An inequality associated with $\mathcal{Q}_p$ functions

TL;DR

This paper proves a key inequality for the $ ext{Q}_p$ function spaces when p>1, confirming a conjecture, but shows it does not hold for 0<p≤1, impacting the understanding of multipliers and Carleson measures.

Contribution

It establishes the validity of a conjectured inequality for $ ext{Q}_p$ spaces when p>1 and demonstrates its failure for 0<p≤1, clarifying the structure of these spaces.

Findings

Inequality holds for p>1.

Inequality does not hold for 0<p≤1.

Implications for multipliers and Carleson measures.

Abstract

The M\"obius invariant space $\mathcal{Q}_p$, $0<p<\infty$, consists of functions $f$ which are analytic in the open unit disk $\mathbb{D}$ with $$ \|f\|_{\mathcal{Q}_p}=|f(0)|+\sup_{w\in \D} \left(\int_\D |f'(z)|^2(1-|\sigma_w(z)|^2)^p dA(z)\right)^{1/2}<\infty, $$ where $\sigma_w(z)=(w-z)/(1-\overline{w}z)$ and $dA$ is the area measure on $\mathbb{D}$. It is known that the following inequality $$ |f(0)|+\sup_{w\in \D} \left(\int_\D \left|\frac{f(z)-f(w)}{1-\overline{w}z}\right|^2 (1-|\sigma_w(z)|^2)^p dA(z)\right)^{1/2} \lesssim \|f\|_{\mathcal{Q}_p} $$ played a key role to characterize multipliers and certain Carleson measures for $\mathcal{Q}_p$ spaces. The converse of the inequality above is a conjectured-inequality in [14]. In this paper, we show that this conjectured-inequality is true for $p>1$ and it does not hold for $0<p\leq 1$.