The Global Wave Front Set of Tempered Oscillatory Integrals with Inhomogeneous Phase Functions

Abstract

Abstract. We study certain families of oscillatory integrals I ϕ (a), parametrised by phase functions ϕ and amplitude functions a globally defined on R d , which give rise to tempered distributions, avoiding the standard homogeneity requirement on the phase function. The singularities of I ϕ (a) are described both from the point of view of the lack of smoothness as well as with respect to the decay at infinity. In particular, the latter will depend on a version of the set of stationary points of ϕ, including elements lying at the boundary of the radial compactification of R d . As applications, we consider some properties of the two-point function of a free, massive, scalar relativistic field and of classes of global Fourier integral operators on R d , with the latter defined in terms of kernels of the form I ϕ (a).

Lagrangian submanifolds at infinity and their parametrization

“…Tempered oscillatory integrals. In this subsection we give a brief summary of the results we obtained in [18]. In that paper we have associated to a given (inhomogeneous) SG-phase function ϕ a family of tempered distributions, denoted by I ϕ (a), parametrized by amplitudes that are SG-symbols and established a bound on their singularities.…”

“…Using the notion of admissible SG-phase function, we can now recall the definition of tempered oscillatory integrals given in [18]. Theorem 1.21.…”

“…In the present paper we follow a notation close to the one used in[28], different from the one we adopted in[18]. In particular, in the original statement of Theorem 1.24 proved there, C ϕ was denoted by M ϕ , andΛ ϕ by SP ϕ , respectively.…”

Fourier integral operators algebra and fundamental solutions to hyperbolic systems with polynomially bounded coefficients on $$\mathbb {R}^n$$ R n

“…To mention a few, see, e.g., Andrews [1], Ruzhansky and Sugimoto [28], Cordero et al [9], and the recent works by Coriasco and Ruzhansky [18], Coriasco and Schulz [19,20]. Concerning applications to SG hyperbolic problems and propagation of singularities, see, e.g., Ascanelli and Cappiello [2][3][4], Cappiello [8], Coriasco et al [13], Coriasco and Maniccia [14].…”

Characterization of the SG-Wave Front Set in Terms of the FBI-Transform

“…The study of such singularities and of associated classes of global Fourier Integral distributions is subject to active research, with several recent contributions, see e.g. [6][7][8]10,15].…”