A Generalization of a Logarithmic Sobolev Inequality to the Hölder Class

H Ibrahim

doi:10.1155/2012/148706

Abstract

In a recent work of the author, a parabolic extension of the elliptic Ogawa type inequality has been established. This inequality is originated from the Brézis-Gallouët-Wainger logarithmic type inequalities revealing Sobolev embeddings in the critical case. In this paper, we improve the parabolic version of Ogawa inequality by allowing it to cover not only the class of functions from Sobolev spaces, but also the wider class of Hölder continuous functions.

Highlights

Introduction and Main ResultsIn 1, a generalization of the Ogawa type inequality 2 to the parabolic framework has been shown
Ogawa inequality can be considered as a generalized version in the LizorkinTriebel spaces of the remarkable estimate of Brezis-Gallouet-Wainger 3, 4 that holds in a limiting case of the Sobolev embedding theorem
1.1 where W22m,m is the parabolic Sobolev space we refer to 5 for the definition and further properties, and BMO is the parabolic bounded mean oscillation space defined via parabolic Journal of Function Spaces and Applications balls instead of Euclidean ones 1, Definition 2.1

Summary

Introduction and Main Results

In 1 , a generalization of the Ogawa type inequality 2 to the parabolic framework has been shown. Ogawa inequality can be considered as a generalized version in the LizorkinTriebel spaces of the remarkable estimate of Brezis-Gallouet-Wainger 3, 4 that holds in a limiting case of the Sobolev embedding theorem. The inequality showed in 1, Theorem 1.1 provides an estimate of the L∞ norm of a function in terms of its parabolic BMO norm, with the aid of the square root of the logarithmic dependency of a higher order Sobolev norm. Inequalities 1.1 and 1.3 can be applied to a wider class of Holder continuous functions f ∇g ∈ Cγ,γ/2, 0 < γ < 1 vector-valued case , or f ∈ Cγ,γ/2 scalar-valued case. The second theorem deals with functions defined on the bounded domain ΩT. Theorem 1.2 Logarithmic Holder inequality on a bounded domain.

Definitions

Proofs of Theorems