Geometric Hardy's inequalities with general distance functions

Abstract

We establish in this paper general geometric Hardy's identities and inequalities on domains in RN in the spirit of their celebrated works by Brezis-Vázquez and Brezis-Marcus. Hardy's identities are powerful tools in establishing more precise and significantly stronger inequalities than those Hardy's inequalities in the literature. More precisely, we use the notion of Bessel pairs introduced by Ghoussoub and Moradifam to establish several Hardy identities and inequalities with the general distance functions. Our distance functions can be understood as the distance to the surfaces of codimension α∈R, and include the distance to the origin (α=N), the distance to the boundary (α=1), and even the distance to surfaces of codimension k∈N with 1≤k≤N, as special cases. Our Hardy's identities for general Bessel pairs and their special cases improve the Hardy inequalities with general distance functions d(x) by Barbatis, Filippas and Tertikas. We also establish the best constant and extremal functions for a new type of the Hardy-Sobolev-Maz'ya inequality on the domain Σ={xi>0,i=1,...,N} with the distance functiond1(x)=d(x,∂Σ)=min⁡{x1,...,xN}. Our Hardy's identities on the domain {0<d1(x)<R} with the distance function d1(x)=min⁡{x1,...,xN} are in the spirit of Brezis-Vázquez and Brezis-Marcus. More applications of our main theorems on Hardy's identities and inequalities with general distance functions are given and these results improve and sharpen many Hardy's inequalities in the literature. Interesting examples of Bessel pairs and distance functions are given as well.