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The goal of this article is to establish L^p-estimates for maximal functionsassociated with nonisotropic dilations of hypersurfaces in R^3. Several resultshave already been obtained by Greenleaf, Iosevich-Sawyer-Seeger,Ikromov-Kempe-Mueller and Zimmermann, but for some situations such as thehypersurface 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 anarbitrary real number a>0, the question was open until recently. In fact, suchproblems do arise already in lower dimensions. For instance, we consider thecurve $\gamma(x)=(x,x^2(1+\phi(x)))$ and associated dilations$\delta_t(x)=(tx_1,t^2x_2)$. If $\phi\equiv 0$, then the corresponding maximalfunction is the maximal function along parabolas in the plane, which is verywell 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 openuntil recently, however, the corresponding maximal function shows featuresrelated to the Bourgain circular maximal function, which required deep ideasand local smoothing estimates established by Mockenhaupt-Seeger-Sogge forFourier integral operators satisfying the so-called "cinematic curvature"condition. However, we observe that in the study of M related to the mentionedcurve $\gamma(x)$ and associated dilations, we will consider a family ofcorresponding Fourier integral operators which fail to satisfy the "cinematiccurvature condition" uniformly, which means that classical local smoothingestimates could not be directly applied to our problem. In this article, wedevelop new ideas in order to overcome the above difficulty and finallyestablish sharp L^p-estimates for the maximal function related to the curve$\gamma(x)$

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