Fourier series, mean Lipschitz spaces
and bounded mean oscillation


Paul S. Bourdon, Joel H. Shapiro and William T. Sledd


Analysis at Urbana, Vol. 1 (1986) LMS Lecture Note Series #137, 81--110, 1989
 Abstract: Hardy and Littlewood showed in 1928 that if f is in the mean Lipschitz space L(p, 1/p) and the Fourier series for f is Cesaro summable at a point, then the series converges at that point. The principal result in the present paper is a proof that the same conclusion holds under the weaker hypothesis of Abel summability. A simple direct proof is given of the fact that L(p, 1/p) lies in BMO, (first proved for p = 2, by Cima and Petersen). The paper concludes with a survey of a few places in which these spaces arise naturally.


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