## Real Analysis Exchange

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

### 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.

#### Article information

**Source**

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

**Dates**

First available in Project Euclid: 14 May 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.rae/1337001356

**Mathematical Reviews number (MathSciNet)**

MR1640019

**Zentralblatt MATH identifier**

0943.28002

**Subjects**

Primary: 28A99: None of the above, but in this section

Secondary: 39A70: Difference operators [See also 47B39] 42A28 43A15: $L^p$-spaces and other function spaces on groups, semigroups, etc. 43A46: Special sets (thin sets, Kronecker sets, Helson sets, Ditkin sets, Sidon sets, etc.)

**Keywords**

{\(L_p\) function} {measurable function} {difference function} {circle group} {pseudo-Dirichlet set} {\(N\)-set}

#### Citation

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