与R^3中超曲面的各向异性扩张相关的极大函数
Maximal functions associated with nonisotropic dilations of hypersufaces in R^3
摘要：
本文的目的是建立与R^3中超曲面的各向异性伸缩相关的极大函数的L^p-估计。Greenleaf，Iosevich-Sawyer-Seeger，Ikromov-Kempe-Mueller和Zimmermann已经得到了一些结果，但是对于某些情况，例如参数化为原点附近光滑函数$Φ(x_1,x_2)=x_2^d(1+\mathcal{O}(x_2^m))$的图的超曲面，其中$d\geq 2$，$m\geq 1$和相关的膨胀$δ_t(x)=(t^ax_1,tx_2,t^dx_3)$对于任意实数a>0，这个问题直到最近才得到解决。事实上，这样的问题已经在较低维度中出现了。例如，我们考虑曲线$γ(x)=(x,x^2(1+ϕ(x)))$和相关的扩张$δ_t(x)=(tx_1,t^2x_2)$。如果是$ϕ\equiv 0$，那么对应的极大函数是沿着平面中的抛物线的极大函数，由于Nagel-Riviere-Wainger等人的工作，这是非常好理解的。如果$ϕ\neq 0$和$ϕ(x)=\mathcal{O}(x^m)$，$m\geq 1$，问题是开放的，直到最近，然而，相应的极大函数显示有关的布尔增益循环极大函数，这需要深入的想法和局部光滑估计建立的Mockenhaupt-Seeger-Sogge傅立叶积分算子满足所谓的“电影曲率”的条件。然而，我们观察到，在研究M与上述曲线$γ(x)$和相关的伸缩，我们将考虑一个家庭的相应的傅里叶积分算子，不满足“电影曲率条件”一致，这意味着经典的局部光滑估计不能直接应用于我们的问题。在这篇文章中，我们提出了新的想法，以克服上述困难，并最终建立了尖锐的L^p估计的极大函数有关的曲线$γ(x)$.
英文摘要：
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 $Φ(x_1,x_2)=x_2^d(1+\mathcal{O}(x_2^m))$ near the origin, where $d\geq 2$, $m\geq 1$, and associated dilations $δ_t(x)=(t^ax_1,tx_2,t^dx_3)$ 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 $γ(x)=(x,x^2(1+ϕ(x)))$ and associated dilations $δ_t(x)=(tx_1,t^2x_2)$. If $ϕ\equiv 0$, 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 $ϕ\neq 0$ and $ϕ(x)=\mathcal{O}(x^m)$, $m\geq 1$, 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 $γ(x)$ 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 $γ(x)$ ......
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