Tail estimates in Grand Lebesgue Spaces with applications to Riesz transforms

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