A strong convergence theorem for $H^1(\mathbb{T}^{n})$

Feng Dai¹

¹ Department of Mathematical and Statistical Sciences CAB 632, University of Alberta Edmonton, Alberta, T6G 2G1, Canada

Studia Mathematica, Tome 173 (2006) no. 2, pp. 167-184

Résumé

Let $\mathbb{T}^{n} $ denote the usual $n$-torus and let $ \widetilde {S}_u^\delta (f)$, $u>0$, denote the Bochner–Riesz means of order $\delta >0$ of the Fourier expansion of $ f\in L^1(\mathbb{T}^{n} )$. The main result of this paper states that for $f\in H^1(\mathbb{T}^{n} )$ and the critical index $\alpha:={(n-1)}/2$, $$ \lim_{R\to \infty} \frac 1 {\log R} \int_0^R \frac { \|\widetilde {S}^\alpha_u(f) -f\|_{H^1(\mathbb{T}^{n} )}} { u+1}\, du =0. $$

Détail

DOI : 10.4064/sm173-2-4

Keywords: mathbb denote usual n torus widetilde delta denote bochner riesz means order delta fourier expansion mathbb main result paper states mathbb critical index alpha n lim infty frac log int frac widetilde alpha f mathbb