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A generalized three lines lemma in Hardy-like spaces

Thiago Carvalho Corso

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.

A generalized three lines lemma in Hardy-like spaces

Abstract

In this paper we address the following question: given a holomorphic function with prescribed and norm (with ) 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.
Paper Structure (13 sections, 25 theorems, 173 equations)

This paper contains 13 sections, 25 theorems, 173 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Applications
  4. Outline of the paper
  5. Hardy-like spaces on the strip
  6. Duality
  7. A dual formulation over meromorphic functions
  8. Duality relation
  9. The Euler-Lagrange equations
  10. Measures with one-sided exponentially decaying Fourier transform
  11. The Poisson kernel on the strip
  12. Analytic formula for optimal values
  13. Weigthed three-lines lemma

Key Result

Theorem 1.2

Let $\alpha \in(0,1)$ and $1\leq p,q \leq \infty$, then the variational problem defined in eq:pq3lineproblem admits an unique optimizer $h \in \mathbb{H}^{p,q}(S)$ up to the transformation $h(z) \mapsto h(z-\tau) \mathrm{e}^{\imath \lambda z + \beta}$ with $\tau, \lambda \in \mathbb R$ and $\beta \i where $1\leq p^\ast, q^\ast \leq \infty$ denote the Hölder conjugate exponents of $p$ and $q$, i.e.

Theorems & Definitions (57)

  • Definition 1.1: $\mathbb{H}^{p,q}(S)$ spaces
  • Theorem 1.2: Duality relation
  • Theorem 1.3: Optimal value
  • Remark 1.4: Analytic formula with dilogarithm function
  • Theorem 1.5: Optimizers
  • Remark
  • Theorem 1.6: Weighted three-lines inequality
  • Theorem 1.7: Generalized Stein interpolation theorem
  • Remark : Controlled growth assumption
  • Remark : Further extensions
  • ...and 47 more