Tail estimates in Grand Lebesgue Spaces with applications to Riesz transforms

We study the tail behavior of measurable functions under operators satisfying suitable Lp bounds in the framework of Grand Lebesgue Spaces. By exploiting the connection between the growth of Lp 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 growth of the operator interacts with the intrinsic tail behavior of the input function.