The global wave front set of tempered oscillatory integrals with inhomogeneous phase functions

Abstract

We study certain families of oscillatory integrals $I_{\phi }(a)$, parametrised by phase functions $\phi$ and amplitude functions $a$ globally defined on ${\mathbb{R}}^d$, which give rise to tempered distributions, avoiding the standard homogeneity requirement on the phase function. The singularities of $I_{\phi }(a)$ 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 $\phi$, including elements lying at the boundary of the radial compactification of ${\mathbb{R}}^d$. 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 ${\mathbb{R}}^d$, with the latter defined in terms of kernels of the form $I_{\phi }(a)$.

30 pages, 2 figures, mistakes and typos correction

References in corpus (5)

• Global Lp continuity of Fourier integral operators
• Global wave-front sets of Banach, Fr{é}chet and Modulation space types, and pseudo-differential operators
• Wodzicki Residue for Operators on Manifolds with Cylindrical Ends
• On the Spectral Asymptotics of Operators on Manifolds with Ends
• The wave front set of oscillatory integrals with inhomogeneous phase function

Cited by in corpus (6)

• On the characterisations of wave front sets via the short-time Fourier transform
• Calculus, continuity and global wave-front properties for Fourier integral operators on ${\mathbf{R}}^d$
• Microlocal analysis of quasianalytic Gelfand-Shilov type ultradistributions
• Fourier integral operators algebra and fundamental solutions to hyperbolic systems with polynomially bounded coefficients on R^n
• Calculus for Fourier Integral Operators in generalized SG classes
• SG-Lagrangian submanifolds and their parametrization