Estimates for tail functions under Riesz transforms in Grand Lebesgue Spaces

TL;DR

This paper derives explicit tail estimates for functions under Riesz-type operators within Grand Lebesgue Spaces, linking $L^p$ norm growth to tail behaviour.

Contribution

It introduces a method to obtain tail estimates from $L^p$ bounds using Young--Fenchel transform in the context of Riesz transforms and Grand Lebesgue Spaces.

Findings

Derived explicit tail estimates from $L^p$ bounds.

Applied results to classical Riesz transforms.

Illustrated the interaction between $L^p$ growth and tail behaviour.

Abstract

We study the tail behaviour of measurable functions under generalized Riesz-type operators in the framework of Grand Lebesgue Spaces. By exploiting the connection between the growth of $L^p$ norms and the Young--Fenchel transform, we derive explicit tail estimates from suitable $L^p$ bounds. We also present model examples and apply the abstract result to the classical Riesz transforms, showing how the $L^p$ growth of the operator interacts with the intrinsic tail behaviour of the input function.