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 \[\|f\|_{α,p}^p := |f(0)|^p+\int_0^1 \left(\frac{d}{dr} M_p^p(r,f)\right) (1-r^2)^{α-1} \,dr < \infty,\] 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 $\Vert f{\Vert }_{\beta ,q}\le \Vert f{\Vert }_{\alpha ,p}$, 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 ($1\le \alpha <\beta$) and Llinares ($\beta =1$ and $p=2$). The extension of results from the classical range $1\le \alpha <\infty$ to the Dirichlet range $0<\alpha <1$ uses arguments relying on analytic continuation.