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Boundedness of the Bergman projection on some weighted mixed norm Lebesgue spaces of the upper-half space

Jean-Marcel Tanoh Dje, Felix Ofori, Benoit F. Sehba

TL;DR

This work characterizes the boundedness of the Bergman projection on weighted mixed-norm Bergman spaces over the upper-half plane by weights built from a growth function and logarithmic terms. The main finding is that the boundedness condition reduces to the unweighted-type constraint $\alpha+1<q(\beta+1)$, independent of the weight, and this is connected to a one-dimensional Hilbert-type operator $H_\beta$ via a Forelli–Rudin-type reduction. The authors develop a weighted theory for $H_\beta$ using interval estimates and Schur’s test, addressing $p=1$ and $p>1$ separately, and then transfer these results to the Bergman projection $P_\beta$ (and $P_\beta^+$) on mixed-norm spaces. Applications include reproducing formulas and duality, a derivative characterization of weighted Bergman spaces, and an atomic decomposition obtained through a Whitney-type decomposition and kernel-based reconstruction, extending classical results to the weighted mixed-norm setting.

Abstract

In this paper, we prove the boundedness of the Bergman projection on weighted mixed norm spaces of the upper-half space for some weights that are constructed using the logarithm function and growth functions. Our necessary and sufficient condition is the same as in the unweighted case, that is it involves only the parameters and not the weight. We then provide some applications in terms of dual and derivative characterization, and an atomic decomposition of the corresponding Bergman spaces.

Boundedness of the Bergman projection on some weighted mixed norm Lebesgue spaces of the upper-half space

TL;DR

This work characterizes the boundedness of the Bergman projection on weighted mixed-norm Bergman spaces over the upper-half plane by weights built from a growth function and logarithmic terms. The main finding is that the boundedness condition reduces to the unweighted-type constraint , independent of the weight, and this is connected to a one-dimensional Hilbert-type operator via a Forelli–Rudin-type reduction. The authors develop a weighted theory for using interval estimates and Schur’s test, addressing and separately, and then transfer these results to the Bergman projection (and ) on mixed-norm spaces. Applications include reproducing formulas and duality, a derivative characterization of weighted Bergman spaces, and an atomic decomposition obtained through a Whitney-type decomposition and kernel-based reconstruction, extending classical results to the weighted mixed-norm setting.

Abstract

In this paper, we prove the boundedness of the Bergman projection on weighted mixed norm spaces of the upper-half space for some weights that are constructed using the logarithm function and growth functions. Our necessary and sufficient condition is the same as in the unweighted case, that is it involves only the parameters and not the weight. We then provide some applications in terms of dual and derivative characterization, and an atomic decomposition of the corresponding Bergman spaces.
Paper Structure (12 sections, 23 theorems, 111 equations)

This paper contains 12 sections, 23 theorems, 111 equations.

Table of Contents

  1. Introduction
  2. Boundedness of the Hilbert-type operator $H_\beta$
  3. Weighted estimates on the positive line
  4. Proofs of Theorem \ref{['thm:main2']}
  5. Boundedness of the Bergman projection
  6. Proofs of Theorem \ref{['thm:main1']}
  7. Applications
  8. Reproducing formulas and duality
  9. Derivative characterization of weighted mixed norm Bergman spaces
  10. Atomic decomposition of functions in $A_{\omega^{k}}^{p,q}({\mathbb C}_+)$
  11. A Whitney decomposition of ${\mathbb C}_+$
  12. Atomic decomposition

Key Result

Theorem 1

Let $1\leq p,q<\infty$, $\alpha, \beta>-1$. Assume that $\Phi\in \mathscr{L}\cup\mathscr{U}$. Put $\omega=\mathcal{T}_\Phi^{\Vec{\varepsilon}}$, and let $k\in\mathbb{R}$. Then the following assertions are equivalent.

Theorems & Definitions (41)

  • Theorem 1
  • Theorem 2
  • Remark 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • ...and 31 more