A sharp Wirtinger inequality and some related functional spaces

TL;DR

This paper establishes the best constant and extremals for a generalized Wirtinger inequality involving weighted integrals and periodic functions, and characterizes the associated functional space.

Contribution

It provides the explicit best constant, all extremals, and characterizes the functional space for a generalized weighted Wirtinger inequality.

Findings

Explicit best constant for the inequality.

Complete characterization of extremal functions.

Description of the functional space where the inequality holds.

Abstract

We consider the generalized Wirtinger inequality \[ (\int_{0}^{T} a |u|^q )^{1/q} \le C \biggm(\int_{0}^{T} a^{1-p} |u'|^{p}\biggm)^{1/p}, \] with $p,q>1$, $T>0$, $a\in L^1[0,T]$, $a\ge0$, $a\not\equiv0$ and where $u$ is a $T$-periodic function satisfying the constraint \[ \int_{0}^{T} a |u|^{q-2}u =0. \] We provide the best constant $C>0$ as well as all extremals. Furthermore, we characterize the natural functional space where the inequality is defined.