Class Of Fourier Integral Operators Research Articles

H s -Boundedness of a class of Fourier Integral Operators

Abstract In this paper, we define a particular class of Fourier Integral Operators (FIO for short). These FIO turn out to be bounded on the spaces S (ℝ n ) of rapidly decreasing functions (or Schwartz space) and S′ (ℝ n ) of temperate distributions. Results about the composition of FIO with its L 2-adjoint are proved. These allow to obtain results about the continuity on the Sobolev Spaces.

Analysis of resolution of tomographic-type reconstruction from discrete data for a class of distributions * *This work was supported in part by NSF Grants DMS-1615124 and DMS-1906361.

Let f(x), , be a piecewise smooth function with a jump discontinuity across a smooth surface . Let f Λϵ denote the Lambda tomography (LT) reconstruction of f from its discrete Radon data . The sampling rate along each variable is ∼ϵ. First, we compute the limit for a generic . Once the limiting function is known (which we call the discrete transition behavior, or DTB for short), the resolution of reconstruction can be easily found. Next, we show that straight segments of lead to non-local artifacts in f Λϵ , and that these artifacts are of the same strength as the useful singularities of f Λϵ . We also show that f Λϵ (x) does not converge to its continuous analogue f Λ = (−Δ)1/2 f as ϵ → 0 even if . Results of numerical experiments presented in the paper confirm these conclusions. We also consider a class of Fourier integral operators with the same canonical relation as the classical Radon transform adjoint, and a class of distributions , , and obtain easy to use formulas for the DTB when is computed from discrete data . Exact and LT reconstructions are particular cases of this more general theory.

Square function inequality for a class of Fourier integral operators satisfying cinematic curvature conditions

Abstract In this paper, we establish an improved variable coefficient version of the square function inequality, by which the local smoothing estimate L α p → L p {L^{p}_{\alpha}\to L^{p}} for the Fourier integral operators satisfying cinematic curvature condition is further improved. In particular, we establish almost sharp results for 2 < p ⩽ 3 {2<p\leqslant 3} and push forward the estimate for the critical point p = 4 {p=4} . As a consequence, the local smoothing estimate for the wave equation on the manifold is refined. We generalize the results in [S. Lee and A. Vargas, On the cone multiplier in ℝ 3 \mathbb{R}^{3} , J. Funct. Anal. 263 2012, 4, 925–940; J. Lee, A trilinear approach to square function and local smoothing estimates for the wave operator, preprint 2018, https://arxiv.org/abs/1607.08426v5] to its variable coefficient counterpart. The main ingredients in the argument includes multilinear oscillatory integral estimate [J. Bennett, A. Carbery and T. Tao, On the multilinear restriction and Kakeya conjectures, Acta Math. 196 2006, 2, 261–302] and decoupling inequality [D. Beltran, J. Hickman and C. D. Sogge, Variable coefficient Wolff-type inequalities and sharp local smoothing estimates for wave equations on manifolds, Anal. PDE 13 2020, 2, 403–433].

Variable coefficient Wolff-type inequalities and sharp local smoothing estimates for wave equations on manifolds

The sharp Wolff-type decoupling estimates of Bourgain--Demeter are extended to the variable coefficient setting. These results are applied to obtain new sharp local smoothing estimates for wave equations on compact Riemannian manifolds, away from the endpoint regularity exponent. More generally, local smoothing estimates are established for a natural class of Fourier integral operators; at this level of generality the results are sharp in odd dimensions, both in terms of the regularity exponent and the Lebesgue exponent.

A local-to-global boundedness argument and Fourier integral operators

We give a criterion for the global boundedness of integral operators which are known to be locally bounded. As an application, we discuss the global $L^p$-boundedness for a class of Fourier integral operators. While the local $L^p$-boundedness of Fourier integral operators is known from the work of Seeger, Sogge and Stein, not so many results are available for the global boundedness on $L^p({\mathbb R}^n)$. We give several natural sufficient conditions for them.

On Fourier integral operators with Hölder-continuous phase

We study continuity properties in Lebesgue spaces for a class of Fourier integral operators arising in the study of the Boltzmann equation. The phase has a Hölder-type singularity at the origin. We prove boundedness in [Formula: see text] with a precise loss of decay depending on the Hölder exponent, and we show by counterexamples that a loss occurs even in the case of smooth phases. The results can be seen as a quantitative version of the Beurling–Helson theorem for changes of variables with a Hölder singularity at the origin. The continuity in [Formula: see text] is studied as well by providing sufficient conditions and relevant counterexamples. The proofs rely on techniques from time-frequency analysis.

Approximate Green’s function for the conductivity equation with conormal coefficient

We construct an approximate Green’s function for \(L_{\gamma }:={\nabla } \cdot \gamma (x) {\nabla } u(x)\), which belongs to a class of Fourier integral operators (FIOs) associated to two canonical relations. This leads to analysis of the composition of two FIOs, associated to a canonical relation with a zero section problem. The resulting composition is a sum of two FIOs, each associated to two intersecting canonical relations.

On the unboundedness of a class of Fourier integral operators on [formula omitted

In this paper, we give an example of a Fourier integral operator with a symbol belonging to ⋂0<ρ<1Sρ,10 that cannot be extended as a bounded operator on L2(Rn).

Lebesgue space estimates for a class of Fourier integral operators associated with wave propagation

AbstractWe prove Lq estimates related to Sogge's conjecture for a class of Fourier integral operators associated with wave equations.

Smoothing properties of evolution equations via canonical transforms and comparison principle

This paper describes a new approach to global smoothing problems for dispersive and non-dispersive evolution equations based on the global canonical transforms and the underlying global microlocal analysis. For this purpose, the Egorov-type theorem is established with canonical transformations in the form of a class of Fourier integral operators, and their weighted L-2-boundedness properties are derived. This allows us to globally reduce general dispersive equations to normal forms in one or two dimensions. Then, a new comparison principle for evolution equations is introduced. In particular, it allows us to relate different smoothing estimates by comparing certain expressions involving their symbols. As a result, it is shown that the majority of smoothing estimates for different equations are equivalent to each other. Moreover, new estimates as well as several refinements of known results are obtained. The proofs are considerably simplified. A comprehensive analysis is presented for smoothing estimates for dispersive equations. Applications are given to the detailed description of smoothing properties of the Schrodinger, relativistic Schrodinger, wave, Klein-Gordon and other equations.

Weighted Sobolev L2 estimates for a class of Fourier integral operators

In this paper we develop elements of the global calculus of Fourier integral operators in under minimal decay assumptions on phases and amplitudes. We also establish global weighted Sobolev L2 estimates for a class of Fourier integral operators that appears in the analysis of global smoothing problems for dispersive partial differential equations. As an application, we exhibit a new type of weighted estimates for hyperbolic equations, where the decay of data in space is quantitatively translated into the time decay of solutions.

On the global boundedness of Fourier integral operators

We consider a class of Fourier integral operators, globally defined on R d , with symbolsandphasessatisfyingproducttypeestimates(theso-called SGorscatteringclasses). We prove a sharp continuity result for such operators when acting on the modulation spaces M p . The minimal loss of derivatives is shown to be d|1/2 − 1/ p|. This global perspective produces a loss of decay as well, given by the same order. Strictly related, striking examples of unboundedness on L p spaces are presented.

A class of Fourier integral operators with complex phase related to the Gevrey classes

In this paper we discuss about Fourier integral operators with complex phase functions belonging to \({S^{\kappa}_{\rho,\delta}, 0 < \delta < \rho\leq 1, 0 < \kappa < \rho-\delta}\) where the positivity of imaginary part of the phase functions is not required. In particular we prove composition formulae for 0 and 1 quantization of Fourier integral operators with phase \({\phi}\) and \({-\phi}\). These results are applied to reduce the Cauchy problem for noneffectively hyperbolic operators in the Gevrey classes to the Cauchy problem in Sobolev spaces.

A Mathematical Justification for the Herman-Kluk Propagator

A class of Fourier Integral Operators which converge to the unitary group of the Schrodinger equation in the semiclassical limit e → 0 in the uniform operator norm is constructed. The convergence allows for an error bound of order O(e), which can be improved to arbitrary order in e upon the introduction of corrections in the symbol. On the Ehrenfest-timescale, the result holds with a slightly weaker error bound. In the chemical literature the approximation is known as the Herman-Kluk propagator.

Representation of Fourier Integral Operators Using Shearlets

Traditional methods of time-frequency and multiscale analysis, such as wavelets and Gabor frames, have been successfully employed for representing most classes of pseudodifferential operators. However, these methods are not equally effective in dealing with Fourier Integral Operators in general. In this article, we show that the shearlets, recently introduced by the authors and their collaborators, provide very efficient representations for a large class of Fourier Integral Operators. The shearlets are an affine-like system of well-localized waveforms at various scales, locations and orientations, which are particularly efficient in representing anisotropic functions. Using this approach, we prove that the matrix representation of a Fourier Integral Operator with respect to a Parseval frame of shearlets is sparse and well-organized. This fact recovers a similar result recently obtained by Candes and Demanet using curvelets, which illustrates the benefits of directional multiscale representations (such as curvelets and shearlets) in the study of those functions and operators where traditional multiscale methods are unable to provide the appropriate geometric analysis in the phase space.

Linearized Zverev Transform and its application for modeling radio occultations

The multiple phase screens technique is often used for modeling wave propagation and radio occultation sounding of the atmosphere. The last step of this procedure is the propagation from the last phase screen to the observation orbit of the spaceborne receiver. This step was formerly performed by the computation of multiple diffractive integrals, which impairs the numerical efficiency of the algorithm. We introduce an asymptotic method of wave propagation in vacuum from the last phase screen to a generic observation path. The exact solution is written in the form of the Zverev Transform, which belongs to the class of Fourier Integral Operators (FIO). The phase function of such an operator can be derived from the geometric optical equations. We construct an approximation for the Zverev Transform, utilizing the linearization of the equation of the geometric optical ray propagation in the vicinity of a smooth model of the ray structure. This permits the design of a fast numerical algorithm based on an FFT. Numerical simulations are performed for microwave propagation through a model atmosphere based on realistic gridded global fields of meteorological parameters. The new method based on the Linearized Zverev Transform is compared with the standard combination of multiple phase screens and diffractive integrals and with the asymptotic forward modeling for the propagation of centimeter waves in the atmosphere on limb paths. This method is especially useful for modeling LEO–LEO microwave occultations at frequencies of 9–30 GHz. The simulations demonstrate a high accuracy and numerical efficiency of the proposed method.

Elements of functional calculus andL 2 regularity for some classes of fourier integral operators

We employ the method of slices to develop a rudimentary calculus describing the nature of operators T*T (respectively, TT*), where T are Fourier integral operators with one-sided right (respectively, left) singularities; this idea has its roots in the work of Greenleaf and Seeger. Such a result allows us to reduce the L2 regularity problem for operators in n dimensions with one-sided singularities to the L2 regularity problem for operators with two-sided singularities in n − 1 dimensions. As a consequence we deduce almost sharp L2-Sobolev estimates for operators in three-dimensions; an interesting special case is provided by certain restricted X-ray transforms associated to line complexes which are not well curved. We also provide a proof of almost-sharpness by looking at a restricted X-ray transform associated to the line complex generated by the curve t → (t, tk). Appropriate notions of singularity, strong singularity, and type are also developed.

Global L 2-Boundedness Theorems for a Class of Fourier Integral Operators

ABSTRACT The local L 2-mapping property of Fourier integral operators has been established in Hörmander (1971) and in Eskin (1970). In this article, we treat the global L 2-boundedness for a class of operators that appears naturally in many problems. As a consequence, we improve known global results for several classes of pseudodifferential and Fourier integral operators, as well as extend previous results of Asada and Fujiwara (1978) or Kumano-go (1976). As an application, we show a global smoothing estimate for generalized Schrödinger equations which extends the results of Ben-Artzi and Devinatz (1991) and Walther (1999); (2002).

Fourier integral operators of infinite order and applications to SG-hyperbolic equations

In this work, we develop a global calculus for a class of Fourier integral operators with symbols $a(x, \xi)$ having exponential growth in $R_{x,\xi}^{2n}$. The functional frame is given by the spaces of type S of Gelfand and Shilov. As an application, we construct a parametrix and prove the existence of a solution for the Cauchy problem associated to SG-hyperbolic operators with one characteristic of constant multiplicity.

Wave Front Set at Infinity and Hyperbolic Linear Operators with Multiple Characteristics

We discuss a notion of wave front set which allows us tocontrol the behaviour `at infinity' of temperate distributions. Weobtain the microlocality and microellipticity properties with respect toa class of global pseudodifferential operators and a propagation theoremfor the corresponding class of Fourier Integral Operators. Through theseresults, we prove an adapted global version of the classical theoremconcerning the singularities of solutions of hyperbolic Cauchy problemsfor linear operators with multiple characteristics of constantmultiplicities. Finally, we make a comparison with the scattering wavefront set introduced by Melrose.