Tail estimates in Grand Lebesgue Spaces with applications to Riesz transforms

Abstract

<jats:p> We study the tail behavior of measurable functions under operators satisfying suitable <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$L^p$$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:math> </jats:alternatives> </jats:inline-formula> bounds in the framework of Grand Lebesgue Spaces. By exploiting the connection between the growth of <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$L^p$$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:math> </jats:alternatives> </jats:inline-formula> 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 <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$L^p$$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:math> </jats:alternatives> </jats:inline-formula> growth of the operator interacts with the intrinsic tail behavior of the input function. </jats:p>