Maximal functions associated with nonisotropic dilations of hypersufaces in R^3

Abstract

The goal of this article is to establish L^p-estimates for maximal functions associated with nonisotropic dilations of hypersurfaces in R^3. Several results have already been obtained by Greenleaf, Iosevich-Sawyer-Seeger, Ikromov-Kempe-Mueller and Zimmermann, but for some situations such as the hypersurface parameterized as the graph of a smooth function $\Phi(x_1,x_2)=x_2^d(1+\mathcal{O}(x_2^m))$ near the origin, where $d\geq 2$, $m\geq 1$, and associated dilations $\delta_t(x)=(t^ax_1,tx_2,t^dx_3)$ for an arbitrary real number a>0, the question was open until recently. In fact, such problems do arise already in lower dimensions. For instance, we consider the curve $\gamma(x)=(x,x^2(1+\phi(x)))$ and associated dilations $\delta_t(x)=(tx_1,t^2x_2)$. If $\phi\equiv 0$, then the corresponding maximal function is the maximal function along parabolas in the plane, which is very well understood due to the work by Nagel-Riviere-Wainger and others. If $\phi\neq 0$ and $\phi(x)=\mathcal{O}(x^m)$, $m\geq 1$, the problem was open until recently, however, the corresponding maximal function shows features related to the Bourgain circular maximal function, which required deep ideas and local smoothing estimates established by Mockenhaupt-Seeger-Sogge for Fourier integral operators satisfying the so-called "cinematic curvature" condition. However, we observe that in the study of M related to the mentioned curve $\gamma(x)$ and associated dilations, we will consider a family of corresponding Fourier integral operators which fail to satisfy the "cinematic curvature condition" uniformly, which means that classical local smoothing estimates could not be directly applied to our problem. In this article, we develop new ideas in order to overcome the above difficulty and finally establish sharp L^p-estimates for the maximal function related to the curve $\gamma(x)$ ......