Hardy-type and Heisenberg's inequality in Morrey spaces

TL;DR

This paper develops new inequalities involving the fractional Laplacian and Hardy-type bounds within Morrey spaces, expanding the mathematical understanding of these operators in such function spaces.

Contribution

It introduces novel interpolation and Hardy-type inequalities for the fractional Laplacian in Morrey spaces, leading to a Heisenberg-type inequality.

Findings

Established a Morrey norm estimate for the imaginary power of the Laplacian.

Proved a new interpolation inequality for fractional Laplacian in Morrey spaces.

Derived a Heisenberg-type inequality in Morrey spaces.

Abstract

We use the Morrey norm estimate for the imaginary power of the Laplacian to prove an interpolation inequality for the fractional power of the Laplacian on Morrey spaces. We then prove a Hardy-type inequality and use it together with the interpolation inequality to obtain a Heisenberg-type inequality in Morrey spaces.