Several Integral Estimates and Some Applications

TL;DR

This paper develops integral estimates and applies them to characterize pointwise multipliers between certain function spaces on the unit ball in complex space, providing necessary and sufficient conditions for these multipliers.

Contribution

It introduces new bidirectional integral estimates and applies them to fully characterize pointwise multipliers between specific normal weight function spaces and Bloch type spaces.

Findings

Derived bidirectional integral estimates.

Characterized pointwise multipliers for general normal functions.

Provided explicit conditions for special normal functions.

Abstract

In this paper, the authors first consider the bidirectional estimates of several typical integrals. As some applications of these integral estimates, the authors investigate the pointwise multipliers from the normal weight general function space $F(p,\mu,s)$ to the normal weight Bloch type space $\mathcal{B_{\nu}}(B_{n})$ on the unit ball $B_{n}$ of $\mathbb{C}^{n}$, where $\mu$ and $\nu$ are two normal functions on $[0,1)$. For the special normal function $\displaystyle{\mu(r)=(1-r^{2})^{\alpha}\log^{\beta}\frac{e}{1-r^{2}}}$ ($\alpha>0$, $-\infty<\beta<\infty$), the authors give the necessary and sufficient conditions of pointwise multipliers from $F(p,\mu,s)$ to $\mathcal{B_{\nu}}(B_{n})$ for all cases.