A generalized three lines lemma in Hardy-like spaces

TL;DR

This paper extends the classical three lines lemma to Hardy-like spaces, providing explicit formulas for optimizers and optimal values for holomorphic functions with prescribed norms along parallel lines, with applications to interpolation and inequalities.

Contribution

It introduces a generalized three lines lemma in Hardy-like spaces, offering explicit solutions for extremal problems involving prescribed norms.

Findings

Explicit formulas for optimizers in Hardy-like spaces.

Complete characterization of maximum function values between lines.

Applications to interpolation theory and Lieb-Thirring inequalities.

Abstract

In this paper we address the following question: given a holomorphic function with prescribed $L^p(\mathbb{R})$ and $L^q(\mathbb{R})$ norm (with $1\leq p,q \leq \infty$) along two parallel lines in the complex plane, then what is the maximum value that this function can achieve at a given point between these lines. Here we show that this problem is well-posed in suitable Hardy-like spaces on the strip. Moreover, in this setting we completely solve this problem by providing not only an explicit formula for the optimizers but also for the optimal values. In addition, we briefly discuss some applications of these results to interpolation theory and to Lieb-Thirring inequalities.