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A class of maximal operators of Bochner-Riesz means of multiple Fourier series in the fractional integral space. (Chinese. English summary) Zbl 0774.42007

Let \(S^ \alpha_ R(f,x)\) denote the Bochner-Riesz means of the Fourier series of \(f(x)\). Let \(C^ p_ \lambda\) be the Riesz potential spaces corresponding to Bessel potential spaces in the nonperiodic case. The purpose of this paper is to set up an approximation result: If \(\alpha>((n-1)/2+\lambda)| 2/p-1|\), \(f\in C^ p_ \lambda\), \(1<p<\infty\), and \(0\leq \lambda\leq 2\), then \(\| S^ \alpha_ R f- f\|_ p=O(R^{-\lambda})\). However, the above result is not sharp. In fact, the lower bound of \(\alpha\) can be reduced.

MSC:

42B05 Fourier series and coefficients in several variables
42B25 Maximal functions, Littlewood-Paley theory
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
26A33 Fractional derivatives and integrals
47B38 Linear operators on function spaces (general)
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