Allan Greenleaf, Andreas Seeger
We mostly survey results concerning the $L^2$ boundedness of oscillatory and Fourier integral operators. This article does not intend to give a broad overview; it mainly focusses on a few topics directly related to the work of the authors.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
In this paper, we consider the degenerate and singular oscillatory integral operator with a singular kernel which is not a Calderón-Zygmund kernel and satisfies suitable size and derivative conditions related to a real parameter $μ$. For any given homogeneous polynomial phases, except monomial phases, of degreee $n$, we give the range of $p$ for which the sharp decay rate $-\frac{1-μ}{n}$ on $L^2$ spaces can be preserved on $L^p$ spaces.
Some sharp $L^2 \to L^p$ decay estimates for $(2+1)$-dimensional degenerate oscillatory integral operators
本文研究(2+1)维多形式相位振荡积分算子，利用Stein复插值方法建立了其锐利 $L^2\ o L^p$ 衰减估计。
We prove sharp endpoint results for the Fourier restriction operator associated to nondegenerate curves in $\Bbb R^d$, $d\ge 3$, and related estimates for oscillatory integral operators. Moreover, for some larger classes of curves in $\Bbb R^d$ we obtain sharp uniform $L^p\to L^q$ bounds with respect to affine arclength measure, thereby resolving a problem of Drury and Marshall.
Some new decay estimates for $(2+1)$-dimensional degenerate oscillatory integral operators
针对具有三次齐次相位的(2+1)维退化振荡积分算子，该研究将L^2→L^2衰减率改进至3/8，并利用分数次积分方法建立尖锐的L^2→L^6衰减估计。
We obtain new estimates for a class of oscillatory integral operators with folding canonical relations satisfying a curvature condition. The main lower bounds showing sharpness are proved using Kakeya set constructions. As a special case of the upper bounds we deduce optimal $L^p(mathbb{S}^2)\to L^q(R \mathbb{S}^2)$ estimates for the Fourier extension operator on large spheres in $\mathbb{R}^3$, which are uniform in the radius $R$. Two appendices are included, one concerning an application to Lorentz space bounds for averaging operators along curves in $R^3$, and one on bilinear estimates.
We use broad-narrow method to estabish the sharp $L^4$ decay estimate for a class of degenerate oscillatory integral operators in $(2+1)$ dimensions. Especially, the model phase function is \[xt^2+y^2t,\] a cubic homogeneous polynomial which is degenerate in the sense of [28].
We consider the following model of degenerate and singular oscillatory integral operators: \begin{equation*} Tf(x)=\int_{\mathbb{R}} e^{iλS(x,y)}K(x,y)ψ(x,y)f(y)dy, \end{equation*} where the phase functions are homogeneous polynomials of degree $n$ and the singular kernel $K(x,y)$ satisfies suitable conditions related to a real parameter $μ$. We show that the sharp decay estimates on $L^2$ spaces, obtained in \cite{liu1999model}, can be preserved on more general $L^p$ spaces with an additional condition imposed on the singular kernel. In fact, we obtain that \begin{equation*} \|Tf\|_{L^p}\leq …
A theorem of Varchenko gives the order of decay of the leading term of the asymptotic expansion of a degenerate oscillatory integral with real-analytic phase in two dimensions. His theorem expresses this order of decay in a simple geometric way in terms of its Newton polygon once one is in certain coordinate systems called adapted coordinate systems. In this paper, we give explicit formulas that not only provide the order of decay of the leading term, but also the coefficient of this term. There are three rather different formulas corresponding to three different types of Newton polygon. Analo…
Bilinear oscillatory Fourier multipliers
本文研究含振荡因子的双线性 Fourier 乘子，建立其在 Lebesgue 空间间（含端点）的有界性，并讨论所得结果的尖锐性。
Fourier transform inequalities, lattice point discrepancy, asymptotic behavior, oscillatory integrals
本文研究了$ l^p $-单位球在$ 1<p\le 2 $时傅里叶变换的精确不等式，揭示了其渐进行为及格点偏差的界。
In this note, by using the result in one variable, we obtain asymptotic expansions of oscillatory integrals for certain multivariable phase functions with {\bf degenerate} singular points. Moreover by using this result, we have asymptotic expansions of oscillatory integrals with phase function of type $A_{k}$, $E_6$, $E_8$-function germs.
We consider non oscillatory functions and prove an everywhere Fourier Inversion Theorem for functions of very moderate decrease. The proofs rely on some ideas in nonstandard analysis.
Sharp L^2 estimates for oscillatory integral operators and Fourier integral operators associated with canonical relations having two-sided cusp or one-sided swallowtail singularities are obtained.
In this article we prove a sharp decay estimate for certain multilinear oscillatory integral operators of a form inspired by the general framework of Christ, Li, Tao, and Thiele [6]. A key purpose of this work is to determine when such estimates are stable under smooth perturbations of both the phase and corresponding projections, which are typically only assumed to be linear. The proof is accomplished by a novel decomposition which mixes features of Gabor or windowed Fourier bases with features of wavelet or Littlewood-Paley decompositions. This decomposition very nearly diagonalizes the prob…
As to methods for expanding an oscillatory integral into an asymptotic series with respect to the parameter, the method of stationary phase for the non-degenerate phases and the method of using resolution of singularities for degenerate phases are known. The aim of this paper is to extend the former for degenerate phases with positive real powers without using resolution of singularities. For this aim, we first generalize Fresnel integrals as oscillatory integrals with phase functions of positive real powers. Next, by using this result, we have asymptotic expansions of oscillatory integrals fo…
Analog version of Hausdorff--Young's theorem for quadratic Fourier transforms and boundedness of oscillatory integral operator
本文基于 Riesz--Thorin 插值定理证明了二次 Fourier 变换和线性正则变换的 Hausdorff--Young 型不等式，并研究了多项式相位振荡积分算子的有界性。
In this paper, we first generalize the work of Bourgain and state a curvature condition for Hörmander-type oscillatory integral operators, which we call Bourgain's condition. This condition is notably satisfied by the phase functions for the Fourier restriction problem and the Bochner-Riesz problem. We conjecture that for Hörmander-type oscillatory integral operators satisfying Bourgain's condition, they satisfy the same $L^p$ bounds as in the Fourier Restriction Conjecture. To support our conjecture, we show that whenever Bourgain's condition fails, then the $L^{\infty} \to L^q$ boundedness a…
This paper establishes the optimal decay rate for scalar oscillatory integrals in $n$ variables which satisfy a nondegeneracy condition on the third derivatives. The estimates proved are stable under small linear perturbations, as encountered when computing the Fourier transform of surface-carried measures. The main idea of the proof is to construct a nonisotropic family of balls which locally capture the scales and directions in which cancellation occurs.
Using some resolution of singularities methods of the author, a generalization of a well-known theorem of Varchenko relating decay of oscillatory integrals to the Newton polyhedron is proven. They are derived from analogous results for sublevel integrals, proven here. Varchenko's theorem requires a certain nondegeneracy condition on the faces of the Newton polyhedron on the phase. In this paper, it is shown that the estimates of Varchenko's theorem also hold for a significant class of phase functions for which this nondegeneracy condition does not hold. Thus in problems where one wants to swit…
We study the restriction of the Fourier transform to quadratic surfaces in vector spaces over finite fields. In two dimensions, we obtain the sharp result by considering the sums of arbitrary two elements in the subset of quadratic surfaces on two dimensional vector spaces over finite fields. For higher dimensions, we estimate the decay of the Fourier transform of the characteristic functions on quadratic surfaces so that we obtain the Tomas-Stein exponent. Using incidence theorems, we also study the extension theorems in the restricted settings to sizes of sets in quadratic surfaces. Estimate…
We control a broad class of singular (or "rough") Fourier multipliers by geometrically-defined maximal operators via general weighted $L^2(\mathbb{R})$ norm inequalities. The multipliers involved are related to those of Coifman--Rubio de Francia--Semmes, satisfying certain weak Marcinkiewicz-type conditions that permit highly oscillatory factors of the form $e^{i|ξ|^α}$ for both $α$ positive and negative. The maximal functions that arise are of some independent interest, involving fractional averages associated with tangential approach regions (related to those of Nagel and Stein), and more no…
Sharp $L^2$ decay rate for (1+2)-dimensional oscillatory integral operators with cubic polynomial phases
本文研究了(1+2)维振荡积分算子中三阶多项式相位的$L^2$衰减率，证明了文献中给出的3/8衰减率是精确的。
The sharp range of $L^p$-estimates for the class of Hörmander-type oscillatory integral operators is established in all dimensions under a positive-definite assumption on the phase. This is achieved by generalising a recent approach of the first author for studying the Fourier extension operator, which utilises polynomial partitioning arguments.
We study the family of Fourier-Laplace transforms $$ F_{α,β}(z)= \operatorname*{F.p.} \int_{0}^{\infty} t^β\exp(\mathrm{i} t^α-\mathrm{i} z t)\:\mathrm{d} t, \quad \operatorname*{Im} z<0, $$ for $α>1$ and $β\in\mathbb{C}$, where Hadamard finite part is used to regularize the integral when $\operatorname*{Re} β\leq -1$. We prove that each $F_{α,β}$ has analytic continuation to the whole complex plane and determine its asymptotics along any line through the origin. We also apply our ideas to show that some of these functions provide concrete extremal examples for the Wiener-Ikehara theorem and a…
Let $T$ be a Fourier integral operator on $\R^n$ of order $-(n-1)/2$. It was shown by Seeger, Sogge, and Stein that $T$ mapped the Hardy space $H^1$ to $L^1$. In this note we show that $T$ is also of weak-type $(1,1)$. The main ideas are a decomposition of $T$ into non-degenerate and degenerate components, and a factorization of the non-degenerate portion.
The theory of Fourier integral operators is surveyed, with an emphasis on local smoothing estimates and their applications. After reviewing the classical background, we describe some recent work of the authors which established sharp local smoothing estimates for a natural class of Fourier integral operators. We also show how local smoothing estimates imply oscillatory integral estimates and obtain a maximal variant of an oscillatory integral estimate of Stein. Together with an oscillatory integral counterexample of Bourgain, this shows that our local smoothing estimates are sharp in odd spati…
Oscillatory integrals and weighted gradient flows
本研究提出沿加权梯度方向分部积分估计标量振荡积分的方法，证明了关于相位线性扰动一致的估计定理，推导了拟齐次相位下的曲面测度傅里叶变换估计，并给出Varchenko振荡积分估计推广到曲面测度傅里叶变换的新证明。
In this paper we establish sharp estimates (up to logarithmic losses) for the multilinear oscillatory integral operator studied by Phong, Stein, and Sturm and Carbery and Wright on any product $\prod_{j=1}^d L^{p_j}(\mathbb R)$ with each $p_j \geq 2$, expanding the known results for this operator well outside the previous range $\sum_{j=1}^d p_j^{-1} = d-1$. Our theorem assumes second-order nondegeneracy condition of Varchenko type, and as a corollary reproduces Varchenko's theorem and implies Fourier decay estimates for measures of smooth density on degenerate hypersurfaces in $\mathbb R^d$.
In this paper, we first generalize the Fresnel integrals by changing of a path for integration in the proof of the Fresnel integrals by Cauchy's integral theorem. Next, according to oscillatory integral, we also obtain further generalization of the extended Fresnel integrals. Moreover by using this result, we have an asymptotic expansion of an oscillatory integral with a positive real parameter, for a phase function with a degenerate critical point expressed by positive real power, including a moderate oscillation, and for a suitable amplitude function. This result gives a finer extension of t…
In this note, a general formula is proved. It expresses the integral on the line of the product of a function $f$ and a periodic function $g$ in terms of the Fourier transform of $f$ and the Fourier coefficients of $g$. This allows the evaluation of some oscillatory integrals.
We prove an elementary multilinear identity for the Fourier extension operator on $\mathbb{R}^n$, generalising to higher dimensions the classical bilinear extension identity in the plane. In the particular case of the extension operator associated with the paraboloid, this provides a higher dimensional extension of a well-known identity of Ozawa and Tsutsumi for solutions to the free time-dependent Schrödinger equation. We conclude with a similar treatment of more general oscillatory integral operators whose phase functions collectively satisfy a natural multilinear transversality condition. T…
We prove sparse bounds for pseudodifferential operators associated to Hörmander symbol classes. Our sparse bounds are sharp up to the endpoint and rely on a single scale analysis. As a consequence, we deduce a range of weighted estimates for pseudodifferential operators. The results naturally apply to the context of oscillatory Fourier multipliers, with applications to dispersive equations and oscillatory convolution kernels.
We study oscillatory integrals of the type ${\mathcal F}^{-1}(e^{ita(\cdot)}ψ(\cdot))$ where $a$ is a general function satisfying some elliptic type and non-degenerate conditions at both the origin and infinity, and $ψ$ belongs to some symbol class. Point-wise estimates in space-time are gained with partial sharpness. As applications, global smoothing effects of $L^p-L^q$ as well as Strichartz type for dispersive equations are studied. An application to fractional Schrödinger equations is also given.
Lower bounds for parameter-dependent oscillatory integrals and Fourier frames on hypersurfaces
本文证明了有限型凸超曲面上曲面测度Fourier变换的一致下界，并利用参数依赖振荡积分的新工具，推广了Fourier框架不存在性的结果至更弱曲率条件。
Around the Fejér-Jackson inequality: Tight bounds for certain oscillatory functions via Laplace transform representations
本文通过拉普拉斯变换表示法，为特定振荡函数提供了更紧的界，研究了锯齿函数的傅里叶多项式近似误差以及对数函数的泰勒级数余项的渐近紧致不等式。
Oscillatory integral operators with homogeneous phase functions
本文针对满足凸性条件的齐次相函数振荡积分算子，结合多项式划分与Guth-Hickman-Iliopoulou分析，证明尖锐Lp估计，并推广至紧致黎曼流形上的波动方程局部平滑估计。
We study an operator analogue of the classical problem of finding the rate of decay of an oscillatory integral on the real line. This particular problem arose in the analysis of oscillatory Riemann-Hilbert problems associated with partial differential equations in the Ablowitz-Kaup-Newell-Segur hierarchy, but is interesting in its own right as a question in harmonic analysis and oscillatory integrals. As was the case in earlier work of the first author, the approach is general and purely real-variable. The resulting estimates we achieve are strongly uniform as a function of the phase and can s…
We consider the question of when it is possible to force a degenerate scalar oscillatory integral to decay as fast as a nondegenerate one by restricting the support to the region where the Hessian determinant of the phase is bounded below. We show in two dimensions that the desired outcome is not always possible, but does occur for a broad class of phases which may be described in terms of the Newton polygon. The estimates obtained are uniform with respect to linear perturbation of the phase and uniform in the cutoff value of the Hessian determinant. In the course of the proof, we investigate …
We propose a new method of estimating oscillatory integrals, which we call a stationary set method. We use it to obtain the sharp convergence exponents of Tarry's problems in dimension two for every degree $k\ge 2$. As a consequence, we obtain sharp Fourier extension estimates for a family of monomial surfaces.
This paper investigates higher order wave-type equations of the form $\partial_{tt}u+P(D_{x})u=0$, where the symbol $P(ξ)$ is a real, non-degenerate elliptic polynomial of the order $m\ge4$ on ${\bf R}^n$. Using methods from harmonic analysis, we first establish global pointwise time-space estimates for a class of oscillatory integrals that appear as the fundamental solutions to the Cauchy problem of such wave equations. These estimates are then used to establish (pointwise-in-time) $L^p-L^q$ estimates on the wave solution in terms of the initial conditions.
Bilinear Sparse Domination for Oscillatory Integral Operators
本文证明了一般秩 Fourier 积分算子和 Hörmander 符号类振荡积分算子的双线性稀疏控制界，由此推导出弱(1,1)估计、向量值估计及加权范数不等式。
A sharp Hörmander condition for bilinear Fourier multipliers with Lipschitz singularities
该研究针对带有限Lipschitz曲线奇异集的双线性Fourier乘子，提出关于Sobolev指数精确的Hörmander条件，证明其局部L²范围内的L^p有界性，推广了非退化双线性Hilbert变换并规避退化附近一致有界性难题。
In this paper, we investigate the behavior of the Fourier transform on finite dimensional 2-step Lie groups and develop a general theory akin to that of the whole space or the torus. We provide a familiar framework in which computations are made sensibly easier than with the usual representation-theoretic Fourier transform. In addition, we study the 'singular frequencies' of the group, at which the canonical bilinear antisymmetric form degenerates. We also exhibit a specific example for which partial degeneracy of the canonical form occurs, as opposed to the full degeneracy at the origin. We t…
Estimates for a certain bilinear Fourier integral operator
本文研究了具有非退化相位函数及特定振幅的双线性 Fourier 积分算子从 $H^{1} \times L^{\infty}$ 到 $L^{1}$ 的有界性，并改进了已有结果。
We call a function "constructible" if it has a globally subanalytic domain and can be expressed as a sum of products of globally subanalytic functions and logarithms of positively-valued globally subanalytic functions. Our main theorem gives uniform bounds on the decay of parameterized families of oscillatory integrals with a constructible amplitude function and a globally subanalytic phase function, assuming that the amplitude function is integrable and that the phase function satisfies a certain natural condition called the hyperplane condition. As a simple application of this theorem, we al…
We define a scale of Hardy spaces $\mathcal{H}^{p}_{FIO}(\mathbb{R}^{n})$, $p\in[1,\infty]$, that are invariant under suitable Fourier integral operators of order zero. This builds on work by Smith for $p=1$. We also introduce a notion of off-singularity decay for kernels on the cosphere bundle of $\mathbb{R}^{n}$, and we combine this with wave packet transforms and tent spaces over the cosphere bundle to develop a full Hardy space theory for oscillatory integral operators. In the process we extend the known results about $L^{p}$-boundedness of Fourier integral operators, from local boundednes…
A locally compact group $G$ is said to be weakly amenable if the Fourier algebra $A(G)$ admits completely bounded approximative units. Consider the family of groups $G_n=SL(2,\Bbb R)\ltimes H_n$ where $n\ge 2$, $H_n$ is the $2n+1$ dimensional Heisenberg group and $SL(2,\Bbb R)$ acts via the irreducible representation of dimension $2n$ fixing the center of $H_n$. We show that these groups fail to be weakly amenable. Following an idea of Haagerup for the case $n=1$ one can reduce matters to the problem of obtaining nontrivial uniform bounds for a family of singular oscillatory integral operators…
Limiting behavior of principal eigenvalues and eigenfunctions for a class of elliptic operators with degenerate large advection
本文通过数值与解析方法研究了一类退化强对流椭圆算子的主特征函数在参数趋于无穷时的渐近行为，并在最小正则性条件下推导了主特征值的渐近行为，揭示了$m(x)$的振荡与正则性对极限行为的严重影响。
Counting integral points near space curves: a Fourier analytic approach
本文利用傅里叶分析方法（特别是 Arkhipov--Chubarikov--Karatsuba 型振荡积分估计），建立了 $\mathbb{R}^n$（$n\geq 3$）中光滑非退化曲线邻域内整点数量的上下界，不依赖平面曲线的精确计数结果，并恢复了 $n=3$ 时的已知结论。
In this article, we study the problem of obtaining Lebesgue space inequalities for the Fourier restriction operator associated to rectangular pieces of the paraboloid and perturbations thereof. We state a conjecture for the dependence of the operator norms in these inequalities on the sidelengths of the rectangles, prove that this conjecture follows from (a slight reformulation of the) restriction conjecture for elliptic hypersurfaces, and prove that, if valid, the conjecture is essentially sharp. Such questions arise naturally in the study of restriction inequalities for degenerate hypersurfa…
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。