Contractive Hardy--Littlewood inequalities in the Dirichlet range

Abstract

The class $A_\alpha^p$ consists of those analytic functions $f$ in the unit disc such that \[\|f\|_{\alpha,p}^p := |f(0)|^p+\int_0^1 \left(\frac{d}{dr} M_p^p(r,f)\right) (1-r^2)^{\alpha-1} \,dr < \infty,\] where $M_p^p(r,f)$ is the radial integral mean of $|f|^p$ and $0<\alpha, p <\infty$. For $\alpha>1$, $A_\alpha^p$ is the standard weighted Bergman space, and $A_1^p=H^p$. We consider $A_\alpha^p$ for $0<\alpha<1$ and show that (weighted) isometric conformal invariance extends to this range, and we also clarify the relation between $A_{\alpha}^p$ and the classical Besov spaces. Our main result is the contractive inequality $\|f\|_{\beta,q} \leq \|f\|_{\alpha,p}$, valid when $0<\alpha<\beta<\infty$ and $\alpha/p=\beta/q$. We also identify the functions for which equality is attained. We thus extend recent results of the second-named author ($1\leq \alpha<\beta$) and Llinares ($\beta=1$ and $p=2$). The extension of results from the classical range $1\leq \alpha < \infty$ to the Dirichlet range $0<\alpha <1$ uses arguments relying on analytic continuation.