# Criteria for Membership of the Mean Lipschitz Spaces

### D. Walsh

National University, Maynooth, Ireland

## Abstract

Our aim is to characterize the elements in certain function spaces by means of the Ces\'aro means and/or partial sums of their Fourier series. Firstly, we seek to extend known results for the Besov spaces $B^s_{pq}\ \ (1 \le p,q \less \infty)$ to the case where $q = \infty$. Secondly, we consider the Mean Lipschitz spaces $\Lambda(p,s)$. We confine attention to the values $1 \le p \less \infty$ and $0 \less s \le 1$ for the parameters. For $s \less 1$, the spaces $B^s _{p\infty}$ and $\Lambda(p,s)$ coincide. For the case $p = 1$ certain counter-examples are provided; some positive results are also given. We then treat the case $s = 1$ and consider the spaces $B^1_{p\infty}$ and $\Lambda(p,1)$ separately. Analogues of some known results for the spaces $\Lambda_s$ are given.

## Cite this article

D. Walsh, Criteria for Membership of the Mean Lipschitz Spaces. Z. Anal. Anwend. 22 (2003), no. 2, pp. 339–355

DOI 10.4171/ZAA/1149