Wirtinger-type inequalities for some rearrangement invariant spaces

TL;DR

This paper extends classical Sobolev inequalities to a broad class of rearrangement invariant spaces, specifically moment spaces, providing a generalized framework for norm inequalities involving functions and their derivatives.

Contribution

It introduces generalized Sobolev-type inequalities for moment rearrangement invariant spaces, broadening the scope beyond classical Lebesgue spaces.

Findings

Generalized Sobolev inequalities established for moment rearrangement invariant spaces

Extension of classical inequalities to new functional space classes

Potential applications in analysis involving these broader spaces

Abstract

In this short paper we generalize the classical inequality between the norms in Lebesgue spaces of the functions and its derivatives, which in the multidimensional case are called Sobolev's inequalities, on the many popular classes pairs of rearrangement invariant (r.i.) spaces, namely, on the so-called moment rearrangement invariant spaces.