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On high dimensional maximal functions associated to convex bodies. (English) Zbl 0621.42015

The author’s summary: ”For a centrally symmetric convex body B in \({\mathbb{R}}^ n\), define the corresponding maximal function \[ Mf=M_ Bf=\sup_{r>0}(\int_{B_ r}| f(x+y)| dy/Vol_ n Br) \] where \(B_ r=r\cdot B\). It is proved that there is a numerical constant D (independent of B and the dimension n), such that \[ \| M_ B\|_{2\to 2}=^{def}\| M_ B\|_{L^ 2({\mathbb{R}}^ n)\to L^ 2({\mathbb{R}}^ n)}\leq D. \] This provides the solution to a problem considered by E. Stein in Proc. Natl. Acad. Sci. U.S.A. 73, 2174- 2175 (1976; Zbl 0332.42018) (see also the recent paper by E. Stein and J. O. Stromberg in Ark. Mat.), in the \(L^ 2\) case. Our argument combines geometrical facts such as Brunn’s theorem on convex sets and methods from Fourier analysis. Some remarks on the method are included.”
Reviewer: M.Milman

MSC:

42B25 Maximal functions, Littlewood-Paley theory

Citations:

Zbl 0332.42018
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