## Annals of Functional Analysis

### 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 $p$ ($p\geq2$). 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 $\delta$.

#### Article information

Source
Ann. Funct. Anal., Volume 10, Number 1 (2019), 29-45.

Dates
Accepted: 11 February 2018
First available in Project Euclid: 16 January 2019

https://projecteuclid.org/euclid.afa/1547629221

Digital Object Identifier
doi:10.1215/20088752-2018-0006

Mathematical Reviews number (MathSciNet)
MR3899954

Zentralblatt MATH identifier
07045483

#### Citation

Zhao, Junyan; Fan, Dashan. The rate of almost-everywhere convergence of Bochner–Riesz means on Sobolev spaces. Ann. Funct. Anal. 10 (2019), no. 1, 29--45. doi:10.1215/20088752-2018-0006. https://projecteuclid.org/euclid.afa/1547629221

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