A strong convergence theorem for $H^1(\mathbb{T}^{n})$
Tom 173 / 2006
Studia Mathematica 173 (2006), 167-184
MSC: Primary 42B08, 42B30.
DOI: 10.4064/sm173-2-4
Streszczenie
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 $\alpha:={(n-1)}/2$, $$ \lim_{R\to \infty} \frac 1 {\log R} \int_0^R \frac { \|\widetilde {S}^\alpha_u(f) -f\|_{H^1(\mathbb{T}^{n} )}} { u+1}\, du =0. $$
