[Paper Review] Reproducing kernel estimates, bounded projections and duality on large weighted Bergman spaces

Hicham Arroussi, Jordi Pau|RECERCAT (Consorci de Serveis Universitaris de Catalunya)|Sep 24, 2013

Holomorphic and Operator Theory11 references4 citations

TL;DR

This paper establishes sharp pointwise and integral estimates for reproducing kernels in large weighted Bergman spaces with rapidly decreasing radial weights, such as exponential-type weights. Using these estimates, it proves the boundedness of the Bergman projection on $ L^p(\omega^{p/2}) $ for $ 1 \leq p < \infty $, and identifies dual spaces of $ A^p(\omega^{p/2}) $ with $ A^{p'}(\omega^{p'/2}) $, resolving a key duality problem for exponential weights.

ABSTRACT

We obtain ceratin estimates for the reproducing kernels of large weighted Bergman spaces. Applications of these estimates to boundedness of the Bergman projection on $L^p(\D,ω^{p/2})$, complex interpolation and duality of weighted Bergman spaces are given.

Motivation & Objective

• To understand the behavior of reproducing kernels in large weighted Bergman spaces with rapidly decreasing radial weights, such as exponential-type weights.
• To establish pointwise and integral estimates for the reproducing kernels $ K_z $, which are crucial for operator theory in these spaces.
• To prove the boundedness of the Bergman projection $ P_\omega $ from $ L^p(\omega^{p/2}) $ to $ A^p(\omega^{p/2}) $ for $ 1 \leq p < \infty $.
• To characterize the dual space of $ A^p(\omega^{p/2}) $ as $ A^{p'}(\omega^{p'/2}) $ under the natural integral pairing.
• To resolve the duality problem for exponential-type weights, which had remained open despite known results for standard weights.

Proposed method

• Derives pointwise estimates for the reproducing kernel $ |K_z(\xi)| $ using the geometry of the weight and the Laplacian of the associated potential $ \varphi $, where $ \omega = e^{-2\varphi} $.
• Uses the class $ \mathcal{L}^* $ of weights satisfying $ (\Delta\varphi)^{-1/2} \asymp \tau(z) $ for $ \tau \in \mathcal{L} $, ensuring regularity and doubling-type properties.
• Applies a generalized sub-mean value inequality (Lemma A) to control $ |f(z)|^p \omega(z) $ via integrals over $ \tau(z) $-scaled disks.
• Establishes an integral estimate for $ \int_{\mathbb{D}} |K_z(\xi)| \omega(\xi) \, dA(\xi) $, which is central to proving boundedness of the Bergman projection.
• Uses complex interpolation and duality techniques to identify dual spaces, relying on the density of finite linear combinations of reproducing kernels in $ A^p(\omega^{p/2}) $.
• Applies the Riesz representation theorem and Fubini’s theorem to construct the dual function $ g $ from a functional $ \Lambda $, showing $ g \in A^{p'}(\omega^{p'/2}) $.

Experimental results

Research questions

• RQ1 Can sharp pointwise estimates for the reproducing kernel $ K_z $ be obtained in large weighted Bergman spaces with rapidly decreasing weights?
• RQ2 Is the Bergman projection $ P_\omega $ bounded on $ L^p(\omega^{p/2}) $ for $ 1 \leq p < \infty $ when $ \omega $ is a rapidly decreasing radial weight?
• RQ3 What is the dual space of $ A^p(\omega^{p/2}) $ under the natural integral pairing $ \langle f,g \rangle_\omega $ for such weights?
• RQ4 How does the lack of explicit kernel formulas affect the duality and boundedness theory in these spaces?
• RQ5 Can the duality results for standard weights be extended to exponential-type weights, which are not in the standard $ (1-|z|^2)^\alpha $ class?

Key findings

• The reproducing kernel $ K_z $ satisfies the pointwise estimate $ |K_z(\xi)| \lesssim \omega(z)^{-1/2} \omega(\xi)^{-1/2} $ under the given weight conditions, which is sharp and of independent interest.
• The Bergman projection $ P_\omega $ is bounded from $ L^p(\omega^{p/2}) $ to $ A^p(\omega^{p/2}) $ for all $ 1 \leq p < \infty $, extending known results from $ p=2 $.
• The dual space of $ A^p(\omega^{p/2}) $ is isometrically isomorphic to $ A^{p'}(\omega^{p'/2}) $ under the natural integral pairing, with equivalent norms.
• The set of finite linear combinations of reproducing kernels is dense in $ A^p(\omega^{p/2}) $ for $ 1 \leq p < \infty $, which supports the duality results.
• The predual of $ A_0(\omega^{1/2}) $ is identified as $ A^1(\omega^{1/2}) $, extending classical duality results to the exponential weight setting.
• The results resolve a long-standing open problem regarding duality for exponential-type weights, previously unresolved despite progress in related settings.