Periodic \(L_p\) Functions with \(L_q\) Difference Functions

Abstract

Let \(0\lt p\lt q\lt \infty\). We investigate the following question: For which subsets \(H\) of the circle group \(\mathbb{T}=\mathbb{R}/\mathbb{Z}\) is it true that if \(f\in L_p\) and \(\Delta_h f(x)=f(x+h)-f(x)\in L_q\) for any \(h\in H\), then \(f\in L_q\)? We prove that this is not true for pseudo-Dirichlet sets. Evidence is gathered for the conjecture that the class of counter-examples is precisely the class of \(N\)-sets.