Almost everywhere convergence of the inverse Jacobi transform and endpoint results for a disc multiplier

Troels Roussau Johansen ¹

¹ Mathematisches Seminar Christian-Albrechts Universität zu Kiel Ludewig-Meyn-Strasse 4 D-24098 Kiel, Germany

Studia Mathematica, Tome 205 (2011) no. 2, pp. 101-137

Résumé

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 $\frac{4\alpha+4}{2\alpha+3} p\leq 2$.

Détail

DOI : 10.4064/sm205-2-1

Keywords: maximal operator * spherical summation operator disc multiplier associated jacobi transform through defining relation widehat lambda vert lambda vert leq widehat function mathbb shown bounded mathbb mathbb mathbb frac alpha alpha leq moreover * bounded mathbb infty mathbb mathbb particular converges almost everywhere towards mathbb whenever frac alpha alpha leq