1,720,991 research outputs found
SG-pseudodifferential operators and Gelfand-Shilov spaces
No full textIn this paper, we study a class of pseudo-differential operators of SG type in the functional setting of the Gelfand-Shilov spaces . As an application we prove a result of hypoellipticity in the same classes. In the last of the paper, we define a notion of wave front set for tempered ultradistributions which allows to describe both the local regularity and the behaviour at infinity of the elements of the dual space
Pseudo differential operators with multiple characteristics and Gevrey singularities
No full textLet be a classical analytic pseudodifferential operator with principal symbol in some cone for a fixed , where is an elliptic symbol, homogeneous of order , and the first order symbol is real-valued and of principal type, i.e., never vanishes and is not parallel to . Suppose that , where is a classical analytic pseudodifferential operator with principal symbol and are classical analytic pseudodifferential operators of order , . Let , , be the Gevrey class, let , and let be the -singular support of . If , then means that and is the factor space
Semilinear pseudo-differential equations and travelling waves
No full textWe consider semilinear perturbations of the SG-elliptic pseudodifferential equations. We prove for them a theorem on the regularity of the solutions in the functional frame of the Gelfand-Shilov classes. Applications are given to the study of travelling solitary waves
Exponential decay and regularity for SG-elliptic operators with polynomial coefficients
No full textWe study the exponential decay and the regularity for solutions of elliptic partial differential equations ,
globally defined in In particular, we consider linear operators with
polynomial coefficients which are SG-elliptic at infinity. Starting from in the so-called Gelfand-Shilov spaces, the solutions of the equation are proved to belong to the same classes.
Proofs are based on a priori estimates and arguments on the Newton polyhedron associated to the operator
Gelfand-Shilov spaces, pseudo-differential operators and localization operators
No full textWe present new results concerning pseudo-differential operators in the function spaces of Gelfand and Shilov. In particular we discuss -regularity of solutions to SG-elliptic pseudo-differential equations, allowing lower order semilinear perturbations.
The results apply to SG-elliptic partial differential equations with polynomial coefficients. We also study the action of Weyl operators and localization operators on $S^{\mu}_{\mu}(\R^n).
Super-exponential decay and holomorphic extensions for semilinar equations with polynomial coefficients
No full textWe show that all eigenfunctions
of linear partial differential operators in with polynomial coefficients of Shubin type are extended to entire functions in \C^n of finite exponential type
and decay like for in conic neighbourhoods of the form
. We also show that under semilinear polynomial perturbations all nonzero
homoclinics %(if exist)
keep the super-exponential decay of the above type, whereas a loss of the
holomorphicity occurs, namely we show holomorphic extension into
a strip \{z\in \C^n: \, |\Im z| \leq T\} for some .
The proofs are based on geometrical and perturbative methods in
Gelfand--Shilov spaces.
The results apply in particular to semilinear Schr\"{o}dinger equations of the form
\begin{equation}
\label{schro}
-\Delta u + |x|^2u -\lambda u = F(x,u, \nabla u).
\end{equation}
Our estimates on homoclinics are sharp. In fact, we
exhibit examples of solutions of \eqref{schro} with super-exponential decay, which are
meromorphic functions, the key point of our argument being the celebrated great
Picard theorem in complex analysis
An Introduction to the Gabor Wave Front Set
Full text linkIn this expository note we present an introduction to the Gabor wave front set. As is often the case, this tool in microlocal analysis has been introduced and reinvented in different forms which turn out to be equivalent or intimately related. We provide a short review of the history of this notion and then focus on some recent variations inspired by function spaces in time-frequency analysis. Old and new results are presented, together with a number of concrete examples and applications to the problem of propagation of singularities
- …
