Maximal averages associated to families of finite type surfaces

Abstract

We study the boundedness problem for maximal operators $\mathcal{M}$ associated to averages along families of hypersurfaces $S$ of finite type in ${\mathbb{R}}^n.$ In this paper, we prove that if $S$ is a finite type hypersurface which is of finite type $k$ at $x_0\in {\mathbb{R}}^n$, then the associated maximal operator is bounded on $L^p({\mathbb{R}}^n)$ for $p>k.$ We shall also consider a variable coefficient version of maximal theorem and we obtain the same $L^p-$ boundedness result for $p>k.$ We also discuss the consequence of this result. In particular, we verify a conjecture by E. M. Stein and its generalization by A. Iosevich and E. Sawyer on the connection between the decay rate of the Fourier transform of the surface measure on $S$ and the $L^p-$ boundedness of the associated maximal operator $\mathcal{M}.$

arXiv comment17 pages

Cited by in corpus (1)

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