Exact Evaluation Of Some Highly Oscillatory Integrals

Oscillatory integral operators with low-order degeneracies

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.

Global Range Estimates for Maximal Oscillatory Integrals with Radial Testfunctions

We consider the maximal function of oscillatory integrals and prove a global estimate for radial test functions which is almost sharp with respect to the Sobolev regularity.

Oscillatory Integrals and Fractal Dimension

We study geometrical representation of oscillatory integrals with an analytic phase function and a smooth amplitude with compact support. Geometrical properties of the curves defined by the oscillatory integral depend on the type of a critical point of the phase. We give explicit formulas for the box dimension and the Minkowski content of these curves. Methods include Newton diagrams and the resolution of singularities.

Damping estimates for oscillatory integral operators with real-analytic phases and its applications

In this paper, we investigate sharp damping estimates for a class of one dimensional oscillatory integral operators with real-analytic phases. By establishing endpoint estimates for suitably damped oscillatory integral operators, we are able to give a new proof of the sharp $L^p$ estimates which have been proved by Xiao in Endpoint estimates for one-dimensional oscillatory integral operators, \emph{Advances in Mathematics}, \textbf{316}, 255-291 (2017). The damping estimates obtained in this paper are of independent interest.

Uniform estimates for oscillatory integrals with smooth phase

We consider the problem on uniform estimates for an oscillatory integrals with the smooth phase functions having singularities $D_{\infty} $. The estimate is sharp and analogy to estimates of the work of V.N.Karpushkin.

Special cases of power decay in multilinear oscillatory integrals

We use tools from the multilinear oscillatory integral program developed by Christ, Li, Thiele, and Tao to treat special cases which are not covered by existing theory. Consideration of special cases leads to an extended class of examples for which $λ$-power decay holds.

具有奇异相位的振荡积分的渐近展开

Asymptotic expansion of oscillatory integrals with singular phases

本文研究相位具有奇异性的一维振荡积分，给出其奇异性的渐近展开描述，并将类似结果推广到拉普拉斯积分。

Decay Rate of n-Linear Oscillatory Integral Operators in $\mathbb{R}^2$

In this paper, we prove $L^p$ decay estimates for multilinear oscillatory integrals in $\mathbb{R}^2$, establishing sharpness through a scaling argument. The result in this paper is a generalization of the previous work by Gressman and Xiao (2016).

A sharp decay estimate for degnerate oscillatory integral operators using broad-narrow method

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].

Multilinear Oscillatory Integrals Via Reduction of Dimension

Christ, Li, Tao, and Thiele have established multilinear oscillatory integral operator inequalities under severe dimensional restrictions. These restrictions are relaxed substantially in the present paper, but are by no means eliminated. The simple method used here has the advantage of yielding power type bounds, but cannot be expected to extend to the general case.

Negative results for Nikodym maximal functions and related oscillatory integrals in curved space

We expand on counterxamples of Bourgain showing how Nikodym maximal estimates and oscillatory integral estimates can break down in the non-Euclidean case. Our examples show the role that the non-existence of totally geodesic submanifolds can play in these problems.

关于满足电影曲率条件的振荡积分算子

On Oscillatory Integral Operators Satisfying the cinematic curvature condition

本文提出了一种基于Wolff双端减少和改进分解不等式的新的振荡积分算子估计方法，实现了从L^r到L^p的几乎精确估计。

On multilinear oscillatory integrals, nonsingular and singular

Basic questions concerning nonsingular multilinear operators with oscillatory factors are posed and partially answered. Lebesgue space norm inequalities are established for multilinear integral operators of Calderon-Zygmund type which incorporate oscillatory factors exp(iP), where P is a real-valued polynomial with large coefficients. A related problem concerning upper bounds for measures of sublevel sets is solved.

Weighted norm inequalities for oscillatory integrals with finite type phases on the line

We obtain two-weighted $L^2$ norm inequalities for oscillatory integral operators of convolution type on the line whose phases are of finite type. The conditions imposed on the weights involve geometrically-defined maximal functions, and the inequalities are best-possible in the sense that they imply the full $L^p(\mathbb{R})\rightarrow L^q(\mathbb{R})$ mapping properties of the oscillatory integrals. Our results build on work of Carbery, Soria, Vargas and the first author.

具有Dini核的振荡奇异性积分的加权弱类型(1,1)估计

Weighted weak type $(1,1)$ estimates for oscillatory singular integrals with Dini kernels

本文研究了A1权重下的振荡奇异性积分，证明了满足Dini条件的核的加权弱类型(1,1)估计。

Sharp estimates for trilinear oscillatory integrals and an algorithm of two-dimensional resolution of singularities

We obtain sharp estimates for certain trilinear oscillatory integrals. In particular, we extend Phong and Stein's seminal result to a trilinear setting. This result partially answers a question raised by Christ, Li, Tao and Thiele concerning the sharp estimates for certain multilinear oscillatory integrals. The method in this paper relies on a self-contained algorithm of resolution of singularities in $\mathbb R^2$, which may be of independent interest.

Uniform Oscillatory Integral estimates for Convex Phases via Sublevel Set estimates

We examine the relation between oscillatory integral estimates and sublevel set estimates associated to convex functions. Whilst the former implies the latter in many cases, the reverse requires additional assumptions. Under finite (line) type assumptions, Bruna, Nagel & Wainger were able to demonstrate a very precise control of oscillatory integrals with convex phases via their sublevel sets. Without the finite type assumption, certain erratic behaviour can force this precise control to fail (Bak, McMichael, Vance & Wainger). We establish the same precise control under an alternative qualitat…

具有 $D$ 型奇点的相位的振荡积分估计

Estimates for oscillatory integrals with phase having $D$ type singularities

本文研究二维振荡积分在 $D$ 型奇点相位下的估计，通过 Randol 型极大函数获得尖锐 $L^p_{loc}$ 估计，并探讨指数 $p$ 与线性适配坐标系的关系。

Sharp Bounds for Oscillatory Integral Operators with Homogeneous Polynomial Phases

We obtain sharp $L^p$ bounds for oscillatory integral operators with generic homogeneous polynomial phases in several variables. The phases considered in this paper satisfy the rank one condition which is an important notion introduced by Greenleaf, Pramanik and Tang. Under certain additional assumptions, we can establish sharp damping estimates with critical exponents to prove endpoint $L^p$ estimates.

Oscillatory and Fourier Integral operators with degenerate canonical relations

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.

Scalar oscillatory integrals in smooth spaces of homogeneous type

We consider a generalization of the notion of spaces of homogeneous type, inspired by recent work of Street [21] on the multi-parameter Carnot-Caratheodory geometry, which imbues such spaces with differentiability structure. The setting allows one to formulate estimates for scalar oscillatory integrals on these spaces which are uniform and respect the underlying geometry of both the space and the phase function. As a corollary we obtain a generalization of a theorem of Bruna, Nagel, and Wainger [1] on the asymptotic behavior of scalar oscillatory integrals with smooth, convex phase of finite t…

振荡积分与 Hankel 变换的正性

Positivity of oscillatory integrals and Hankel transforms

本文利用 Sturm 理论分析核为二阶线性微分方程振荡解的积分变换的正性，并给出 Hankel 变换与正实轴上三角积分的正性判据。

Sharp $L^p$ estimates for oscillatory integral operators of arbitrary signature

The sharp range of $L^p$-estimates for the class of Hörmander-type oscillatory integral operators is established in all dimensions under a general signature assumption on the phase. This simultaneously generalises earlier work of the authors and Guth, which treats the maximal signature case, and also work of Stein and Bourgain--Guth, which treats the minimal signature case.

Sharp $L^{p}$-Boundedness of Oscillatory Integral Operators with Polynomial Phases

In this paper, we shall prove the $L^{p}$ endpoint decay estimates of oscillatory integral operators with homogeneous polynomial phases $S$ in $\mathbb{R} \times \mathbb{R}$. As a consequence, sharp $L^{p}$ decay estimates are also obtained when polynomial phases have the form $S(x^{m_{1}},y^{m_{2}})$ with $m_1$ and $m_2$ being positive integers.

Uniform estimates for cubic oscillatory integrals

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.

振荡奇异积分端点问题的一种方法

An Approach To Endpoint Problems in Oscillatory Singular Integrals

本文用初等鸽笼原理和驻相法，证明了振荡奇异积分极大截断从 L^1 到弱 L^1 的有界性。

$L^2$-estimates for singular oscillatory integral operators

In this note we study singular oscillatory integrals with linear phase function over hypersurfaces which may oscillate, and prove estimates of $L^2 \mapsto L^2$ type for the operator, as well as for the corresponding maximal function. If the hypersurface is flat, we consider a particular class of a nonlinear phase functions, and apply our analysis to the eigenvalue problem associated with the Helmholtz equation in $\mathbb{R}^3$.

Bounds on oscillatory integral operators based on multilinear estimates

We apply the Bennett-Carbery-Tao multilinear restriction estimate in order to bound restriction operators and more general oscillatory integral operators. We get improved L^p estimates in the Stein restriction problem for dimension at least 5 and a small improvement in dimension 3. We prove similar estimates for Hormander-type oscillatory integral operators when the quadratic term in the phase function is positive definite, getting improvements in dimension at least 5. We also prove estimates for Hormander-type oscillatory integral operators in even dimensions. These last oscillatory estimates…

Sharp estimates for oscillatory integral operators via polynomial partitioning

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.

Singular Oscillatory Integrals on R^n

Let Pd,n denote the space of all real polynomials of degree at most d on R^n. We prove a new estimate for the logarithmic measure of the sublevel set of a polynomial P in Pd,1. Using this estimate, we prove a sharp estimate for a singular oscillatory integral on R^n.

(1+2)维振荡积分算子中三阶多项式相位的$L^2$衰减率的精确性

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。

Algebraic Stability of Oscillatory Integral Estimates: A Calculus for Uniform Estimates

Oscillatory integrals arise in many situations where it is important to obtain decay estimates which are stable under certain perturbations of the phase. Examining the structural problems underpinning these estimates leads one to consider sublevel set estimates, which behave nicely under certain algebraic operations such as composition with a polynomial. This motivates us to ask how oscillatory integral estimates behave under such transformations of the phase, and under some natural higher order convexity assumptions we obtain stable estimates under composition with polynomial phases in one di…

A theory of complex oscillatory integrals: A case study

In this paper we develop a theory for oscillatory integrals with complex phases. When $f:{\mathbb C}^n \to {\mathbb C}$, we evaluate this phase function on the basic character ${\rm e}(z) := e^{2πi x} e^{2πi y}$ of ${\mathbb C} \simeq {\mathbb R}^2$ (here $z = x+iy \in {\mathbb C}$ or $z = (x,y) \in {\mathbb R}^2$) and consider oscillatory integrals of the form $$ I \ = \ \int_{{\mathbb C}^n} {\rm e}(f({\underline{z}})) \, ϕ({\underline{z}}) \, d{\underline{z}} $$ where $ϕ\in C^{\infty}_c({\mathbb C}^n)$. Unfortunately basic scale-invariant bounds for the oscillatory integrals $I$ do not hold …

Sharp $L^p$ decay estimates for degenerate and singular oscillatory integral operators: Homogeneous polynomial phases

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.

Maximal Decay Inequalities for Trilinear oscillatory integrals of convolution type

In this paper we prove sharp $L^\infty$-$L^\infty$-$L^\infty$ decay for certain trilinear oscillatory integral forms of convolution type on $\mathbb R^2$. These estimates imply earlier $L^2$-$L^2$-$L^2$ results obtained by the second author as well as corresponding sharp, stable sublevel set estimates of the form studied by Christ and Christ, Li, Tao, and Thiele. New connections to the multilinear results of Phong, Stein, and Sturm are also considered.

(2+1)维退化振荡积分算子的若干锐利 $L^2 \ o L^p$ 衰减估计

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。

Restriction Theorem for Oscillatory Integral Operator with Certain Polynomial Phase

We consider the following oscillatory integral operator \begin{equation}\label{opera-defi-1} T_{α,m}f(x)=\int_{\mathbb R^n}e^{i(x_1^{α_1} y_1^m+\cdots+x_n^{α_n} y_n^m)}f(y)dy, \end{equation} where the function $f$ is a Schwartz function. In this paper, the restriction theorem on $\mathbb{S}^{n-1}$ for this operator is obtained. Moreover, we obtain a necessary condition which ensures the restriction theorem hold.

Some Oscillatory Integral Estimates via Real Analysis

We study oscillatory integrals in several variables with analytic, smooth, or $C^k$ phases satisfying a nondegeneracy condition attributed to Varchenko. With only real analytic methods, Varchenko's estimates are rediscovered and generalized. The same methods are pushed further to obtain full asymptotic expansions of such integrals with analytic and smooth phases, and finite expansions with error assuming the phase is only $C^k$. The Newton polyhedron appears naturally in the estimates; in particular, we show precisely how the exponents appearing in the asymptotic expansions depend only on the …

Resolution of singularities, asymptotic expansions of oscillatory integrals, and related phenomena

The elementary resolution of singularities algorithm of the author's earlier paper (math.CA/0609217) is developed further, replacing the quasibump functions in the blown up coordinates with the characteristic function of a rectangle times a smooth function. Such functions are easier to deal with, and as application the existence of asymptotic expansions for oscillatory integrals and related objects is given an elementary proof. In addition, some more detailed information about these expansions is given.

Restriction of Fourier transforms to curves and related oscillatory integrals

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.

Oscillatory integrals related to Carleson's theorem: fractional monomials

Stein and Wainger proved the $L^p$ bounds of the polynomial Carleson operator for all integer-power polynomials without linear term. In the present paper, we partially generalise this result to all fractional monomials in dimension one. Moreover, the connections with Carleson's theorem and the Hilbert transform along vector fields or (variable) curves are also discussed in details.

Higher decay inequalities for multilinear oscillatory integrals

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 $\mathbb R^d$.

Toric resolution of singularities in a certain class of $C^{\infty}$ functions and asymptotic analysis of oscillatory integrals

In a seminal work of A. N. Varchenko, the behavior at infinity of oscillatory integrals with real analytic phase is precisely investigated by using the theory of toric varieties based on the geometry of the Newton polyhedron of the phase. The purpose of this paper is to generalize his results to the case that the phase is contained in a certain class of $C^{\infty}$ functions. The key in our analysis is a toric resolution of singularities in the above class of $C^{\infty}$ functions. The properties of poles of local zeta functions, which are closely related to the behavior of oscillatory integ…

Damped oscillatory integrals and boundedness of maximal operators associated to mixed homogeneous hypersurfaces

We study the boundedness problem for maximal operators in 3-dimensional Euclidean space associated to hypersurfaces given as the graph of $c+f$, where $f$ is a mixed homogeneous function which is smooth away from the origin and $c$ is a constant. Our result generalizes a corresponding theorem on mixed homogeneous polynomial functions by A. Iosevich and E. Sawyer.

关于(2+1)维退化振荡积分算子的若干新衰减估计

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。

Estimating Oscillatory Integrals of Convolution Type in $\mathbb{R}^d$

In this paper, we prove an $L^2-L^2-L^2$ decay estimate for a trilinear oscillatory integral of convolution type in $\mathbb{R}^d,$ which recovers the earlier result of Li (2013) when $d=1.$ We discuss the sharpness of our result in the $d=2$ case. Our main hypothesis has close connections to the property of simple nondegeneracy studied by Christ, Li, Tao and Thiele (2005).

Multilinear oscillatory integral operators and geometric stability

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…

Weak Type Bound for Oscillatory Singular Integrals

Let $ T _{P} f (x) = \int e ^{i P (y)} K (y) f (x-y) \, dy $, where $ K (y)$ is a smooth Calderón-Zygmund kernel on $ \mathbb R ^{n}$, and $ P$ be a polynomial. The maximal truncations of $ T_P$ satisfy the weak $ L ^{1}$ inequality, our proof simplifying and extending the argument of Chanillo and Christ for the weak type bound for $ T_P$.

The Fourier extension operator on large spheres and related oscillatory integrals

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.

Endpoint estimates for one-dimensional oscillatory integral operator

The one-dimensional oscillatory integral operator associated to a real analytic phase $S$ is given by $$ T_λf(x) =\int_{-\infty}^\infty e^{iλS(x,y)} χ(x,y) f(y) dy. $$ In this paper, we obtain a complete characterization for the mapping properties of $T_λ$ on $L^p(\mathbb R)$ spaces, namely we prove that $\|T_λ\|_p \lesssim |λ|^{-α}\|f\|_p$ for some $α>0$ if and only if the point $(\frac 1 {αp} , \frac 1 {αp'})$ lies in the reduced Newton polygon of $S$, and this estimate is sharp if and only if it lies on the reduced Newton diagram.

Oscillatory integral operators with homogeneous polynomial phases in several variables

We obtain $L^2$ decay estimates in $λ$ for oscillatory integral operators whose phase functions are homogeneous polynomials of degree m and satisfy various genericity assumptions. The decay rates obtained are optimal in the case of (2+2)--dimensions for any m while, in higher dimensions, the result is sharp for m sufficiently large. The proof for large $m$ follows from essentially algebraic considerations. For cubics in (2+2)--dimensions, the proof involves decomposing the operator near the conic zero variety of the determinant of the Hessian of the phase function, using an elaboration of the …

A sharp bound for the Stein-Wainger oscillatory integral

Let Pd denote the space of all real polynomials of degree at most d. It is an old result of Stein and Wainger that for every polynomial P in Pd: |p.v.\int_R {e^{iP(t)} dt/t} | < C(d) for some constant C(d) depending only on d. On the other hand, Carbery, Wainger and Wright claim that the true order of magnitude of the above principal value integral is log d. We prove this conjecture.

The Hilbert transform along the parabola, the polynomial Carleson theorem and oscillatory singular integrals

We make progress on an interesting problem on the boundedness of maximal modulations of the Hilbert transform along the parabola. Namely, if we consider the multiplier arising from it and restrict it to lines, we prove uniform $L^p$ bounds for maximal modulations of the associated operators. Our methods consist of identifying where to use effectively the polynomial Carleson theorem, and where we can take advantage of the presence of oscillation to obtain decay through the $TT^*$ method.

Some one-sided estimates for oscillatory singular integrals

The purpose of this paper is to establish some one-sided estimates for oscillatory singular integrals. The boundedness of certain oscillatory singular integral on weighted Hardy spaces $H^{1}_{+}(w)$ is proved. It is here also show that the $H^{1}_{+}(w)$ theory of oscillatory singular integrals above cannot be extended to the case of $H^{q}_{+}(w)$ when $0<q<1$ and $w\in A_{p}^{+}$, a wider weight class than the classical Muckenhoupt class. Furthermore, a criterion on the weighted $L^{p}$-boundednesss of the oscillatory singular integral is given.