Junyan Zhao, Dashan Fan. Source: Annals of Functional Analysis, Volume 10, Number 1, 29--45.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 $p$ ( $p\geq2$ ). 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, $(1-\vert \xi \vert ^{\gamma_{1}})_{+}^{\delta}$ and $(1-\vert \xi \vert ^{\gamma_{2}})_{+}^{\delta}$ , 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 $L^{2}(|x|^{-\beta})$ -boundedness is equivalent for any $\gamma_{1}$ , $\gamma_{2}$ and fixed $\delta$ .