Toshio Nagano, Naoya Miyazaki
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.
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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…
In this note, we generalize the Fresnel integrals using oscillatory integral, and then we obtain an extention of the stationary phase method.
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.
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。
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。
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…
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…
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.
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。
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。
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.
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。
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.
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 $\mathbb{R}^3$.
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.
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。
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.
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$.
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…
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.
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. We show that there is a sparse bound for the bilinear form $ \langle T_P f, g \rangle$. This in turn easily implies $ A_p $ inequalities. The method of proof is applied in a random discrete setting, yielding the first weighted inequalities for operators defined on sparse sets of integers.
We establish endpoint bounds on a Hardy space $H^1$ for a natural class of multiparameter singular integral operators which do not decay away from the support of rectangular atoms. Hence the usual argument via a Journé-type covering lemma to deduce bounds on product $H^1$ is not valid. We consider the class of multiparameter oscillatory singular integral operators given by convolution with the classical multiple Hilbert transform kernel modulated by a general polynomial oscillation. Various characterisations are known which give $L^2$ (or more generally $L^p, 1<p<\infty$) bounds. Here we initi…
We prove sparse bounds for maximal oscillatory rough singular integral operator $$T^{P}_{Ω,*}f(x):=\sup_{ε>0} \left|\int_{|x-y|>ε}e^{ιP(x,y)}\frac{Ω\big((x-y)/|x-y|\big)}{|x-y|^{n}}f(y)dy\right|,$$ where $P(x,y)$ is a real-valued polynomial on $\mathbb{R}^{n}\times \mathbb{R}^{n}$ and $Ω\in L^{\infty}(\mathbb{S}^{n-1})$ is a homogeneous function of degree zero with $\int_{\mathbb{S}^{n-1}}Ω(θ)~dθ=0$. This allows us to conclude weighted $L^p-$estimates for the operator $T^{P}_{Ω,*}$. Moreover, the norm $\|T^P_{Ω,*}\|_{L^p\rightarrow L^p}$ depends only on the total degree of the polynomial $P(x,…
For polynomial $ P (x,y)$, and any Calderón-Zygmund kernel, $K$, the operator below satisfies a $ (1,r)$ sparse bound, for $ 1< r \leq 2$. $$ \sup _{ε>0} \Bigl\lvert \int_{|y| > ε} f (x-y) e ^{2 πi P (x,y) } K(y) \; dy \Bigr\rvert $$ The implied bound depends upon $ P (x,y)$ only through the degree of $ P$. We derive from this a range of weighted inequalities, including weak type inequalities on $ L ^{1} (w)$, which are new, even in the unweighted case. The unweighted weak-type estimate, without maximal truncations, is due to Chanillo and Christ (1987).
We prove sharp $L^2$ regularity results for classes of strongly singular Radon transfoms on the Heisenberg group by means of oscillatory integrals. We show that the problem in question can be effectively treated by establishing uniform estimates for certain oscillatory integrals whose canonical relations project with two-sided fold singularities; this new approach also allows us to treat operators which are not necessarily translation invariant.
Weak type $(1,1)$ bounds for rough oscillatory singular integrals
本文解决了粗糙核振荡奇异积分弱型(1,1)不等式的长期开放问题，证明当Ω属于Llog⁺L时成立，并进一步给出相关粗糙极大振荡算子的弱型界。
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…
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…
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。
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.
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.
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.
$L^p$ boundness of Oscillatory singular integral with Calderón Type Commutators
本文研究了具有Calderón型交换子的振荡奇异积分算子$T_{P,K,A}$的$L^p$有界性，通过vanishing moment和CZ(δ)条件证明其在$R^n$上的统一有界性，扩展了先前结果。
Multilinear Mikhlin Multipliers with Degenerate Singularities
本文证明了符号沿退化子空间奇点的多重线性 Mikhlin 乘子的 $L^p$ 有界性，通过将非退化理论的波包分析推广到温和退化情形，并指出指数范围由奇点几何决定。
Oscillatory integrals with phases arising from algebraic number fields
本文建立了相位为代数数域上多项式迹的振荡积分理论，并将其应用于代数整数Tarry问题的奇异积分研究。
Regularity of solutions to degenerate and singular free boundary problems with volume constraint
本文研究了在预设正性集体积条件下退化和奇异椭圆自由边界问题解的存在性和正则性。
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.
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 the initial value problem and global properties of solutions are studied for the scalar second order ODE: $ (|u'|^{l}u')' + c|u'|^αu' + d|u|^βu=0$, where $α,β,l,c, d$ are positive constants. In particular, existence, uniqueness and regularity as well as optimal decay rates of solutions to 0 are obtained depending on the various parameters, and the oscillatory or non-oscillatory behavior is elucidated.
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 …
In this paper we consider three types of discrete operators stemming from singular Radon transforms. We first extend an $\ell^p$ result for translation invariant discrete singular Radon transforms to a class of twisted operators including an additional oscillatory component, via a simple method of descent argument. Second, we note an $\ell^2$ bound for quasi-translation invariant discrete twisted Radon transforms. Finally, we extend an existing $\ell^2$ bound for a closely related non-translation invariant discrete oscillatory integral operator with singular kernel to an $\ell^p$ bound for all…
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.
A New Proof About Certain Oscillatory Singular Integrals with Nonstandard Kernel
本文提出新的方法研究非标准核的振荡奇异积分算子T_{Q,A}，在Ω∈L(logL)^2(S^{n-1})和∇A∈BMO(R^n)条件下，证明T_{Q,A}在L^p(R^n)上有统一的有界性，改进并扩展了先前结果。
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