Hardy's inequalities with non-doubling weights and sharp remainders

TL;DR

This paper establishes multidimensional Hardy's inequalities with complex non-doubling weights near the boundary of smooth domains, building on sharp one-dimensional inequalities with boundary conditions and remainders.

Contribution

It extends Hardy's inequalities to non-doubling weights in higher dimensions, including weights that vanish or blow up at the boundary, with sharp remainder terms.

Findings

Established n-dimensional Hardy's inequalities with non-doubling weights.

Derived sharp remainders for these inequalities.

Included weights with infinite order vanishing or blow-up.

Abstract

In the present paper we shall establish n-dimensional Hardy's inequalities with non-doubling weight functions of the distance to the boundary, where the boundary is a $C^2$ class bounded domain of $R^N$. This work is essentially based on one dimensional weighted Hardy's inequalities with one-sided boundary condition and sharp remainders. As weights we admit rather general ones that may vanish or blow up in infinite order such as $e^{-1/t}$ or $e^{1/t}$ at $t=0$ in one dimensional case.