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 the stationary phase method in one variable, which is known as a method for an asymptotic expansion of an oscillatory integral of a phase function with a non-degenerate critical point.
The two Fresnel Integrals are real and imaginary part of the integral over complex-valued exp(ix^2) as a function of the upper limit. They are special cases of the integrals over x^m*exp(i*x^n) for integer powers m and n, which are essentially Incomplete Gamma Functions. We generalize one step further and focus on evaluation of the integrals with kernel p(x)*exp[i*phi(x)] and polynomials p and phi. Series reversion of phi seems not to help much, but repeated partial integration leads to a first order differential equation for an auxiliary oscillating function which allows to fuse the integrals…
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…
Special values of the Lommel functions allow the calculation of Fresnel like integrals. These closed form expressions along with their asymptotic values are reported.
In this paper we consider the problem of estimation of oscillatory integrals with Mittag-Leffler functions in two variables. The generalisation is that we replace the exponential function with the Mittag-Leffler-type function, to study oscillatory type integrals.
In this paper we consider the problem on uniform estimates for generalized oscillatory integrals given by Mittag- Leffler functions with the homogeneous polynomial phase. We obtain a variant of Ricci-Stein Lemma and invariant estimates for corresponding integrals.
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.
In this paper, by using asymptotic expansions of oscillatory integrals with positive real power phase functions in one variable, we obtain asymptotic expansions of oscillatory integrals with phase functions expressed by a product of positive real power function and real analytic function in one variable. Moreover we show an example which we can compute all coefficients of terms in asymptotic expansions concretely.
We establish an $L^\infty\times L^2 \to L^2$ norm estimate for a bilinear oscillatory integral along parabolas incorporating oscillatory factors $e^{i|t|^{-β}}$.
This paper is devoted to $L^2$ estimates for trilinear oscillatory integrals of convolution type on $\mathbb{R}^2$. The phases in the oscillatory factors include smooth functions and polynomials. We shall establish sharp $L^2$ decay estimates of trilinear oscillatory integrals with smooth phases, and then give $L^2$ uniform estimates for these integrals with polynomial phases.
We examine a certain class of trilinear integral operators which incorporate oscillatory factors e^{iP}, where P is a real-valued polynomial, and prove smallness of such integrals in the presence of rapid oscillations.
In this paper, we shall prove the uniform sharp $L^p$ decay estimates for a class of oscillatory integral operators with polynomial phases. By this one-dimensional result, we can use the rotation method to obtain uniform sharp $L^p$ estimates of certain higher-dimensional oscillatory integral operators.
We prove that the $L^2$ bound of an oscillatory integral associated with a polynomial depends only on the number of monomials that this polynomial consists of.
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 maximal function of oscillatory integrals and prove a global estimate for radial test functions which is almost sharp with respect to the Sobolev regularity.
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 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.
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 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.
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).
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].
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.
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.
We give an exact result about the asymptotic limit of an oscillatory integral whose phase contains a certain flat term. Corresponding to the real analytic phase case, one can see an essential difference in the behavior of the above oscillatory integral.
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.
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.
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.
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.
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…
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.
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.
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…
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.
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.
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.
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 $\mathbb{R}^3$.
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…
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.
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.
We consider bilinear oscillatory integrals, i.e. pseudo-product operators whose symbol involves an oscillating factor. Lebesgue space inequalities are established, which give decay as the oscillation becomes stronger ; this extends the well-known linear theory of oscillatory integral in some directions. The proof relies on a combination of time-frequency analysis of Coifman-Meyer type with stationary and non-stationary phase estimates. As a consequence of this analysis, we obtain Lebesgue estimates for new bilinear multipliers defined by non-smooth symbols.
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…
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 …
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.
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.
The asymptotic behavior at infinity of oscillatory integrals is in detail investigated by using the Newton polyhedra of the phase and the amplitude. We are especially interested in the case that the amplitude has a zero at a critical point of the phase. The properties of poles of local zeta functions, which are closely related to the behavior of oscillatory integrals, are also studied under the associated situation.
We discuss the asymptotic behaviour for the best constant in L^p-L^q estimates for trigonometric polinomials and for an integral operator which is related to the solution of inhomogeneous Schrodinger equations. This gives us an opportunity to review some basic facts about oscillatory integrals and the method of stationary phase, and also to make some remarks in connection with Strichartz estimates.
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.
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 …
