# Uncertainty Principles for the Continuous Shearlet Transforms in Arbitrary Space Dimensions

## TL;DR

This paper develops new uncertainty principles for continuous shearlet transforms in arbitrary dimensions, including Pitt's, Beckner's, Nazarov's, and local uncertainty principles, expanding the theoretical understanding of shearlet analysis.

## Contribution

It introduces several novel uncertainty principles for shearlet transforms in any space dimension, linking classical inequalities with shearlet analysis.

## Key findings

- Derived Pitt's inequality analogue for shearlet transforms
- Formulated Beckner's uncertainty principle via two approaches
- Established Nazarov's uncertainty principle for shearlet transforms

## Abstract

The aim of this article is to formulate some novel uncertainty principles for the continuous shearlet transforms in arbitrary space dimensions. Firstly, we derive an analogue of the Pitt's inequality for the continuous shearlet transforms, then we formulate the Beckner's uncertainty principle via two approaches: one based on a sharp estimate from Pitt's inequality and the other from the classical Beckner's inequality in the Fourier domain. Secondly, we consider a logarithmic Sobolev inequality for the continuous shearlet transforms which has a dual relation with Beckner's inequality. Thirdly, we derive Nazarov's uncertainty principle for the shearlet transforms which shows that it is impossible for a non-trivial function and its shearlet transform to be both supported on sets of finite measure. Towards the culmination, we formulate local uncertainty principles for the continuous shearlet transforms in arbitrary space dimensions.