Criteria for Membership of the Mean Lipschitz Spaces

Abstract

Our aim is to characterize the elements in certain function spaces by means of the Cesáro means and/or partial sums of their Fourier series. Firstly, we seek to extend known results for the Besov spaces $B_{pq}^s$ $(1\le p,q<\infty )$ to the case where $q=\infty$. Secondly, we consider the Mean Lipschitz spaces $\Lambda (p,s)$. We confine attention to the values $1\le p<\infty$ and $0<s\le 1$ for the parameters. For $s<1$, the spaces $B_{p\infty }^s$ 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 $B_{p\infty }^1$ and $\Lambda (p,1)$ separately. Analogues of some known results for the spaces ${\Lambda }_s$ are given.