# Littlewood–Paley inequalities for fractional derivative on Bergman spaces

## Authors

• José Ángel Peláez Universidad de Málaga, Departamento de Análisis Matemático
• Elena de la Rosa Universidad de Málaga, Departamento de Análisis Matemático

## Keywords:

Bergman space, fractional derivative, radial weight, Littlewood-Paley formula

### 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 $$\omega\in\hat D$$.

Issue
Section
Articles

2022-09-17

## How to Cite

Peláez, J. Ángel, & de la Rosa, E. (2022). Littlewood–Paley inequalities for fractional derivative on Bergman spaces. Annales Fennici Mathematici, 47(2), 1109–1130. https://doi.org/10.54330/afm.121831