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 that a radial weight $\ddot{I}‰$ on the unit disc of the complex plane $\mathbb{D}$ satisfies the Littlewood-Paley equivalence $${\int }_{\mathbb{D}}|f(z){|}^p\ddot{I}‰(z)dA(z)\asymp {\int }_{\mathbb{D}}|f^{(n)}(z){|}^p(1-|z|{)}^{np}\ddot{I}‰(z)dA(z)+\sum _{j=0}^{n-1}|f^{(j)}(0){|}^p,$$ for any analytic function $f$ in $\mathbb{D}$, if and only if $\ddot{I}‰\in \mathcal{D}=\hat{\mathcal{D}}\cap \check{\mathcal{D}}$. A radial weight $\ddot{I}‰$ belongs to the class $\hat{\mathcal{D}}$ if ${\sup }_{0\le r<1}\frac{{\int }_r^1\ddot{I}‰(s)ds}{{\int }_{\frac{1+r}{2}}^1\ddot{I}‰(s)ds}<\infty$, and $\ddot{I}‰\in \check{\mathcal{D}}$ if there exists $k>1$ such that ${\inf }_{0\le r<1}\frac{{\int }_r^1\ddot{I}‰(s)ds}{{\int }_{1-\frac{1-r}{k}}^1\ddot{I}‰(s)ds}>1$. In this paper we extend this result to the setting of fractional derivatives. Being precise, for an analytic function $f(z)={\sum }_{n=0}^{\infty }\hat{f}(n)z^n$ we consider the fractional derivative $D^{\hat{I}}¼(f)(z)=\sum _{n=0}^{\infty }\frac{\hat{f}(n)}{\hat{I}{¼}_{2n+1}}{z}^n$ induced by a radial weight $\hat{I}¼\in \mathcal{D}$, where $\hat{I}{¼}_{2n+1}={\int }_0^1{r}^{2n+1}\hat{I}¼(r)dr$. Then, we prove that for any $p\in (0,\infty )$, the Littlewood-Paley equivalence $${\int }_{\mathbb{D}}|f(z){|}^p\ddot{I}‰(z)dA(z)\asymp {\int }_{\mathbb{D}}|D^{\hat{I}}¼(f)(z){|}^p{\left[{\int }_{|z|}^1\hat{I}¼(s)ds\right]}^p\ddot{I}‰(z)dA(z)$$ holds for any analytic function $f$ in $\mathbb{D}$ if and only if $\ddot{I}‰\in \mathcal{D}$. We also prove that for any $p\in (0,\infty )$, the inequality $${\int }_{\mathbb{D}}|D^{\hat{I}}¼(f)(z){|}^p{\left[{\int }_{|z|}^1\hat{I}¼(s)ds\right]}^p\ddot{I}‰(z)dA(z)\lesssim {\int }_{\mathbb{D}}|f(z){|}^p\ddot{I}‰(z)dA(z)$$ holds for any analytic function $f$ in $\mathbb{D}$ if and only if $\ddot{I}‰\in \hat{\mathcal{D}}$.

Accepted manuscript (postprint)

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