Capacity and the Corresponding Heat Semigroup Characterization from Dunkl-Bounded Variation

Xiang-Ling Meng ,Yu Liu,Xiang-Yun Xie

doi:10.3390/fractalfract5040280

Abstract

In this paper, we study some important basic properties of Dunkl-bounded variation functions. In particular, we derive a way of approximating Dunkl-bounded variation functions by smooth functions and establish a version of the Gauss–Green Theorem. We also establish the Dunkl BV capacity and investigate some measure theoretic properties, moreover, we show that the Dunkl BV capacity and the Hausdorff measure of codimension one have the same null sets. Finally, we develop the characterization of a heat semigroup of the Dunkl-bounded variation space, thereby giving its relation to the functions of Dunkl-bounded variation.

Highlights

IntroductionDunkl operator is found to play an increasingly important role in the study of many special functional problems with reflective symmetry
We develop the characterization of a heat semigroup of the Dunkl-bounded variation space, thereby giving its relation to the functions of Dunkl-bounded variation
Throughout this paper, Ω stands for an open subset of R N with N ≥ 2

Summary

Introduction

Dunkl operator is found to play an increasingly important role in the study of many special functional problems with reflective symmetry. Dunkl operator is a parameterized differential difference operator related to the finite reflection group, which operates in the Euclidean space In recent years, these operators and their generalizations have gained considerable interest in various fields of mathematics and physics.

Dunkl BV Space

Basic Facts of Dunkl BV Capacity

Heat Semigroups Characterization of Dunkl Bounded Variation Functions

Conclusions