Littlewood-Paley inequalities for fractional derivative on Bergman spaces

TL;DR

The work addresses Littlewood-Paley type inequalities for fractional derivatives on Bergman spaces with radial weights. It introduces the fractional derivative $D^{\mu}$ induced by a weight $\mu \in \mathcal{D}$ and establishes two main results: (i) a full LP equivalence $\int_{\mathbb{D}} |f|^p \omega \,dA \asymp \int_{\mathbb{D}} |D^{\mu}f|^p [\int_{|z|}^1 \mu(s)\,ds]^p \omega \,dA$ holds for all analytic $f$ if and only if $\omega \in \mathcal{D}$, and (ii) a one-sided bound $\int_{\mathbb{D}} |D^{\mu}f|^p [\int_{|z|}^1 \mu(s)\,ds]^p \omega \,dA \lesssim \int_{\mathbb{D}} |f|^p \omega \,dA$ holds if and only if $\omega \in \widehat{\mathcal{D}}$. The authors develop a universal Cesàro basis to decompose and compare norms, characterize the weight classes involved, and connect the sufficiency/necessity parts through recent results on radial weight classes. This extends classical LP theory to fractional derivatives on weighted Bergman spaces and clarifies the weight conditions required for these inequalities.}

Abstract

For any pair $(n,p)$, $n\in\mathbb{N}$ and $0<p<\infty$, it has been recently proved that a radial weight $ω$ on the unit disc of the complex plane $\mathbb{D}$ satisfies the Littlewood-Paley equivalence $$ \int_{\mathbb{D}}|f(z)|^p\,ω(z)\,dA(z)\asymp\int_\mathbb{D}|f^{(n)}(z)|^p(1-|z|)^{np}ω(z)\,dA(z)+\sum_{j=0}^{n-1}|f^{(j)}(0)|^p,$$ for any analytic function $f$ in $\mathbb{D}$, if and only if $ω\in\mathcal{D}=\widehat{\mathcal{D}} \cap \check{\mathcal{D}}$. A radial weight $ω$ belongs to the class $\widehat{\mathcal{D}}$ if $\sup_{0\le r<1} \frac{\int_r^1 ω(s)\,ds}{\int_{\frac{1+r}{2}}^1ω(s)\,ds}<\infty$, and $ω\in \check{\mathcal{D}}$ if there exists $k>1$ such that $\inf_{0\le r<1} \frac{\int_{r}^1ω(s)\,ds}{\int_{1-\frac{1-r}{k}}^1 ω(s)\,ds}>1$. In this paper we extend this result to the setting of fractional derivatives. Being precise, for an analytic function $f(z)=\sum_{n=0}^\infty \widehat{f}(n) z^n$ we consider the fractional derivative $ D^μ(f)(z)=\sum\limits_{n=0}^{\infty} \frac{\widehat{f}(n)}{μ_{2n+1}} z^n $ induced by a radial weight $μ\in \mathcal{D}$, where $μ_{2n+1}=\int_0^1 r^{2n+1}μ(r)\,dr$. Then, we prove that for any $p\in (0,\infty)$, the Littlewood-Paley equivalence $$\int_{\mathbb{D}} |f(z)|^p ω(z)\,dA(z)\asymp \int_{\mathbb{D}}|D^μ(f)(z)|^p\left[\int_{|z|}^1μ(s)\,ds\right]^pω(z)\,dA(z)$$ holds for any analytic function $f$ in $\mathbb{D}$ if and only if $ω\in\mathcal{D}$. We also prove that for any $p\in (0,\infty)$, the inequality $$\int_{\mathbb{D}} |D^μ(f)(z)|^p \left[\int_{|z|}^1μ(s)\,ds\right]^pω(z)\,dA(z) \lesssim \int_{\mathbb{D}} |f(z)|^p ω(z)\,dA(z) $$ holds for any analytic function $f$ in $\mathbb{D}$ if and only if $ω\in \widehat{\mathcal{D}}$.

Theorems & Definitions (25)

• Theorem 1
• Theorem 2
• Lemma A
• proof
• Lemma B
• proof
• Lemma C
• Lemma 3
• proof
• Theorem D