Annals of Functional Analysis

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

Junyan Zhao and Dashan Fan


We investigate the convergence rate of the generalized Bochner–Riesz means SRδ,γ on Lp-Sobolev spaces in the sharp range of δ and p (p2). We give the relation between the smoothness imposed on functions and the rate of almost-everywhere convergence of SRδ,γ. As an application, the corresponding results can be extended to the n-torus Tn by using some transference theorems. Also, we consider the following two generalized Bochner–Riesz multipliers, (1|ξ|γ1)+δ and (1|ξ|γ2)+δ, where γ1, γ2, δ are positive real numbers. We prove that, as the maximal operators of the multiplier operators with respect to the two functions, their L2(|x|β)-boundedness is equivalent for any γ1, γ2 and fixed δ.

Article information

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

Received: 3 November 2017
Accepted: 11 February 2018
First available in Project Euclid: 16 January 2019

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Zentralblatt MATH identifier

Primary: 42B15: Multipliers
Secondary: 41A35: Approximation by operators (in particular, by integral operators) 46E35: Sobolev spaces and other spaces of "smooth" functions, embedding theorems, trace theorems 47B38: Operators on function spaces (general)

Bochner–Riesz means Sobolev spaces almost-everywhere convergence saturation of approximation maximal functions Fourier series


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.

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