Atomic decomposition and Carleson measures for weighted mixed norm spaces

TL;DR

This paper develops an atomic decomposition for weighted mixed norm spaces in the unit disc, enabling characterization of Carleson measures and bounded differentiation operators across a broad parameter range.

Contribution

It introduces an atomic decomposition for $A^{p,q}_bla$ spaces with radial weights, and applies it to characterize Carleson measures and bounded differentiation operators.

Findings

Atomic decomposition for $A^{p,q}_bla$ spaces established.

Characterization of Carleson measures for these spaces.

Boundedness of differentiation operators from $A^{p,q}_bla$ to $L^p_mbda$.

Abstract

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