Variable Exponent Bergman Spaces

• Variable exponent Bergman spaces are holomorphic function spaces with spatially varying integrability defined via Luxemburg norms on the unit disc or ball.
• The theory integrates modular analysis, weighted inequalities, and bounded projection operators to establish fundamental embedding and operator-boundedness results.
• These spaces exhibit unique boundary determination and operator interactions that distinguish norm inequalities from more rigid modular inequalities.

Searching arXiv for recent and foundational papers on variable exponent Bergman spaces, operator theory, and weighted Bergman projections. I’m unable to invoke an external arXiv search tool in this interface, so I will rely on the arXiv records and detailed source material provided, citing the relevant arXiv papers directly. Variable exponent Bergman spaces are holomorphic function spaces in which the integrability index is allowed to vary with the spatial variable. On the unit disc or unit ball, they are defined as analytic subspaces of variable Lebesgue spaces endowed with a Luxemburg-type norm, and their modern theory combines several strands: variable-exponent modular analysis, Bergman projection theory, weighted inequalities of Békolle–Bonami type, Carleson measure embeddings, and operator-theoretic characterizations for Toeplitz and weighted composition operators. Recent work has placed these spaces within a Carleson-measure-based operator framework on the unit ball, while complementary disc results emphasize boundary determination, equivalence phenomena, and the sharp distinction between norm inequalities and modular inequalities (Tong et al., 18 Jul 2025, Ferguson, 2018, Izuki et al., 2019, BÉkollÈ et al., 2023).

1. Definitions, measures, and kernels

On the unit ball $B_n \subset \mathbb{C}^n$, with normalized Lebesgue measure $dv$, a standard weighted measure is

$$dv_\alpha(z)=c_\alpha (1-|z|^2)^\alpha dv(z), \qquad \alpha>-1.$$

The corresponding weighted Bergman kernel is

$$K_\alpha(z,w)=\frac{c_{n,\alpha}}{(1-\langle z,w\rangle)^{n+1+\alpha}},$$

and the normalized reproducing kernel may be written

$$k_{z,\alpha}(w)=\frac{(1-|z|^2)^{(n+1+\alpha)/2}}{(1-\langle w,z\rangle)^{n+1+\alpha}},$$

up to a unimodular constant. On the unit disc $\mathbb{D}$, one also uses

$$dA_\alpha(z)=(\alpha+1)(1-|z|^2)^\alpha dA(z), \qquad \alpha>-1,$$

with $dA$ the normalized area measure (Tong et al., 18 Jul 2025, Ferguson, 2018).

A variable exponent is a measurable function $p(\cdot)$ with essential bounds

$$p_-:=\operatorname*{ess\,inf} p(\cdot), \qquad p_+:=\operatorname*{ess\,sup} p(\cdot).$$

In the unit-ball theory developed for operator results, the standing hypotheses are

$dv$0

together with log-Hölder continuity: $dv$1 For the weighted Bergman-projector theory on the ball, the condition is stated as $dv$2, with $dv$3 and log-Hölder continuity expressed in terms of a pseudo-distance $dv$4 (Tong et al., 18 Jul 2025, BÉkollÈ et al., 2023).

The variable exponent Bergman space on $dv$5 is

$dv$6

where

$dv$7

and the Luxemburg norm is

$dv$8

On the disc, the analogous weighted modular is

$dv$9

with the associated Luxemburg–Nakano quasi-norm (Tong et al., 18 Jul 2025, Ferguson, 2018).

Different papers use different normalizations for the weighted measure and kernel. In the weighted variable-$$dv_\alpha(z)=c_\alpha (1-|z|^2)^\alpha dv(z), \qquad \alpha>-1.$$0 theory for the Bergman projector on the unit ball, one works with

$$dv_\alpha(z)=c_\alpha (1-|z|^2)^\alpha dv(z), \qquad \alpha>-1.$$1

and kernel

$$dv_\alpha(z)=c_\alpha (1-|z|^2)^\alpha dv(z), \qquad \alpha>-1.$$2

This is a normalization issue rather than a change of underlying geometric paradigm (BÉkollÈ et al., 2023).

2. Basic functional-analytic structure

Under $$dv_\alpha(z)=c_\alpha (1-|z|^2)^\alpha dv(z), \qquad \alpha>-1.$$3 and log-Hölder continuity, $$dv_\alpha(z)=c_\alpha (1-|z|^2)^\alpha dv(z), \qquad \alpha>-1.$$4 is a Banach space, and it is reflexive. In the same regime, polynomials are dense and point evaluations are bounded. The modular and norm are tightly linked: if $$dv_\alpha(z)=c_\alpha (1-|z|^2)^\alpha dv(z), \qquad \alpha>-1.$$5, then $$dv_\alpha(z)=c_\alpha (1-|z|^2)^\alpha dv(z), \qquad \alpha>-1.$$6, and modular convergence is equivalent to norm convergence (Tong et al., 18 Jul 2025).

The weighted Bergman projection on $$dv_\alpha(z)=c_\alpha (1-|z|^2)^\alpha dv(z), \qquad \alpha>-1.$$7 is

$$dv_\alpha(z)=c_\alpha (1-|z|^2)^\alpha dv(z), \qquad \alpha>-1.$$8

When $$dv_\alpha(z)=c_\alpha (1-|z|^2)^\alpha dv(z), \qquad \alpha>-1.$$9 and $$K_\alpha(z,w)=\frac{c_{n,\alpha}}{(1-\langle z,w\rangle)^{n+1+\alpha}},$$0, $$K_\alpha(z,w)=\frac{c_{n,\alpha}}{(1-\langle z,w\rangle)^{n+1+\alpha}},$$1 is bounded $$K_\alpha(z,w)=\frac{c_{n,\alpha}}{(1-\langle z,w\rangle)^{n+1+\alpha}},$$2. Its absolute-kernel variant

$$K_\alpha(z,w)=\frac{c_{n,\alpha}}{(1-\langle z,w\rangle)^{n+1+\alpha}},$$3

is also bounded on $$K_\alpha(z,w)=\frac{c_{n,\alpha}}{(1-\langle z,w\rangle)^{n+1+\alpha}},$$4 (Tong et al., 18 Jul 2025).

A basic consequence is the pointwise estimate

$$K_\alpha(z,w)=\frac{c_{n,\alpha}}{(1-\langle z,w\rangle)^{n+1+\alpha}},$$5

Its proof relies on sub-mean value estimates, log-Hölder continuity, and a localized Jensen-type inequality on Bergman balls. For fixed $$K_\alpha(z,w)=\frac{c_{n,\alpha}}{(1-\langle z,w\rangle)^{n+1+\alpha}},$$6, the weighted volume satisfies

$$K_\alpha(z,w)=\frac{c_{n,\alpha}}{(1-\langle z,w\rangle)^{n+1+\alpha}},$$7

and on such balls one has the comparabilities

$$K_\alpha(z,w)=\frac{c_{n,\alpha}}{(1-\langle z,w\rangle)^{n+1+\alpha}},$$8

for $$K_\alpha(z,w)=\frac{c_{n,\alpha}}{(1-\langle z,w\rangle)^{n+1+\alpha}},$$9 near $$k_{z,\alpha}(w)=\frac{(1-|z|^2)^{(n+1+\alpha)/2}}{(1-\langle w,z\rangle)^{n+1+\alpha}},$$0. These geometric facts underlie both embedding theorems and operator estimates (Tong et al., 18 Jul 2025).

3. Weighted Bergman projection and generalized Békolle–Bonami classes

For weighted variable Lebesgue spaces on the unit ball, one considers

$$k_{z,\alpha}(w)=\frac{(1-|z|^2)^{(n+1+\alpha)/2}}{(1-\langle w,z\rangle)^{n+1+\alpha}},$$1

and

$$k_{z,\alpha}(w)=\frac{(1-|z|^2)^{(n+1+\alpha)/2}}{(1-\langle w,z\rangle)^{n+1+\alpha}},$$2

The relevant geometry is encoded by pseudo-balls $$k_{z,\alpha}(w)=\frac{(1-|z|^2)^{(n+1+\alpha)/2}}{(1-\langle w,z\rangle)^{n+1+\alpha}},$$3 defined using the pseudo-distance

$$k_{z,\alpha}(w)=\frac{(1-|z|^2)^{(n+1+\alpha)/2}}{(1-\langle w,z\rangle)^{n+1+\alpha}},$$4

for $$k_{z,\alpha}(w)=\frac{(1-|z|^2)^{(n+1+\alpha)/2}}{(1-\langle w,z\rangle)^{n+1+\alpha}},$$5, and by the family $$k_{z,\alpha}(w)=\frac{(1-|z|^2)^{(n+1+\alpha)/2}}{(1-\langle w,z\rangle)^{n+1+\alpha}},$$6 of pseudo-balls touching the boundary, equivalently $$k_{z,\alpha}(w)=\frac{(1-|z|^2)^{(n+1+\alpha)/2}}{(1-\langle w,z\rangle)^{n+1+\alpha}},$$7 (BÉkollÈ et al., 2023).

The generalized Békolle–Bonami class $$k_{z,\alpha}(w)=\frac{(1-|z|^2)^{(n+1+\alpha)/2}}{(1-\langle w,z\rangle)^{n+1+\alpha}},$$8 is defined for $$k_{z,\alpha}(w)=\frac{(1-|z|^2)^{(n+1+\alpha)/2}}{(1-\langle w,z\rangle)^{n+1+\alpha}},$$9 by

$\mathbb{D}$0

where

$\mathbb{D}$1

This reduces to the classical $\mathbb{D}$2 condition when $\mathbb{D}$3. The class is dual-symmetric: $\mathbb{D}$4 and the paper proves the equivalences

$\mathbb{D}$5

Moreover, any $\mathbb{D}$6 belongs to $\mathbb{D}$7 (BÉkollÈ et al., 2023).

The central weighted projection theorem states that $\mathbb{D}$8 is well-defined and bounded on $\mathbb{D}$9 if and only if $$dA_\alpha(z)=(\alpha+1)(1-|z|^2)^\alpha dA(z), \qquad \alpha>-1,$$0. If $$dA_\alpha(z)=(\alpha+1)(1-|z|^2)^\alpha dA(z), \qquad \alpha>-1,$$1, then the positive Bergman operator

$$dA_\alpha(z)=(\alpha+1)(1-|z|^2)^\alpha dA(z), \qquad \alpha>-1,$$2

is also bounded on $$dA_\alpha(z)=(\alpha+1)(1-|z|^2)^\alpha dA(z), \qquad \alpha>-1,$$3 (BÉkollÈ et al., 2023).

The proof architecture is methodologically significant. A boundary-adapted maximal operator

$$dA_\alpha(z)=(\alpha+1)(1-|z|^2)^\alpha dA(z), \qquad \alpha>-1,$$4

is shown to be bounded on $$dA_\alpha(z)=(\alpha+1)(1-|z|^2)^\alpha dA(z), \qquad \alpha>-1,$$5 when $$dA_\alpha(z)=(\alpha+1)(1-|z|^2)^\alpha dA(z), \qquad \alpha>-1,$$6. The argument uses the regularization operator

$$dA_\alpha(z)=(\alpha+1)(1-|z|^2)^\alpha dA(z), \qquad \alpha>-1,$$7

together with the fact that $$dA_\alpha(z)=(\alpha+1)(1-|z|^2)^\alpha dA(z), \qquad \alpha>-1,$$8. Sufficiency is then obtained through a weighted extrapolation theorem adapted to $$dA_\alpha(z)=(\alpha+1)(1-|z|^2)^\alpha dA(z), \qquad \alpha>-1,$$9, using Rubio de Francia iteration and the Bergman $dA$0-factorization (BÉkollÈ et al., 2023).

4. Carleson embeddings and operator theory on $dA$1

Carleson-measure testing is the organizing principle for the operator theory developed on the unit ball. For a positive Borel measure $dA$2, define

$dA$3

This quantity is independent of the Bergman-ball radius $dA$4 up to comparable constants. A measure $dA$5 is a $dA$6-Carleson measure for $dA$7 if and only if $dA$8, and it is vanishing $dA$9-Carleson if and only if

$p(\cdot)$0

(Tong et al., 18 Jul 2025).

For weighted composition operators

$p(\cdot)$1

with $p(\cdot)$2, the variable exponent forces an additional weight

$p(\cdot)$3

When $p(\cdot)$4, this weight is identically $p(\cdot)$5. If $p(\cdot)$6 is bounded on $p(\cdot)$7, then $p(\cdot)$8, the pointwise bound

$p(\cdot)$9

holds, and the pull-back measure $$p_-:=\operatorname*{ess\,inf} p(\cdot), \qquad p_+:=\operatorname*{ess\,sup} p(\cdot).$$0 is $$p_-:=\operatorname*{ess\,inf} p(\cdot), \qquad p_+:=\operatorname*{ess\,sup} p(\cdot).$$1-Carleson. Conversely, if $$p_-:=\operatorname*{ess\,inf} p(\cdot), \qquad p_+:=\operatorname*{ess\,sup} p(\cdot).$$2 and $$p_-:=\operatorname*{ess\,inf} p(\cdot), \qquad p_+:=\operatorname*{ess\,sup} p(\cdot).$$3 is $$p_-:=\operatorname*{ess\,inf} p(\cdot), \qquad p_+:=\operatorname*{ess\,sup} p(\cdot).$$4-Carleson, then $$p_-:=\operatorname*{ess\,inf} p(\cdot), \qquad p_+:=\operatorname*{ess\,sup} p(\cdot).$$5 is bounded. Compactness is characterized by vanishing $$p_-:=\operatorname*{ess\,inf} p(\cdot), \qquad p_+:=\operatorname*{ess\,sup} p(\cdot).$$6-Carleson behavior together with the boundary limit

$$p_-:=\operatorname*{ess\,inf} p(\cdot), \qquad p_+:=\operatorname*{ess\,sup} p(\cdot).$$7

If $$p_-:=\operatorname*{ess\,inf} p(\cdot), \qquad p_+:=\operatorname*{ess\,sup} p(\cdot).$$8 almost everywhere, the Carleson and vanishing Carleson conditions are also sufficient (Tong et al., 18 Jul 2025).

For the difference $$p_-:=\operatorname*{ess\,inf} p(\cdot), \qquad p_+:=\operatorname*{ess\,sup} p(\cdot).$$9, the pseudo-hyperbolic distance

$dv$00

enters explicitly. If the difference is bounded or compact, then the measures $dv$01, $dv$02, $dv$03, and $dv$04 are $dv$05-Carleson or vanishing $dv$06-Carleson, respectively. Under the monotonicity assumption

$dv$07

the finiteness of the corresponding testing quantities is sufficient for boundedness (Tong et al., 18 Jul 2025).

Toeplitz operators admit an equally sharp characterization. For a positive Borel measure $dv$08 and $dv$09,

$dv$10

Then $dv$11 is bounded on $dv$12 if and only if

$dv$13

and it is compact if and only if

$dv$14

Here $dv$15 is the kernel exponent and is distinct from the Bergman weight parameter $dv$16 (Tong et al., 18 Jul 2025).

5. Boundary determination, equivalence, and disc-specific phenomena

On the unit disc, variable exponent Bergman spaces exhibit a precise dependence on boundary data. If $dv$17 is uniformly radially log-Hölder continuous and

$dv$18

then

$dv$19

with equivalent quasi-norms. In the Hardy setting, the same principle yields $dv$20 with equivalent norms. Under full log-Hölder continuity, one has an exact boundary-value criterion: $dv$21 If $dv$22 and $dv$23 are continuous and differ at one boundary point, then the spaces are different (Ferguson, 2018).

A deeper structural result concerns Hardy spaces $dv$24 when $dv$25 is harmonic, log-Hölder continuous, and has bounded harmonic conjugate. In that regime, the theory reduces many questions to $dv$26 by decomposing $dv$27 into pieces with controlled argument and comparing $dv$28 with $dv$29. This yields a Carleson measure theorem: for $dv$30, a positive measure $dv$31 satisfies

$dv$32

for all $dv$33 if and only if

$dv$34

for all Carleson boxes $dv$35 (Ferguson, 2018).

The same framework gives a Fejér–Riesz-type embedding

$dv$36

a Littlewood subordination analogue for $dv$37 with $dv$38, and boundedness of composition operators $dv$39 on $dv$40 under the harmonic log-Hölder/bounded-conjugate hypothesis (Ferguson, 2018).

In the radial case, there is a complete equivalence criterion between $dv$41 and a constant-exponent Bergman space $dv$42. If $dv$43 is radial with $dv$44, then

$dv$45

if and only if the tail averages of

$dv$46

are uniformly bounded; equivalently,

$dv$47

This permits highly nontrivial exponents. There exists $dv$48 with $dv$49 for any $dv$50 such that $dv$51, and there also exists $dv$52 for all $dv$53 sufficiently close to $dv$54 with the same equality; one explicit example is

$dv$55

These results show that radial oscillation near the boundary can be large without changing the Bergman space as a set (Ferguson, 2018).

6. Modular inequalities, examples, and the constant-exponent limit

A recurrent point of confusion is the distinction between norm inequalities and modular inequalities. For Bergman-type projections, the variable-exponent theory allows norm boundedness under log-Hölder hypotheses, but the modular inequality is much more rigid. On the unit disc, if

$dv$56

holds for all $dv$57, then $dv$58 must equal a constant almost everywhere. The same phenomenon holds for the analytic Bergman projection on the upper half-plane and the harmonic Bergman projection on the upper half-space (Izuki et al., 2019).

The obstruction is produced by lower pointwise bounds for projections of characteristic functions. For the disc, fixing $dv$59, there exist a compact neighborhood $dv$60 and $dv$61 such that

$dv$62

for all measurable $dv$63 and all $dv$64. This makes the modular inequality incompatible with nonconstant exponents on disjoint subsets of $dv$65. Thus norm inequalities and modular inequalities are fundamentally different in variable exponent Bergman analysis (Izuki et al., 2019).

Variable exponents also create genuinely new operator-theoretic behavior. On the unit ball, if

$dv$66

then $dv$67 is bounded on $dv$68 for every constant $dv$69. However, for

$dv$70

one has

$dv$71

so $dv$72 is unbounded on $dv$73. This exhibits a mechanism absent from the constant-exponent theory: the interaction between the local oscillation of $dv$74 and the geometry of $dv$75 (Tong et al., 18 Jul 2025).

In the constant-exponent limit, the variable-exponent criteria reduce to the classical Bergman theory. The additional ingredients that disappear in that limit are precisely the variable-exponent weights such as $dv$76, the dependence on $dv$77, and the need for log-Hölder control to stabilize local averages and projection estimates. This places variable exponent Bergman spaces as a strict extension of classical $dv$78, rather than a mere reparameterization of it (Tong et al., 18 Jul 2025).