## 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

### Citations:

Zbl 0215.18303; Zbl 0522.42015; Zbl 1068.42011
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### References:

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