Hardy inequalities with Aharonov-Bohm type magnetic field on the Heisenberg group

Yingxiong Xiao

doi:10.1186/s13660-015-0617-4

Abstract

We introduce an Aharonov-Bohm type magnetic field on three-dimensional Heisenberg group and show this quadratic form satisfy an improved Hardy inequality with weights.

Highlights

The classical Hardy inequality states that, for N ≥ and for all u ∈ C ∞(RN ), |∇u| dx ≥ RN N– u RN |x| dx ( . ) and ( N – )is the best constant in ( . )
We introduce an Aharonov-Bohm type magnetic field on three-dimensional Heisenberg group and show this quadratic form satisfy an improved Hardy inequality with weights
A version of the Aharonov-Bohm magnetic field for a Grushin subelliptic operator has been introduced by Aermark and Laptev [ ]. Such quadratic form satisfies an improved Hardy inequality. They asked the following question: does there exist a similar result for the Heisenberg quadratic form?

Summary

Introduction

). for some magnetic forms in dimension two, the Hardy inequality becomes possible. A version of the Aharonov-Bohm magnetic field for a Grushin subelliptic operator has been introduced by Aermark and Laptev [ ]. Such quadratic form satisfies an improved Hardy inequality. They asked the following question: does there exist a similar result for the Heisenberg quadratic form?. Recall that the three-dimension Heisenberg group H = (R × R, ◦) is a step-two nilpotent group whose group structure is given by (x, y, t) ◦ x , y , t x + x , y + y , t + t – xy – yx

The vector fields

Given any ξ

If we represent u by the Fourier series