Littlewood–Paley inequalities for fractional derivative on Bergman spaces

Abstract

For any pair $(n,p)$, $n\in \mathbb{N}$ and $0<p<\mathrm{\infty }$, 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 $\underset{0\le r<1}{sup}\frac{{\int }_r^1\omega (s)ds}{{\int }_{\frac{1+r}{2}}^1\omega (s)ds}<\mathrm{\infty }$, and $\omega \in \check{\mathcal{D}}$ if there exists $k>1$ such that $\underset{0\le r<1}{inf}\frac{{\int }_r^1\omega (s)ds}{{\int }_{1-\frac{1-r}{k}}^1\omega (s)ds}>1.$   In this paper we extend this result to the setting of fractional derivatives. Being precise, for an analytic function $f(z)=\sum _{n=0}^{\mathrm{\infty }}\hat{f}(n)z^n$ we consider the fractional derivative $D^{\mu }(f)(z)=\sum _{n=0}^{\mathrm{\infty }}\frac{\hat{f}(n)}{{\mu }_{2n+1}}{z}^n$ 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 $p\in (0,\mathrm{\infty })$, the Littlewood-Paley equivalence

${\int }_{\mathbb{D}}|f(z){|}^p\omega (z)dA(z)\asymp {\int }_{\mathbb{D}}|D^{\mu }(f)(z){|}^p{[{\int }_{|z|}^1\mu (s)ds]}^p\omega (z)dA(z)$


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


${\int }_{\mathbb{D}}|D^{\mu }(f)(z){|}^p{[{\int }_{|z|}^1\mu (s)ds]}^p\omega (z)dA(z)\lesssim {\int }_{\mathbb{D}}|f(z){|}^p\omega (z)dA(z)$

holds for any analytic function $f$ in $\mathbb{D}$ if and only if $\omega \in \hat{D}$.