arXiv subjects

Atte Pennanen

Publications and source records attributed to Atte Pennanen.

3 recordsLinked to original sources

arXiv2026

Optimal off-diagonal upper estimates for Bergman reproducing kernels

In this paper, we establish a sharp off-diagonal pointwise upper estimate for the Bergman reproducing kernel associated with a radial weight on the unit disc. Our proof is self-contained and assumes the weight satisfies a natural one-sided doubling condition on its moments. The standard kernels demonstrate that this estimate is sharp, up to a multiplicative constant, at every point. We also show that the estimate in fact characterizes the class of radial doubling weights under consideration. As applications, we first obtain optimal $L^p$-mean estimates for certain modified Bergman kernels. This approach recovers the key estimates in [Pel\'aez et al., J. Math. Pures Appl. 105(2016), 102--130] and [Pel\'aez et al., arxiv.org/pdf/2407.04645] via a novel and more direct proof. Second, we establish novel connections between the non-tangential maximal function of the Berezin transform, the H\"ormander maximal function, and Carleson measures. Notably, these connections are new even in the setting of the standard weighted Bergman spaces. Finally, we extend our main results to higher dimensions, harmonic Bergman kernels, and two-weight fractional derivatives of kernels, the latter of which yields sharp estimates for Dirichlet reproducing kernels.

math.CV↗

arXiv2025

Embedding theorems for Bergman-Zygmund spaces induced by doubling weights

Let $0<p<\infty$ and $\Psi: [0,1) \to (0,\infty)$, and let $\mu$ be a finite positive Borel measure on the unit disc $\mathbb{D}$ of the complex plane. We define the Lebesgue-Zygmund space $L^p_{\mu,\Psi}$ as the space of all measurable functions $f$ on $\mathbb{D}$ such that $\int_{\mathbb{D}}|f(z)|^p\Psi(|f(z)|)\,d\mu(z)<\infty$. The weighted Bergman-Zygmund space $A^p_{\omega,\Psi}$ induced by a weight function $\omega$ consists of analytic functions in $L^p_{\mu,\Psi}$ with $d\mu=\omega\,dA$. Let $0<q<p<\infty$ and let $\omega$ be radial weight on $\mathbb{D}$ which has certain two-sided doubling properties. In this study, we will characterize the measures $\mu$ such that the identity mapping $I: A^p_{\omega,\Psi} \to L^q_{\mu,\Phi}$ is bounded and compact, when we assume $\Psi,\Phi$ to be almost monotonic and to satisfy certain doubling properties. In addition, we apply our result to characterize the measures for which the differentiation operator $D^{(n)}: A^p_{\omega,\Psi} \to L^q_{\mu,\Phi}$ is bounded and compact.

math.CV↗

arXiv2024

Carleson measures for weighted Bergman--Zygmund spaces

For $0<p<\infty$, $Ψ:[0,\infty)\to(0,\infty)$ and a finite positive Borel measure $μ$ on the unit disc $\mathbb{D}$, the Lebesgue--Zygmund space $L^p_{μ,Ψ}$ consists of all measurable functions $f$ such that $\lVert f \rVert_{L_{μ, Ψ}^{p}}^p =\int_{\mathbb{D}}|f|^pΨ(|f|)\,dμ< \infty$. For an integrable radial function $ω$ on $\mathbb{D}$, the corresponding weighted Bergman-Zygmund space $A_{ω, Ψ}^{p}$ is the set of all analytic functions in $L_{μ, Ψ}^{p}$ with $dμ=ω\,dA$. The purpose of the paper is to characterize bounded (and compact) embeddings $A_{ω,Ψ}^{p}\subset L_{μ, Φ}^{q}$, when $0<p\le q<\infty$, the functions $Ψ$ and $Φ$ are essential monotonic, and $Ψ,Φ,ω$ satisfy certain doubling properties. The tools developed on the way to the main results are applied to characterize bounded and compact integral operators acting from $A^p_{ω,Ψ}$ to $A^q_{ν,Φ}$, provided $ν$ admits the same doubling property as $ω$.

math.CV↗