Shifts of a Measurable Function and Criterion of p-integrability

TL;DR

The paper establishes a criterion linking shifts and multiplications of functions to their p-integrability, revealing a precise condition involving the product of shift and frequency parameters.

Contribution

It provides a new necessary and sufficient condition for p-integrability based on shifts and multiplications of functions, with a specific focus on the product of parameters.

Findings

The conditions involving shifts and multiplications imply p-integrability when the product ab is not in rac{ Z.

The criterion is both necessary and sufficient for p inite integrability.

The result connects harmonic analysis concepts with integrability conditions.

Abstract

It is shown that two conditions $f(a + \cdot) - f(\cdot) \in L^p(R)$, and $(\sin b \cdot) f(\cdot) \in L^p(R)$ guarantee $f \in L^p(R)$, $1 \leq p < \infty$, if and only if $ab$ is not in $(\pi Z)$.