Almost everywhere convergence of the inverse Jacobi transform and endpoint results for a disc multiplier
Volume 205 / 2011
Studia Mathematica 205 (2011), 101-137
MSC: Primary 43A50; Secondary 33C05, 34E05.
DOI: 10.4064/sm205-2-1
Abstract
The maximal operator $S_*$ for the spherical summation operator (or disc multiplier) $S_R$ associated with the Jacobi transform through the defining relation $\widehat{S_Rf}(\lambda) =1_{\{\vert\lambda\vert\leq R\}}\widehat{f}(t)$ for a function $f$ on $\mathbb R$ is shown to be bounded from $L^p(\mathbb R_+,d\mu)$ into $L^p(\mathbb R,d\mu)+L^2(\mathbb R,d\mu)$ for $\frac{4\alpha+4}{2\alpha+3}< p\leq 2$. Moreover $S_*$ is bounded from $L^{p_0,1}(\mathbb R_+,d\mu)$ into $L^{p_0,\infty}(\mathbb R,d\mu)+L^2(\mathbb R,d\mu)$. In particular $\{S_Rf(t)\}_{R>0}$ converges almost everywhere towards $f$, for $f\in L^p(\mathbb R_+,d\mu)$, whenever $\frac{4\alpha+4}{2\alpha+3}< p\leq 2$.
