scholarly journals On the geometry of $Diff(S^1)$-pseudodifferential operators based on renormalized traces.

2021 ◽  
Vol 14 (1) ◽  
pp. 19-47
Author(s):  
Jean-Pierre Magnot
Keyword(s):  
General Linear Group ◽  
Differential Operators ◽  
Pseudodifferential Operators ◽  
Integral Operators ◽  
Fourier Integral ◽  
Riemannian Metric ◽  
Fourier Integral Operators ◽  
Pseudo Differential Operators ◽  
Relationship Of ◽  
The Relationship

In this article, we examine the geometry of a group of Fourier-integral operators, which is the central extension of $Diff(S^1)$ with a group of classical pseudo-differential operators of any order. Several subgroups are considered, and the corresponding groups with formal pseudodifferential operators are defined. We investigate the relationship of this group with the restricted general linear group $GL_{res}$, we define a right-invariant pseudo-Riemannian metric on it that extends the Hilbert-Schmidt Riemannian metric by the use of renormalized traces of pseudo-differential operators, and we describe classes of remarkable connections.

scholarly journals L p boundedness of rough bi-parameter Fourier integral operators

2018 ◽  
Vol 30 (1) ◽  
pp. 87-107 ◽  
Author(s):  
Qing Hong ◽  
Guozhen Lu ◽  
Lu Zhang
Keyword(s):  
Compact Support ◽  
Phase Function ◽  
Differential Operators ◽  
Integral Operators ◽  
Fourier Integral ◽  
Fourier Multipliers ◽  
Fourier Integral Operators ◽  
Pseudo Differential Operators ◽  
Degeneracy Condition ◽  
Phase Functions

Abstract In this paper, we will investigate the boundedness of the bi-parameter Fourier integral operators (or FIOs for short) of the following form: T(f\/)(x)=\frac{1}{(2\pi)^{2n}}\int_{\mathbb{R}^{2n}}e^{i\varphi(x,\xi,\eta)}% \cdot a(x,\xi,\eta)\cdot\widehat{f}(\xi,\eta)\,d\xi\,d\eta, where {x=(x_{1},x_{2})\in\mathbb{R}^{n}\times\mathbb{R}^{n}} and {\xi,\eta\in\mathbb{R}^{n}\setminus\{0\}} , {a(x,\xi,\eta)\in L^{\infty}BS^{m}_{\rho}} is the amplitude, and the phase function is of the form \varphi(x,\xi,\eta)=\varphi_{1}(x_{1},\xi\/)+\varphi_{2}(x_{2},\eta) , with \varphi_{1},\varphi_{2}\in L^{\infty}\Phi^{2}(\mathbb{R}^{n}\times\mathbb{R}^{% n}\setminus\{0\}) , and satisfies a certain rough non-degeneracy condition (see (2.2)). The study of these operators are motivated by the {L^{p}} estimates for one-parameter FIOs and bi-parameter Fourier multipliers and pseudo-differential operators. We will first define the bi-parameter FIOs and then study the {L^{p}} boundedness of such operators when their phase functions have compact support in frequency variables with certain necessary non-degeneracy conditions. We will then establish the {L^{p}} boundedness of the more general FIOs with amplitude {a(x,\xi,\eta)\in L^{\infty}BS^{m}_{\rho}} and non-smooth phase function {\varphi(x,\xi,\eta)} on x satisfying a rough non-degeneracy condition.

Extrapolation to weighted Morrey spaces with variable exponents and applications

2021 ◽  
pp. 1-26
Author(s):  
Kwok-Pun Ho
Keyword(s):  
Differential Operators ◽  
Integral Operators ◽  
Fourier Integral ◽  
Morrey Spaces ◽  
Fourier Integral Operators ◽  
Variable Exponents ◽  
Weighted Morrey Spaces ◽  
Pseudo Differential Operators ◽  
Extrapolation Theory

Abstract This paper establishes the mapping properties of pseudo-differential operators and the Fourier integral operators on the weighted Morrey spaces with variable exponents and the weighted Triebel–Lizorkin–Morrey spaces with variable exponents. We obtain these results by extending the extrapolation theory to the weighted Morrey spaces with variable exponents. This extension also gives the mapping properties of Calderón–Zygmund operators on the weighted Hardy–Morrey spaces with variable exponents and the wavelet characterizations of the weighted Hardy–Morrey spaces with variable exponents.

scholarly journals On the nuclear trace of Fourier integral operators

2019 ◽  
Vol 37 (2) ◽  
pp. 219-249
Author(s):  
Duván Cardona
Keyword(s):  
Symmetric Spaces ◽  
Differential Operators ◽  
Integral Operators ◽  
Fourier Integral ◽  
Fourier Integral Operators ◽  
Weyl Quantization ◽  
Homogeneous Manifolds ◽  
Dimensional Torus ◽  
Lebesgue Spaces ◽  
Pseudo Differential Operators

In this paper we characterise ther-nuclearity of Fourier integraloperators on Lebesgue spaces. Fourier integral operators will be consideredinRn,the discrete groupZn,then-dimensional torus and symmetric spaces(compact homogeneous manifolds). We also give formulae forthe nucleartrace of these operators. Explicit examples will be given onZn,the torusTn,the special unitary group SU(2),and the projective complex planeCP2.Ourmain theorems will be applied to the characterization ofr-nuclear pseudo-differential operators defined by the Weyl quantization procedure.

Boundedness of multi-parameter pseudo-differential operators on multi-parameter local Hardy spaces

2020 ◽  
Vol 32 (4) ◽  
pp. 919-936 ◽  
Author(s):  
Jiao Chen ◽  
Wei Ding ◽  
Guozhen Lu
Keyword(s):  
Hardy Spaces ◽  
Differential Operators ◽  
Integral Operators ◽  
Fourier Multipliers ◽  
Regularity Of Solutions ◽  
Local Version ◽  
Fourier Integral Operators ◽  
Pseudo Differential Operators ◽  
Local Hardy Spaces ◽  
The One

AbstractAfter the celebrated work of L. Hörmander on the one-parameter pseudo-differential operators, the applications of pseudo-differential operators have played an important role in partial differential equations, geometric analysis, harmonic analysis, theory of several complex variables and other branches of modern analysis. For instance, they are used to construct parametrices and establish the regularity of solutions to PDEs such as the {\overline{\partial}} problem. The study of Fourier multipliers, pseudo-differential operators and Fourier integral operators has stimulated further such applications. It is well known that the one-parameter pseudo-differential operators are {L^{p}({\mathbb{R}^{n}})} bounded for {1<p<\infty}, but only bounded on local Hardy spaces {h^{p}({\mathbb{R}^{n}})} introduced by Goldberg in [D. Goldberg, A local version of real Hardy spaces, Duke Math. J. 46 1979, 1, 27–42] for {0<p\leq 1}. Though much work has been done on the {L^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})} boundedness for {1<p<\infty} and Hardy {H^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})} boundedness for {0<p\leq 1} for multi-parameter Fourier multipliers and singular integral operators, not much has been done yet for the boundedness of multi-parameter pseudo-differential operators in the range of {0<p\leq 1}. The main purpose of this paper is to establish the boundedness of multi-parameter pseudo-differential operators on multi-parameter local Hardy spaces {h^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})} for {0<p\leq 1} recently introduced by Ding, Lu and Zhu in [W. Ding, G. Lu and Y. Zhu, Multi-parameter local Hardy spaces, Nonlinear Anal. 184 2019, 352–380].

scholarly journals On the Spectral Asymptotics of Operators on Manifolds with Ends

2013 ◽  
Vol 2013 ◽  
pp. 1-21 ◽  
Author(s):  
Sandro Coriasco ◽  
Lidia Maniccia
Keyword(s):  
Asymptotic Behaviour ◽  
Pseudodifferential Operators ◽  
Counting Function ◽  
Integral Operators ◽  
Fourier Integral ◽  
Spectral Asymptotics ◽  
Fourier Integral Operators ◽  
Noncompact Manifolds ◽  
Adjoint Operators

We deal with the asymptotic behaviour, forλ→+∞, of the counting functionNP(λ)of certain positive self-adjoint operatorsPwith double order(m,μ),m,μ>0,  m≠μ, defined on a manifold with endsM. The structure of this class of noncompact manifolds allows to make use of calculi of pseudodifferential operators and Fourier integral operators associated with weighted symbols globally defined onℝn. By means of these tools, we improve known results concerning the remainder terms of the Weyl Formulae forNP(λ)and show how their behaviour depends on the ratiom/μand the dimension ofM.

scholarly journals Generalized oscillatory integrals and Fourier integral operators

2009 ◽  
Vol 52 (2) ◽  
pp. 351-386 ◽  
Author(s):  
Claudia Garetto ◽  
Günther Hörmann ◽  
Michael Oberguggenberger
Keyword(s):  
Differential Operators ◽  
Integral Operators ◽  
Fourier Integral ◽  
Generalized Functions ◽  
Oscillatory Integrals ◽  
Fourier Integral Operators ◽  
Partial Differential Operators ◽  
Colombeau Algebras ◽  
Smooth Coefficients ◽  
Phase Functions

AbstractIn this paper, a theory is developed of generalized oscillatory integrals (OIs) whose phase functions and amplitudes may be generalized functions of Colombeau type. Based on this, generalized Fourier integral operators (FIOs) acting on Colombeau algebras are defined. This is motivated by the need for a general framework for partial differential operators with non-smooth coefficients and distribution dataffi The mapping properties of these FIOs are studied, as is microlocal Colombeau regularity for OIs and the influence of the FIO action on generalized wavefront sets.

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