Maximal functions associated with nonisotropic dilations of hypersurfaces in [formula omitted

Abstract

The goal of this article is to establish Lp-estimates for maximal functions associated with nonisotropic dilations of hypersurfaces in R3. Several results have already been obtained by Greenleaf [5], Iosevich–Sawyer [9], Ikromov–Kempe–Müller [6] and Zimmermann [28], but for some situations such as the hypersurface parameterized as the graph of a smooth function Φ(x1,x2)=x2d(1+O(x2m)) near the origin, where d≥2, m≥1, and associated dilations δt(x)=(tax1,tx2,tdx3) 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,x2(1+ϕ(x))) and associated dilations δt(x)=(tx1,t2x2). If ϕ≡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 ϕ≠0 and ϕ(x)=O(xm), m≥1, the problem was open until recently. We observe that in the study of the maximal function 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 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 γ(x) with associated dilations in the plane. Later, we generalize the result to curves of finite type d (d≥2) and associated dilations δt(x)=(tx1,tdx2). Furthermore, we also obtain Lp-estimates for the maximal function related to the mentioned hypersurface Φ(x1,x2) in R3 with associated dilations.