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

TL;DR

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

Contribution

It introduces the exact constants for Sobolev and trace inequalities within Grand Lebesgue Spaces with monomial weights, generalizing classical results.

Findings

Derived exact constants for inequalities

Extended inequalities to weighted Grand Lebesgue Spaces

Unified classical and trace Sobolev inequalities

Abstract

We generalize in this article the classical Sobolev's and Sobolev's trace inequalities on the Grand Lebesgue Spaces under monomial weight instead the classical Lebesgue or grand Lebesgue Spaces. We will distinguish the classical Sobolev's inequality and the so-called trace Sobolev's inequality.