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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 [5], Iosevich Sawyer [9], Ikromov-Kempe-Miiller [6] and Zimmermann [28], 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 + O(x(2)(m))) near the origin, where d >= 2, m >= 1, and associated dilations delta(t)(x) = (t(a)x(1) tx(2), t(d)x(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(2)x(2)). If phi = 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 [17] and others. If phi = 0 and phi(x) = O(x(m)), m >>= 1, the problem was open until recently. We observe that in the study of the maximal function 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 established by Mockenhaupt-Seeger-Sogge could not be directly applied to our problem. In this article, we develop new ideas to establish sharp LP-estimates for the maximal function related to the curve gamma(x) with associated dilations in the plane. Later, we generalize the result to curves of finite type d (d > 2) and associated dilations delta(t)(x) = (tx(1), t(d)x(2)). Furthermore, we also obtain L-p-estimates for the maximal function related to the mentioned hypersurface Phi(x(1), x(2)) in R-3 with associated dilations.
- 出版日期2018-5
- 单位西北工业大学
