Difference functions of periodic \(L_p\) functions.

Abstract

We examine for which sets \(H\) of the circle group \(\mathbb{R} / \mathbb{Z}\) can the difference functions \(f(x+h)- f(x)\) of a measurable or \(L_{p}\) function \(f\) belong to an \(L_{q}\) class for every \(h \in H\) without \(f\) itself being in \(L_{q}\). Tamás Keleti conjectured in \cite{Elek1} that these sets are the \(N\)-sets; that is, the sets of absolute convergence of Fourier-series. We prove this conjecture for \(q \leq 2\). For \(q=2\), as a quantitative analogue of this statement, we prove a minimax theorem.