超曲面上的极大平均与牛顿多面体

$L^p$ Sobolev regularity of averaging operators over hypersurfaces and the Newton polyhedron

$L^p$ to $L^p_β$ boundedness theorems are proven for translation invariant averaging operators over hypersurfaces in Euclidean space. The operators can either be Radon transforms or averaging operators with multiparameter fractional integral kernel. In many cases, the amount $β> 0 $ of smoothing proven is optimal up to endpoints, and in such situations this amount of smoothing can be computed explicitly through the use of appropriate Newton polyhedra.

Newton polyhedrons and $L^p$ Sobolev estimations

The aim of this study is to provide a perspective to help understand the singular average operator over polynomial hypersurfaces. In particular, this perspective will provide brevity and the possibility of generalizing previous results dealing with the fundamental problem of determining the precise $L^p$ regularity enhancement for the average operators. In previous studies dealing with polynomials, the Newton polyhedron of a polynomial has been utilized to observe dominant monomials. In this study, we go further by discussing the involvements of other monomials in detail, by introducing severa…

振荡积分衰减、子水平集增长和牛顿多面体

Oscillatory Integral Decay, Sublevel Set Growth, and the Newton Polyhedron

利用作者的一些奇点的分解方法，证明了Varchenko关于振荡积分衰减的一个著名定理在Newton多面体上的推广。他们来自类似的结果，子级积分，证明在这里。瓦尔琴科定理要求牛顿多面体的面在相位上具有一定的非退化性条件。在本文中，它表明，估计的瓦尔琴科定理也持有一个重要的类相函数的非退化条件不成立。因此，在问题中，一个人想要切换坐标到一个坐标系，其中瓦尔琴科的估计是有效的，有更大的灵活性。一些额外的估计也证明了更退化的情况下，包括一些太退化的牛顿多面体的最佳衰减的意义上的Varchenko。

The Newton Polyhedron and positivity of ${}_2F_3$ hypergeometric functions

As for the ${}_2F_3$ hypergeometric function of the form \begin{equation*} {}_2F_3\left[\begin{array}{c} a_1, a_2\\ b_1, b_2, b_3\end{array}\biggr| -x^2\right]\qquad(x>0), \end{equation*} where all of parameters are assumed to be positive, we give sufficient conditions on $(b_1, b_2, b_3)$ for its positivity in terms of Newton polyhedra with vertices consisting of permutations of $\,(a_2, a_1+1/2, 2a_1)\,$ or $\,(a_1, a_2+1/2, 2a_2).$ As an application, we obtain an extensive validity region of $(α, λ, μ)$ for the inequality \begin{equation*} \int_0^x (x-t)^λ\, t^μ J_α(t)\, dt \ge 0\qquad(x>0)…

Oscillatory Loomis-Whitney and Projections of Sublevel Sets

We consider an oscillatory integral operator with Loomis-Whitney multilinear form. The phase is real analytic in a neighborhood of the origin in $\mathbb{R}^d$ and satisfies a nondegeneracy condition related to its Newton polyhedron. Maximal decay is obtained for this operator in certain cases, depending on the Newton polyhedron of the phase and the given Lebesgue exponents. Our estimates imply volumes of sublevel sets of such real analytic functions are small relative to the product of areas of projections onto coordinate hyperplanes.

振荡积分与牛顿多面体

Oscillating integrals and Newton polyhedra

本文建立了一个主值积分公式，用于求与解析芽f相关联的实或复局部zeta函数的最大非平凡候选极点的留数，该解析芽f关于其Newton多面体是非退化的.特别地，直到一个容易的非零因子，该剩余仅取决于f的（tau_0）-主部，其中tau_0是牛顿多面体与对角线相交的最小面。这个公式使我们能够证明剩余的一些消失结果。更确切地说，我们证明了当τ_0不稳定时，剩余为零，并给出了复情形下逆蕴涵的部分证明。当tau_0是余维为1的单形，f在tau_0上的支撑点是它的顶点时，我们还导出了一个关于剩余的显式公式。