping 0 to z.For anya∈Bn, let φa be a biholomorphic map on Bn such that φa(0)=a and φa-1=φa. The explicit formulas are available for φa (see [5]).Letβ be the Bergman metric on Bn, namely, for z,w∈Bn, (6)βz,w≔12log⁡1+φzw1-φzw. It was known that ρ(z,w)≔φzw is also a distance function on Bn. ρ is called the pseudohyperbolic metric on Bn. For any r∈(0,1) and z∈Bn, we let D(z,r)≔{w∈Bn:ρ(w,z)<r}, the pseudohyperbolic ball centered at z with radius r. D(z,r) is Euclidean ball B(x,a)={w∈Bn:w-x<a} with(7)x=1-r21-r2z2z,a=1-z21-r2z2r. In particular, if r is fixed, then the volume of D(z,r) is comparable to (1-z2)n+1.For anyR>0 and z∈Bn, we let E(z,R)≔{w∈Bn:β(w,z)<R}, the hyperbolic ball centered at z with radius R. If r∈(0,1), then D(z,r)=E(z,R) with(8)R=12log⁡1-r1+r.Consequently, if R is fixed, then the volume of E(z,R) is also comparable to (1-z2)n+1.Definition 1.
A functionp:Bn→R is said to be log-Hölder continuous or satisfy the Dini-Lipschitz condition on Bn if there exists a positive constant Clog such that(9)pz-pw≤Cloglog⁡1/z-w,for all z,w∈Bn such that z-w<1/2. We will denote by Plog(Bn) the set of all log-Hölder continuous functions on Bn.Now, the main result of the paper is the following.Theorem 2.
Supposep∈Plog(Bn), and f is holomorphic in Bn. Then the following conditions are equivalent.
(a)f∈Ap(·)(Bn).
(b)∇~f·∈Lp(·)(Bn,dν).
(c)1-·2∇f·∈Lp(·)(Bn,dν).
(d)1-·2Rf·∈Lp(·)(Bn,dν).
(e) There exists a continuous functiong in Lp(·)(Bn,dν) such that, for all z,w∈Bn,(10)fz-fw≤ρz,wgz+gw.
(f) There exists a continuous functiong in Lp(·)(Bn,dν) such that, for all z,w∈Bn,(11)fz-fw≤βz,wgz+gw.
(g) There exists a continuous functiong such that (1-·2)g in Lp(·)(Bn,dν) and for all z,w∈Bn,(12)fz-fw≤z-wgz+gw.In Section2, 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 a≲b means there exists a constant C>0 such that a≤Cb, and a≈b means a≲b and b≲a.


## 2. Preliminaries

In this section, we recall some preliminary results that we shall need in our paper.Lemma 3 ([5], Lemma 2.20).
Letr be a positive number. Then there exists a positive constant Cr such that (13)Cr-1≤1-z21-z,w≤Cr,Cr-1≤1-z21-w2≤Cr, for all z,w∈Bn with β(z,w)<r. Moreover, if r is bounded above, then we may choose Cr to be independent of r.The following Jensen type inequality was proved in [17] in the context of spaces of homogeneous type (SHT).Lemma 4.
Suppose thatp(·)∈Plog(Bn). Then(14)⨍Dz,rfwdνwpy≲⨍Dz,rfwpwdνw+1,
for ally∈D(z,r), z∈Bn, provided that fLp(·)(Bn)≤1.Remark 5.
Usually, Lemma4 holds for Euclidean balls. We shall use it in pseudohyperbolic metric. Since each pseudohyperbolic ball D(z,r) is actually Euclidean ball, we have the above form. And ⨍Dz,rfwdνw=(1/Dz,r)∫Dz,rfwdνw.Definition 6.
Given a functionf∈Lloc1(Bn), the Hardy-Littlewood maximal function of f, denoted by Mf, is defined for any z∈Cn by (15)Mfz≔supr>01Bz,r∫Bz,rfwdw.Lemma 7 ([14], Theorem 3.16).
Letp∈Plog(Bn). Then the Hardy-Littlewood maximal function is bounded in Lp(·)(Bn), and it means that there exists a positive constant C such that for each f∈Lp(·)(Rn)(16)MfLp·Bn≤CfLp·Bn.LetF denote a family of pairs of nonnegative measurable function and A1 denote the Muckenhoupt A1 weight.Lemma 8 ([14], Theorem 5.24).
Suppose that for somep0≥1 the family F is such that, for all w∈A1,(17)∫BnFzp0wzdz≤C0∫BnGzp0wzdz,F,G∈F.Given p(·)∈P(Bn), if p0≤p-≤p+<∞ and the maximal operator is bounded on L(p(·)/p0)′(Bn), then there is a positive constant C independent of (F,G) such that(18)FLp·Bn≤CGLp·Bn.Forz∈Cn, we define the following radial test function: (19)ηz=Cexp1z2-1,ifz<1,0,ifz≥1,where C>0 is the normalizing constant in the sense ∫Bnη(z)dν(z)=1. For r∈[1/2,1), we define ηr(z)≔(4r2/(1-r)2)η(2rz/(1-r)). Notice that ηr is a C∞ function supported on the set ((1-r)/2r)Bn¯ (where ρBn¯ stands for the closed unit ball with radius ρ) and ∫Bnηr(z)dν(z)=1.Definition 9.
Given a functionf∈Ap(·)(Bn), we will define its mollified dilation fr:(1+r/2r)Bn→C as (20)frz≔∫Bnfrwηrz-wdνw,where ρBn stands for the complex ball with radius ρ.The following lemma is the counterpart of Theorem 3.5 in [12] for any integer n.Lemma 10.
Letp(·)∈Plog(Bn) and f∈Ap(·)(Bn). Then there exists a positive constant C for r∈(1/2,1), fr∈Ap(·)(Bn) such that (21)sup1/2≤r<1frAp·Bn≤CfAp·Bn. Moreover, fr-fAp(·)(Bn)→0 as r→1-.Proof.
Leth∈Ap′(·)(Bn), hAp′(·)(Bn)=1, 1/p(x)+1/p′(x)=1. Then by making change of variable and using Fubini’s theorem, we obtain (22)∫Bnfrzhzdνz≤∫Bnhz∫Bz,1-r/2rfrwηrz-wdνwdνz=∫B0,1-r/2rηrw∫Bnfrz-whzdνzdνw=1rn∫B0,1-r/2rηrw∫B-rw,rfzhzr+wdνzdνw=1rn∫B0,1+r/2rfz∫B0,1-r/2rχB-rw,rzηrw∣hzr+wdνwdνz≤1rn∫B0,1+r/2rfz∫Bz/r,1-r/2r∩Bnηrw-zrhwdνwdνz≲∫B0,1+r/2rfz1Bz/r,1-r/2r∫Bz/r,1-r/2rhwdνwdνz.Since B(z/r,(1-r)/2r)⊂B(z,3(1-r)/2r), we have (23)1Bz/r,1-r/2r∫Bz/r,1-r/2rhwdνw≤9Bz,31-r/2r∫Bz,31-r/2rhwdνw. Therefore, considering the maximal function Mh, and using Hölder’s inequality and Lemma 7 we have that (24)∫Bnfrzhzdνz≲∫B0,1+r/2fzMhzdνz≲fAp·BnMhAp′·Bn≲fAp·BnhAp′·Bn which shows the first part of the lemma.
For the second part, fixε>0 and choose a function g with compact support on Bn, such that f-gAp(·)(Bn)<ε (see [14] Theorem 2.72). Then by the previous part of the proof, (25)fr-fAp·Bn≤f-grLp·Bn+gr-gLp·Bn+g-fLp·Bn≲ε+gr-gLp·Bn. Therefore, we only need to prove the convergence in norm for the compactly supported function g. Defining G(z)≔g(z)/(2gL∞(Bn)), we obtain that GL∞(Bn)≤1/2. And (26)Grz≤∫BnGrwηrz-wdνw≤GL∞Bn∫Bnηrz-wdνw≤12. Thus, Gr-gL∞(Bn)≤1 and consequently (27)∫Bngrz-gzpzdνz=∫Bn2gL∞BnpzGrz-gzpzdνz≲2gL∞Bn+1p+∫BnGrz-gzp-dνz. Therefore, we have reduced the convergence to the case of a constant exponent.
In this case, from [3] we know that radial dilations G(r·) converge in p--norm to G. Consequently, if we define the translation operator τzf(w)≔f(z-w), then by Minkowski’s inequality (28)Gr-GLp-Bn≤∫BnτzGr·-GLp-Bnηrzdνz≤∫BnτzGr·-Gr·Lp-Bnηrzdνz+Gr·-GLp-Bn, and the result follows from the continuity of τz on the space Lp-(Bn).From the above lemma, we have the following lemma.Lemma 11.
Letp(·)∈Plog(Bn). Then the set of holomorphic polynomials is dense in Ap(·)(Bn).Lemma 12 ([5], Proposition 1.13).
Ifα is real and f is in L1(Bn,dν), then(29)∫Bnf∘φwdναw=∫Bnfw·1-z2n+1+α1-z,w2n+1+αdναw, where φ is any automorphism of Bn and φ(0)=z.Lemma 13 ([5], Lemma 2.14).
Iff is holomorphic in Bn, then for all z∈Bn(30)1-z2Rfz≤1-z2∇fz≤∇~fz.Lemma 14 ([5], Theorem 1.12).
Letc∈R and t∈(-1,∞). For z∈Bn, define (31)Icz≔∫Sndσξ1-z,ξn+c,Jc,tz≔∫Bn1-w2tdνw1-z,wn+1+t+c.
Whenc<0, then Ic and Jc,t are bounded in B.
Whenc>0, then (32)Icz~1-z2-c~Jc,t.
Finally,(33)I0z~log⁡11-z2~J0,tz.
The notationa(z)~b(z) means that the ratio a(z)/b(z) has a positive finite limit as z→1.The Bergman projection operatorP is defined for functions f on Bn by(34)Pfz≔∫Bnfξ1-z,ξn+1dνξ.To proceed, we need the classBp for p∈(1,∞). Let ω be a positive measurable function, and ω is called belonging to the class Bp if there exists a constant C such that, for every pseudo-ball B⊂Bn, (35)ωBνB1νB∫Bωz-1/p-1dνzp-1≤C, where ω(B)=∫Bnω(z)dν(z).Lemma 15 ([18], Theorem 1).
Letω∈L1(Bn). A necessary and sufficient condition for the Bergman projection to be bounded on Lp(ω) is that ω belongs to the class Bp for p∈(1,∞).Lemma 16.
Letp∈Plog(Bn). Then the Bergman projection operator P is bounded from Lp(·)(Bn) onto Ap(·)(Bn).Proof.
We follow the proof of Theorem 4.4 in [12]. It is clear that Pf is holomorphic function on Bn, so we only need to prove that P is bounded and surjective from Lp(·)(Bn) to Lp(·)(Bn). By the property of the Muckenhoupt A1 (see [19]), if w∈A1 then w∈Aq for q>1. Now we pick p0 such that 1<p0<p-. By the remark 1 in [18], we have that w∈Bp0. By Lemma 15 we conclude that the family F={(Pf,f):f∈Lp0(Bn)} satisfies (17). By the fact that (p(·)/p0)′∈Plog(Bn) and Lemmas 7 and 8, then there exists a constant C>0 such that (36)PfLp·Bn≤Cp·fLp·Bn. This shows that P is bounded in Lp(·)(Bn).
In order to prove thatP is surjective, we use the fact that Pg=g for every g∈A2(Bn). In particular, this equality holds for any polynomial. Thus, if f∈Ap(·)(Bn), we use Lemma 11 to find a sequence of polynomials qm converging in Ap(·)(Bn) to f. But since qm=Pqm→Pf in Ap(·)(Bn) then we have that Pf=f.Lemma 17 ([5], Lemma 2.24).
Supposer>0, q>0, and α>-1. Then there exists a constant C>0 such that(37)fzq≤C1-z2n+1+α∫Dz,rfwqdναw, for all f∈H(Bn) and all z∈Bn.The last result deals with the limits ofβ(z,w) and ρ(z,w) as w tends z in the radial direction.Lemma 18 ([6], Lemma 5.2).
Supposez∈Bn and w=tz, where t is a scalar. Then (38)limw→zβz,wz-w=limw→zρz,wz-w=11-z2.


## 3. Proof of Theorem2

Proof of Theorem2.
We shall divide the proof into 8 steps. In Step 1, by Lemma13 we obtain that (b) implies (c) and (c) implies (d).
In Step 2, we prove (a) implies (b). We firstly consider thatf∈Ap(·)(Bn) such that fLp(·)(Bn)=1. We follow the idea of the proof of Theorem 2.16 in [5] and make some crucial modifications where needed. Fix r∈(0,1). It follows from Lemma 2.4 in [5] that for fixed β>α=0 there exists a constant C>0 such that(39)∇g0≤C∫Dz,rgwdνβw, for all holomorphic g in Bn.
Now, for eachz∈Bn, let φz be the biholomorphic mapping of Bn which interchanges 0 to z, and let g=f∘φz. Then by making an obvious change of variables according to Lemma 12, we obtain (40)∇~fz≤C1-z2n+1+β∫Dz,rfw1-z,w2n+1+βdνβw=C1-z2n+1+β∫Dz,rfwCβ1-w2β1-z,w2n+1+βdνw≤C1-z2n+1∫Dz,rfwdνw.Since the volume of D(z,r) is comparable to (1-z2)n+1, then we have (41)∇~fz≤C⨍Dz,rfwdνw. Then by Lemma 4 we have (42)∇~fzpz≲⨍Dz,rfwpwdνw+1≲∫Dz,rfwpw1-z2n+1+β1-z,w2n+1+βdνβw+1.
Integrating both sides of the above inequality overBn with respect to dν(z) and using Fubini’s theorem, we have that (43)∫Bn∇~fzpzdνz≲∫Bn∫Dz,rfwpw1-z2n+1+β1-z,w2n+1+βdνβwdνz+1≲∫Bnfwpwdνβw∫Bn1-z2n+1+β1-z,w2n+1+βdνz+1≲∫Bnfwpw1-w2βdνw∫Bn1-z2n+1+β1-z,w2n+1+βdνz+1≲∫Bnfwpw1-w2βdνw∫Bn1-z2n+1+β1-z,w2n+1+βdνz+1.By Lemma 14 we have that(44)∫Bn1-z2n+1+β1-z,w2n+1+βdνz~1-w2-β. Thus using Lemma 3 and the above result we obtain that (45)∫Bn∇~fzpzdνz≲∫Bnfwpw1-w2β1-w2-βdνw+1≲∫Bnfwpwdνw+1. Therefore ∇~f∈Lp(·)(Bn,dν).
Finally, for a general0≠f∈Ap(·)(Bn), define h=ffLp(·)(Bn)-1, and we apply the previous result for h and conclude that ∇~f∈Lp(·)(Bn,dν).
In Step 3, to prove (d) implies (a), we assume thatf is a holomorphic function in Bn such that the function 1-z2Rfz∈Lp(·)(Bn,dν). Let β be a sufficiently large positive constant. Then by the proof of Theorem 2.16 in [5] we have that (46)fz-f0≤∫BnRfw1-z,wn+βdνβw=Cβ∫BnRfw1-w2β1-z,wn+βdνw≲∫BnRfw1-w21-z,wn+1dνw≲P1-·2Rf·z, where P is the Bergman projection operator. By Lemma 16 we know that f(·)-f(0)∈Lp(·)(Bn,dv). Therefore f∈Ap(·)(Bn).
In Step 4, we prove (a) implies (e). Iff∈Ap(·)(Bn) and z∈Bn, we have (47)fz-f0=∫01∑k=1nzk∂f∂zktzdt. Fix r∈(0,1). It follows that, for ρ(z,0)<r, we have (48)fz-f0=∫01∑k=1nzk∂f∂zktzdt≤∫01∑k=1nzk∂f∂zktzdt≤∫01∑k=1nzk21/2∑k=1n∂f∂zktz21/2dt≤∫01z∇ftzdt=z∫01∇ftzdt≤zsup∇fw:w∈D0,r. Since in the relatively compact set D(0,r) the Euclidean metric is comparable to the pseudohyperbolic metric, then ∇fw is comparable to ∇~fw in the relatively compact set D(0,r). Thus, there is a constant C>0, which depends only on r, such that (49)fz-f0≤Cρz,0sup∇~fw:w∈D0,r, for all z∈D(0,r). Replacing f by f∘φw, and z by φw(z), respectively, and using the Möbius invariance of the pseudohyperbolic metric and the invariant gradient, we obtain (50)fz-fw≤Cρz,wsup∇~fu:u∈Dz,r, for all z and w in Bn with ρ(z,w)<r.
Let(51)gz≔fzr+hz, where h(z)=Csup{∇~fu:u∈D(z,r)}. Then it is easy to see that the function g is continuous on Bn and (10) holds. Now we only need to prove that g∈Lp(·)(Bn,dν). Since f is already in Lp(·)(Bn,dν), the remainder is to show that h∈Lp(·)(Bn,dν).
Recall that ifr and R satisfy (8), then D(z,r)=E(z,R). So we can choose r′∈(0,1) such that D(z,r′)=E(z,2R) for all z∈D. For any u∈E(z,R), any w1∈E(u,R), we have β(w1,u)<R. Then we have β(u,z)<R due to u∈E(z,R). By the triangle inequality, we have (52)βw1,z≤βw1,u+βu,z<R+R<2R. So w1∈E(z,2R). That is, E(u,R)⊂E(z,2R). Equivalently, we have D(u,r)⊂D(z,r′) whenever u∈D(z,r).
By Lemmas17 and 3 we obtain that (53)hz≲11-z2n+1∫Dz,rhwdνw≲11-z2n+1∫Dz,rCsup∇~fu:u∈Dw,rdνw≲11-z2n+1∫Dz,r′∇~fwdνw≲⨍Dz,r′∇~fwdνw, where we used that the volume of D(z,r′) is comparable to (1-z2)n+1. Without loss of generality, we assume ∇~fLp(·)(Bn,dv)=1. Therefore, using Lemma 4 we obtain that (54)hzpz≲⨍Dz,r′∇~fwpwdνw+1.≲11-z2n+1∫Dz,r′∇~fwpwdνw+1. Integrating both sides of the above inequality over Bn with respect to dν(z) and using Fubini’s theorem, we have that (55)∫Bnhzpzdνz≲∫Bn11-z2n+1∫Dz,r′∇~fwpwdνwdνz+1≲∫Bn∇~fwpwdνw∫Dw,r′11-z2n+1dνz+1≲∫Bn∇~fwpwdνw+1. Due to f∈Ap(·)(Bn), by Step 2 we have that ∇~f·∈Lp(·)(Bn,dν). Thus h∈Lp(·)(Bn,dν).
In Step 5, it is easy to see that (e) implies (f), according toρ(z,w)≤β(z,w), for all z,w∈Bn.
In Step 6, we prove that (f) implies (a). Assume that there exists a continuous functiong∈Lp(·)(Bn,dν) such that (11) holds for all z,w∈Bn. We fix z∈Bn and let w=tz, where t is a scalar. Then(56)fz-fwz-w≤βz,wz-wgz+gw, for all z≠w in Bn. Let t approach 1 and apply Lemma 18; we obtain that 1-z2Rfz≤2g(z) for all z∈Bn. Since g∈Lp(·)(Bn,dν), then 1-·2Rf·∈Lp(·)(Bn,dν), and by Step 3 we have that f∈Ap(·)(Bn).
In Step 7, we prove that (g) implies (a). If there exists a continuous functiong such that (1-·2)g in Lp(·)(Bn,dν) and for all z,w∈Bn,(57)fz-fw≤z-wgz+gw, then(58)fz-fwz-w≤gz+gw. Let z=(z1,…,zk,…,n) and fix z. For each k∈{1,…,n}, let w=(z1,⋯,wk,…,zn). Let wk tend to zk, and we obtain(59)∂f∂zkz≤2gz. Therefore for all z∈Bn, we obtain(60)∇fz≤2ngz. Thus(61)1-z2∇fzpz≤2n1-z2gzpz. Since (1-·2)g∈Lp(·)(Bn,dν), thus 1-·2∇f∈Lp(·)(Bn,dν). Therefore, we have f∈Ap(·)(Bn) by the equivalency of (c) and (a) which we have proved.
In Step 8, we prove that (e) implies (g). If Condition (e) holds in this theorem, Lemma 1.2 in [5] says that(62)1-φzw2=1-z21-w21-z,w2. Thus(63)ρz,w2z-w2=z-w2+z,w2-z2w2z-w21-z,w2. By the Cauchy inequality, z,w2≤z2w2, we obtain that, for all z,w∈Bn,(64)ρz,w≤z-w1-z,w. By condition (e), there exists a continuous function h in Lp(·)(Bn,dν) such that, for all z,w∈Bn,(65)fz-fw≤ρz,whz+hw. Therefore,(66)fz-fw≤z-w1-z,whz+hw. Since (67)1-z,w≥1-z,1-z,w≥1-w, we have (68)fz-fw≤z-whz1-z+hw1-w. Write g(z)=h(z)/(1-z), then (69)fz-fw≤z-wgz+gw. Since z∈Bn, we have (70)11-z≤21-z2,so (71)gz=hz1-z≤2hz1-z2. Since h∈Lp(·)(Bn,dν), we have (1-·2)g∈Lp(·)(Bn,dν).



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