Littlewood–Paley inequalities for fractional derivative on Bergman spaces

Abstract

For any pair \((n,p)\), \(n\in\mathbb{N}\) and \(0<p<\infty\), it has been recently proved by Peláez and Rättyä (2021) that a radial weight \(\omega\) on the unit disc of the complex plane \(\mathbb{D}\) satisfies the Littlewood-Paley equivalence

\(\int_{\mathbb{D}}|f(z)|^p\,\omega(z)\,dA(z)\asymp\int_\mathbb{D}|f^{(n)}(z)|^p(1-|z|)^{np}\omega(z)\,dA(z)+\sum_{j=0}^{n-1}|f^{(j)}(0)|^p,\)

for any analytic function \(f\) in \(\mathbb{D}\), if and only if \(\omega\in\mathcal{D}=\hat{\mathcal{D}} \cap \check{\mathcal{D}}\). A radial weight \(\omega\) belongs to the class \(\hat{\mathcal{D}}\) if \(\sup_{0\le r<1} \frac{\int_r^1 \omega(s)\,ds}{\int_{\frac{1+r}{2}}^1\omega(s)\,ds}<\infty\), and \(\omega \in \check{\mathcal{D}}\) if there exists \(k>1\) such that \(\inf_{0\le r<1} \frac{\int_{r}^1\omega(s)\,ds}{\int_{1-\frac{1-r}{k}}^1 \omega(s)\,ds}>1.\)   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 \(D^{\mu}(f)(z)=\sum_{n=0}^{\infty} \frac{\widehat{f}(n)}{\mu_{2n+1}} z^n\) induced by a radial weight \(\mu \in \mathcal{D}\) where \(\mu_{2n+1}=\int_0^1 r^{2n+1}\mu(r)\,dr\). Then, we prove that for any \(p\in (0,\infty)\), the Littlewood-Paley equivalence

\(\int_\mathbb{D} |f(z)|^p \omega(z)\,dA(z)\asymp \int_\mathbb{D}|D^{\mu}(f)(z)|^p\left[\int_{|z|}^1\mu(s)\,ds\right]^p\omega(z)\,dA(z)\)


holds for any analytic function \(f\) in \(\mathbb{D}\) if and only if \(\omega\in\mathcal{D}\). We also prove that for any \(p\in (0,\infty)\), the inequality


\(\int_\mathbb{D}|D^{\mu}(f)(z)|^p\left[\int_{|z|}^1\mu(s)\,ds\right]^p\omega(z)\,dA(z)\lesssim \int_\mathbb{D} |f(z)|^p \omega(z)\,dA(z)\)

holds for any analytic function \(f\) in \(\mathbb D\) if and only if \(\omega\in\hat D\).