Journal of Pseudo-differential Operators and Applications 

About: Journal of Pseudo-differential Operators and Applications is an academic journal published by Springer Science+Business Media. The journal publishes majorly in the area(s): Operator theory & Partial differential equation. It has an ISSN identifier of 1662-999X. Over the lifetime, 543 publications have been published receiving 2860 citations. The journal is also known as: JPDOA.

The Bargmann transform on modulation and Gelfand–Shilov spaces, with applications to Toeplitz and pseudo-differential operators

Abstract: We investigate mapping properties for the Bargmann transform on an extended family of modulation spaces whose weights and their reciprocals are allowed to grow faster than exponentials. We prove that this transform is isometric and bijective from modulation spaces to convenient Lebesgue spaces of analytic functions. We use this to prove that such modulation spaces fulfill most of the continuity properties which are valid for modulation spaces with moderate weights. Finally we use the results to establish continuity properties of Toeplitz and pseudo-differential operators on these modulation spaces, and on Gelfand–Shilov spaces.

Images of function and distribution spaces under the bargmann transform

Abstract: We consider a broad family of test function spaces and their dual (distribution) spaces. The family includes Gelfand–Shilov spaces, and a family of test function spaces introduced by Pilipovic. We deduce different characterizations of such spaces, especially under the Bargmann transform and the Short-time Fourier transform. The family also include a test function space, whose dual space is mapped by the Bargmann transform bijectively to the set of entire functions.

Ellipticity, Fredholmness and spectral invariance of pseudo-differential operators on $${{\mathbb S}^1}$$

Abstract: Analogues of pseudo-differential operators of the Hormander class for the unit circle \({{\mathbb S}^1}\) centered at the origin are studied. We prove that elliptic pseudo-differential operators on \({{\mathbb S}^1}\) are Fredholm on \({L^p({\mathbb S}^1),1 < p < \infty.}\) Then we prove the spectral invariance of these operators in \({L^2({\mathbb S}^1)}\) and we use the spectral invariance to prove that ellipticity is a necessary condition for Fredholmness on \({L^2({\mathbb S}^1)}\).

Gevrey smoothing effect of solutions for spatially homogeneous nonlinear Boltzmann equation without angular cutoff

Abstract: There have been extensive studies on the regularizing effect of solutions to the Boltzmann equation without angular cutoff assumption, for both spatially homogeneous and inhomogeneous cases, by noticing the fact that non cutoff Boltzmann collision operator behaves like the fractional power of the Laplace operator. As a further study on the problem in the spatially homogeneous situation, in this paper, we consider the Gevrey regularity of C ∞ solutions with the Maxwellian decay to the Cauchy problem of spatially homogeneous Boltzmann equation for modified hard potentials, by using analytic techniques developed in Alexandre et al. (J Funct Anal 255:2013–2066, 2008; Arch Ration Mech Anal, doi: 10.1007/s00205-010-0290-1 , 2010), Huo et al. (Kinet Relat Models 1:453–489, 2008) and Morimoto et al. (Discrete Contin Dyn Syst Ser A 24:187–212, 2009).

Micro-local analysis in Fourier Lebesgue and modulation spaces: part II

Abstract: We consider different types of (local) products f1 f2 in Fourier Lebesgue spaces. Furthermore, we prove the existence of such products for other distributions satisfying appropriate wave-front properties. We also consider semi-linear equations of the form $$P(x,D)f = G(x,J_k f),$$ with appropriate polynomials P and G, where Jk denotes the k-jet of f. If the solution locally belongs to appropriate weighted Fourier Lebesgue space \({\fancyscript{F}L^q_{(\omega )} (\mathbf{R}^d)}\) and P is non-characteristic at (x0, ξ0), then we prove that \({(x_0,\xi_0) ot \in {\rm WF}_{\fancyscript{F}L^q_{(\widetilde {\omega })}} (f)}\), where \({\widetilde{\omega }}\) depends on ω, P and G.