## Real Analysis Exchange

### Periodic $L_p$ Functions with $L_q$ Difference Functions

Tamás Keleti

#### 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.

#### Article information

Source
Real Anal. Exchange, Volume 23, Number 2 (1999), 431-440.

Dates
First available in Project Euclid: 14 May 2012

Keleti, Tamás. Periodic $L_p$ Functions with $L_q$ Difference Functions. Real Anal. Exchange 23 (1999), no. 2, 431--440. https://projecteuclid.org/euclid.rae/1337001356