Contractive Hardy--Littlewood inequalities in the Dirichlet range
Ole Fredrik Brevig, Aleksei Kulikov, Kristian Seip, Ilya Zlotnikov
TL;DR
This paper extends contractive Hardy–Littlewood inequalities to the Dirichlet range 0<α<1 by leveraging analytic continuation from the classical Bergman range and establishing isometric conformal invariance with index κ=α/p. The authors prove a sharp contractive inequality ⎯⎯⎯⎯⎯⎯ ∥f∥_{β,q} ≤ ∥f∥_{α,p} whenever 0<α<β<∞ and α/p=β/q, with equality attained by the extremal function f(z)=C(1- conj(w) z)^{-2α/p}, and they develop a robust framework using Kulikov’s approach, hyperbolic measure, and a Φ-functional representation to derive and extend these results. They also relate A^p_α to Besov spaces, detailing inclusions and strictness in the Dirichlet range, and study the shift operator and inner-function division, showing that for α<1 the shift is a strict expansion while division by inner factors reduces the norm and preserves the outer part in A^p_α. Overall, the work unifies and broadens previous results, providing a coherent picture of conformal invariance, majorant behavior, and operator-theoretic structure in the Dirichlet range with implications for Besov spaces and analytic function theory in the unit disk.
Abstract
The class $A_α^p$ 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 $0<α, p <\infty$. For $α>1$, $A_α^p$ is the standard weighted Bergman space, and $A_1^p=H^p$. We consider $A_α^p$ for $0<α<1$ and show that (weighted) isometric conformal invariance extends to this range, and we also clarify the relation between $A_α^p$ and the classical Besov spaces. Our main result is the contractive inequality $\|f\|_{β,q} \leq \|f\|_{α,p}$, valid when $0<α<β<\infty$ and $α/p=β/q$. We also identify the functions for which equality is attained. We thus extend recent results of the second-named author ($1\leq α<β$) and Llinares ($β=1$ and $p=2$). The extension of results from the classical range $1\leq α< \infty$ to the Dirichlet range $0<α<1$ uses arguments relying on analytic continuation.