Exact constant in Sobolev's and Sobolev's trace inequalities for Grand Lebesgue Spaces

TL;DR

This paper extends classical Sobolev and trace inequalities to Grand Lebesgue Spaces, providing exact constants and broadening their applicability beyond traditional Lebesgue spaces.

Contribution

It introduces the generalization of Sobolev inequalities to Grand Lebesgue Spaces and computes their exact constants, enhancing the theoretical framework.

Findings

Derived exact constants for Sobolev inequalities in Grand Lebesgue Spaces

Extended classical inequalities to a broader functional setting

Provided a unified approach for Sobolev and trace inequalities

Abstract

In this article we generalize the classical Sobolev's and Sobolev's trace inequalities on the Grand Lebesgue Spaces instead the classical Lebesgue Spaces. We will distinguish the classical Sobolev's inequality and the so-called trace Sobolev's inequality. We consider for simplicity only the case of whole space.