The rate of almost-everywhere convergence of Bochner–Riesz means on Sobolev spaces

Abstract

We investigate the convergence rate of the generalized Bochner–Riesz means $S_R^{\delta ,\gamma }$ on $L^p$-Sobolev spaces in the sharp range of $\delta$ and $p$ ($p\ge 2$). We give the relation between the smoothness imposed on functions and the rate of almost-everywhere convergence of $S_R^{\delta ,\gamma }$. As an application, the corresponding results can be extended to the $n$-torus ${\mathbb{T}}^n$ by using some transference theorems. Also, we consider the following two generalized Bochner–Riesz multipliers, $(1-|\xi {|}^{{\gamma }_1}{)}_{+}^{\delta }$ and $(1-|\xi {|}^{{\gamma }_2}{)}_{+}^{\delta }$, where ${\gamma }_1$, ${\gamma }_2$, $\delta$ are positive real numbers. We prove that, as the maximal operators of the multiplier operators with respect to the two functions, their $L^2\left(\right|x{|}^{-\beta })$-boundedness is equivalent for any ${\gamma }_1$, ${\gamma }_2$ and fixed $\delta$.