Fourier series, mean Lipschitz spaces, and bounded mean oscillation. (English) Zbl 0672.42003
Analysis at Urbana. Vol. 1: Analysis in function spaces, Proc. Spec. Year Mod. Anal./Ill. 1986-87, Lond. Math. Soc. Lect. Note Ser. 137, 81-110 (1989).
Summary: [For the entire collection see Zbl 0667.00018.]
Using simple and direct arguments, we: (i) prove, without recourse to duality, that the mean Lipschitz spaces \(\Lambda\) (p,1/p) are contained in BMO, and (ii) improve the Hardy-Littlewood \(\Lambda\) (p,1/p) Tauberian theorem. Along the way we connect the Hardy-Littlewood result with a recent Tauberian theorem for BMO functions due to Ramey and Ullrich, give an exposition of the relevant classical properties of mean Lipschitz spaces; and survey some known function theoretic applications of the spaces \(\Lambda\) (p,1/p).
Using simple and direct arguments, we: (i) prove, without recourse to duality, that the mean Lipschitz spaces \(\Lambda\) (p,1/p) are contained in BMO, and (ii) improve the Hardy-Littlewood \(\Lambda\) (p,1/p) Tauberian theorem. Along the way we connect the Hardy-Littlewood result with a recent Tauberian theorem for BMO functions due to Ramey and Ullrich, give an exposition of the relevant classical properties of mean Lipschitz spaces; and survey some known function theoretic applications of the spaces \(\Lambda\) (p,1/p).
MSC:
| 42A20 | Convergence and absolute convergence of Fourier and trigonometric series |
| 46E39 | Sobolev (and similar kinds of) spaces of functions of discrete variables |
