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arXiv · 1510.08649

Maximal averages associated to families of finite type surfaces

Ramesh Manna
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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 $\mathcal{M}.$

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Ramesh Manna. 2016-09-26. Maximal averages associated to families of finite type surfaces. https://arxiv.org/abs/1510.08649

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