Convergence of Fourier series at or beyond endpoint

Abstract: We consider several problems at or beyond endpoint in harmonic analysis. The solutions of these problems are related to the estimates of some classes of sublinear operators. To do this, we introduce some new functions spaces $RL^{p,s}_{|x|^{\alpha}}({\bf R}^n)$ and $\dot{R}L^{p,s}_{|x|^{\alpha}}({\bf R}^n)$, which play an analogue role with the classical Hardy spaces $H^p({\bf R}^n)$. These spaces are subspaces of $L^p_{|x|^{\alpha}}({\bf R}^n)$ with $1<s<\infty, 0<p\leq s$ and $-n<\alpha<n(p-1)$, and $\dot{R}L^{p,s}_{|x|^{\alpha}}({\bf R}^n) \supset L^s({\bf R}^n)$ when $ -n<\alpha<n(p/s-1)$. We prove the following results. First, $\mu_\alpha$-a.e. convergence and ${L}^{p}_{|x|^{\alpha}}({\bf R})$ -norm convergence of Fourier series hold for all functions in $ RL^{p,s}_{|x|^{\alpha}}({\bf R})$ and $ \dot{R}L^{p,s}_{|x|^{\alpha}}({\bf R})$ with $1<s<\infty, 0<p\leq s$ and $-1<\alpha<p-1$, where $\mu_\alpha(x)=|x|^{\alpha}$; Second, many sublinear operators initially defined for the functions in $L^p({\bf R}^n)$ with $1<p<\infty$, such as Calder\'{o}n-Zygmund operators, C.Fefferman's singular multiplier operator, R.Fefferman's singular integral operator, the Bochner-Riesz means at the critical index, certain oscillatory singular integral operators, and so on, admit extensions which map $RL^{p,s}_{|x|^{\alpha}}({\bf R}^n)$ and $\dot{R}L^{p,s}_{|x|^{\alpha}}({\bf R}^n)$ into $L^p_{|x|^{\alpha}}({\bf R}^n)$ with $1<s<\infty, 0<p\leq s$ and $-n<\alpha<n(p-1)$; Final, Hardy-Littlewood maximal operator is bounded from $RL^{p,s}_{|x|^{\alpha}}({\bf R}^n)$ (or $\dot{R}L^{p,s}_{|x|^{\alpha}}({\bf R}^n)$) to ${L}^{p}_{|x|^{\alpha}}({\bf R}^n)$ for $ 1<s<\infty$ and $0<p\leq s$ if and only if $-n<\alpha<n(p-1)$.