arXiv subjects

Kian Sierra

Publications and source records attributed to Kian Sierra.

3 recordsLinked to original sources

arXiv2017

Atomic decomposition and Carleson measures for weighted mixed norm spaces

The purpose of this paper is to establish an atomic decomposition for functions in the weighted mixed norm space $A^{p,q}_ω$ induced by a radial weight $ω$ in the unit disc admitting a two-sided doubling condition. The obtained decomposition is further applied to characterize Carleson measures for $A^{p,q}_ω$, and bounded differentiation operators $D^{(n)}(f)=f^{(n)}$ acting from $A^{p,q}_ω$ to $L^p_μ$, induced by a positive Borel measure $μ$, on the full range of parameters $0<p,q,s<\infty$.

math.CV↗

arXiv2016

Berezin transform and Toeplitz operators on weighted Bergman spaces induced by regular weights

Given a regular weight $ω$ and a positive Borel measure $μ$ on the unit disc $\mathbb{D}$, the Toeplitz operator associated with $μ$ is $$ \mathcal{T}_μ(f)(z)=\int_{\mathbb{D}} f(ζ)\bar{B_z^ω(ζ)}\,dμ(ζ), $$ where $B^ω_{z}$ are the reproducing kernels of the weighted Bergman space $A^2_ω$. We describe bounded and compact Toeplitz operators $\mathcal{T}_μ:A^p_ω\to A^q_ω$, $1<q,p<\infty$, in terms of Carleson measures and the Berezin transform $$ \widetilde{\mathcal{T}_μ}(z)=\frac{\langle\mathcal{T}_μ(B^ω_{z}), B^ω_{z} \rangle_{A^2_ω}}{\|B_z^ω\|^2_{A^2_ω}}. $$ We also characterize Schatten class Toeplitz operators in terms of the Berezin transform and apply this result to study Schatten class composition operators.

math.FA↗

arXiv2015

Embedding Bergman spaces into tent spaces

Let $A^p_ω$ denote the Bergman space in the unit disc $\mathbb{D}$ of the complex plane induced by a radial weight $ω$ with the doubling property $\int_{r}^1ω(s)\,ds\le C\int_{\frac{1+r}{2}}^1ω(s)\,ds$. The tent space $T^q_s(ν,ω)$ consists of functions such that \begin{equation*} \begin{split} \|f\|_{T^q_s(ν,ω)}^q =\int_{\mathbb{D}}\left(\int_{Γ(ζ)}|f(z)|^s\,dν(z)\right)^\frac{q}sω(ζ)\,dA(ζ) <\infty,\quad 0 0$, by considering a generalized area operator. The results are provided in terms of Carleson measures for $A^p_ω$.

math.CV↗