On some equivalent definitions of $\rho$- Carleson measures on the unit ball

TL;DR

This paper provides equivalent characterizations of $ ho$-Carleson measures with specific logarithmic weights and applies these results to characterize pointwise multipliers and boundedness of certain integral operators between function spaces on the sphere.

Contribution

It introduces new equivalent definitions of $ ho$-Carleson measures for specific logarithmic functions and applies these to analyze multipliers and bounded operators between $LMOA$ and $BMOA$ spaces.

Findings

Characterization of $ ho$-Carleson measures with logarithmic weights.

Identification of pointwise multipliers on $LMOA(S^n)$.

Boundedness results for Cesàro type integral operators.

Abstract

We give in this paper some equivalent definitions of the so called $\rho$-Carleson measures when $\rho(t)=(\log(4/t))^p(\log\log(e^4/t))^q$, $0\le p,q<\infty$. As applications, we characterize the pointwise multipliers on $LMOA(\mathbb S^n)$ and from this space to $BMOA(\mathbb S^n)$. Boundedness of the Ces\`aro type integral operators on $LMOA(\mathbb S^n)$ and from $LMOA(\mathbb S^n)$ to $BMOA(\mathbb S^n)$ is considered as well.