A+ CATEGORY SCIENTIFIC UNIT

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

Volume 173 / 2006

Feng Dai Studia Mathematica 173 (2006), 167-184 MSC: Primary 42B08, 42B30. DOI: 10.4064/sm173-2-4

Abstract

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 $\alpha:={(n-1)}/2$, $$ \lim_{R\to \infty} \frac 1 {\log R} \int_0^R \frac { \|\widetilde {S}^\alpha_u(f) -f\|_{H^1(\mathbb{T}^{n} )}} { u+1}\, du =0. $$

Authors

  • Feng DaiDepartment of Mathematical and Statistical Sciences
    CAB 632, University of Alberta
    Edmonton, Alberta, T6G 2G1, Canada
    e-mail

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