A Logarithmic Uncertainty Principle for Functions with Radial Symmetry

TL;DR

This paper establishes a new uncertainty principle specifically for radially symmetric functions by analyzing a radial Stein-Weiss inequality and addressing the challenge of differentiability at the optimal constant.

Contribution

It introduces a novel uncertainty principle for radial functions and overcomes the difficulty of proving differentiability of the best constant in this context.

Findings

Proved a new uncertainty principle for radial functions.

Analyzed the differentiability of the best constant in the Stein-Weiss inequality.

Extended the understanding of uncertainty principles in symmetric function spaces.

Abstract

In this note, we prove a new uncertainty principle for functions with radial symmetry by differentiating a radial version of the Stein-Weiss inequality. The difficulty is to prove the differentiability in the limit of the best constant that, unlike the general case, it is not known.