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Improved bounds for Bochner-Riesz and maximal Bochner-Riesz operators. (English) Zbl 1072.42009

The Bochner-Riesz operator in \({\mathbb R}^n\), \(n\geq 2\), of order \(\alpha\) is a multiplier operator defined by \[ \widehat{S^\alpha f}(\xi)=(1-| \xi| ^2)^\alpha_+\hat f(\xi),\quad \xi\in{\mathbb R}^n. \] It is conjectured that when \(\alpha>0\), \(S^\alpha\) is bounded on \(L^p(\mathbb R)\) if and only if \[ \alpha>\alpha(p)=\max\left(n\left| \frac1p-\frac12\right| -\frac12,0\right). \] This conjecture was proven in \({\mathbb R}^2\) by L. Carleson and P. Sjölin [Stud. Math. 44, 287-299 (1972; Zbl 0215.18303)]. When \(n\geq3\), only partial results are known. In this paper, the author improves the known results by proving that: Let \(n\geq 3\). If \(p>(2n+4)/n\) or \(p<(2n+4)/(n+4)\), then \(\| S^\alpha f\| _{L^p(\mathbb R^n)}\leq C\| f\| _{L^p(\mathbb R^n)}\), provided \(\alpha>\alpha(p)\). The author also studies the \(L^p({\mathbb R}^n)\)-boundedness of the maximal Bochner-Riesz operator defined by \[ S^\alpha _\ast f(x)=\sup_{t>0}| S^\alpha_tf(x)| , \] where \(\widehat{S^\alpha_t f}(\xi)=(1-| \xi/t| ^2)^\alpha_+\hat f(\xi)\). When \(p\geq 2\), the natural problem is that \(S^\alpha_\ast\) is bounded on \(L^p({\mathbb R}^n)\) on the same range where \(S^\alpha\) is bounded. In \({\mathbb R}^2\), the problem is solved by A. Carbery [Duke Math. J. 50, 409-416 (1983; Zbl 0522.42015)]. When \(n\geq 3\), only partial results are known.
In this paper, the author improves the known results by proving that: Let \(n\geq 3\). If \(p>(2n+4)/n\), then \(\| S^\alpha_\ast f\| _{L^p(\mathbb R^n)}\leq C\| f\| _{L^p(\mathbb R^n)}\), provided \(\alpha>\alpha(p)\). In the proof of these results, the author does not use Kakeya-type estimates. Instead of this, he mainly uses the bilinear restriction estimates for elliptic surfaces due to T. Tao [Geom. Funct. Anal. 13, No.6, 1359-1384 (2003; Zbl 1068.42011)].
Reviewer: Yang Dachun (Kiel)

MSC:

42B15 Multipliers for harmonic analysis in several variables
42B25 Maximal functions, Littlewood-Paley theory
47G10 Integral operators
42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
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