JournalszaaVol. 22, No. 2pp. 339–355

Criteria for Membership of the Mean Lipschitz Spaces

  • D. Walsh

    National University, Maynooth, Ireland
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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 Bpqs  (1p,q\less)B^s_{pq}\ \ (1 \le p,q \less \infty) to the case where q=q = \infty. Secondly, we consider the Mean Lipschitz spaces Λ(p,s)\Lambda(p,s). We confine attention to the values 1p\less1 \le p \less \infty and 0\lesss10 \less s \le 1 for the parameters. For s\less1s \less 1, the spaces BpsB^s _{p\infty} and Λ(p,s)\Lambda(p,s) coincide. For the case p=1p = 1 certain counter-examples are provided; some positive results are also given. We then treat the case s=1s = 1 and consider the spaces Bp1B^1_{p\infty} and Λ(p,1)\Lambda(p,1) separately. Analogues of some known results for the spaces Λs\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