Szegö’s Theorem and a Complete Symbolic Calculus for Pseudo-Differential Operators

1979 ◽  
pp. 261-283 ◽  
Author(s):  
Harold Widom
Keyword(s):  
Differential Operators ◽  
Symbolic Calculus ◽  
Pseudo Differential Operators

Symbolic Calculus and Boundedness of Multi-parameter and Multi-linear Pseudo-differential Operators

2014 ◽  
Vol 14 (4) ◽  
Author(s):  
Qing Hong ◽  
Guozhen Lu
Keyword(s):  
Partial Differential Equations ◽  
Differential Operators ◽  
Complex Variables ◽  
Several Complex Variables ◽  
Regularity Of Solutions ◽  
Symbolic Calculus ◽  
Pseudo Differential Operators ◽  
Analysis Theory ◽  
Modern Analysis ◽  
Satisfactory Theory

AbstractSince the work of Hörmander on linear pseudo-differential operators, the applications of pseudo-differential operators have played an important role in partial differential equations, harmonic analysis, theory of several complex variables and other branches of modern analysis (e.g., they are used to construct parametrices and establish the regularity of solutions to PDEs such as the ∂̅ problem, etc.). The work of Coifman and Meyer on multi-linear Fourier multipliers and pseudo-differential operators has stimulated further such applications. In [2], the authors developed a fairly satisfactory theory of symbolic calculus for multi-linear pseudo-differential operators. Motivated by this work [2] and L

scholarly journals Operator Symbols and Operator Indices

Symmetry ◽  
2019 ◽  
Vol 12 (1) ◽  
pp. 64
Author(s):  
Vladimir Vasilyev
Keyword(s):  
Banach Spaces ◽  
Differential Operators ◽  
Index Theorem ◽  
Symbolic Calculus ◽  
Smooth Manifolds ◽  
Bounded Operators ◽  
Pseudo Differential Operators ◽  
Operator Symbols ◽  
Special Classes

We suggest a certain variant of symbolic calculus for special classes of linear bounded operators acting in Banach spaces. According to the calculus we formulate an index theorem and give applications to elliptic pseudo-differential operators on smooth manifolds with non-smooth boundaries.

scholarly journals Recent developments on Pseudo-Differential Operators (I)

2015 ◽  
Vol 46 (1) ◽  
pp. 1-30 ◽  
Author(s):  
D.-C. Chang ◽  
W. RUNGROTTHEERA ◽  
B.-W. SCHULZE
Keyword(s):  
Boundary Value Problems ◽  
Differential Calculus ◽  
Differential Operators ◽  
Elliptic Boundary ◽  
Higher Order ◽  
Symbolic Calculus ◽  
Elliptic Boundary Value ◽  
The Past ◽  
Pseudo Differential Operators ◽  
Recent Developments

In recent years the analysis of (pseudo-)differential operators on manifolds with second and higher order corners made considerable progress, and essential new structures have been developed. The main objective of this series of paper is to give a survey on the development of this theory in the past twenty years. We start with a brief background of the theory of pseudo-differential operators which including its symbolic calculus on $\R^n$. Next we introduce pseudo-differential calculus with operator-valued symbols. This allows us to discuss elliptic boundary value problems on smooth domains in $\R^n$ and elliptic problems on manifolds. This paper is based on the first part of lectures given by the authors while they visited the National Center for Theoretical Sciences in Hsinchu, Taiwan during May-July of 2014.

scholarly journals A Radial Basis Function Finite Difference Scheme for the Benjamin–Ono Equation

Mathematics ◽  
2020 ◽  
Vol 9 (1) ◽  
pp. 65
Author(s):  
Benjamin Akers ◽  
Tony Liu ◽  
Jonah Reeger
Keyword(s):  
Radial Basis Function ◽  
Hilbert Transform ◽  
Basis Function ◽  
Differential Operators ◽  
Convergence Rates ◽  
Traveling Wave Solutions ◽  
Algebraic Decay ◽  
Pseudo Differential Operators ◽  
Radial Basis ◽  
The Hilbert Transform

A radial basis function-finite differencing (RBF-FD) scheme was applied to the initial value problem of the Benjamin–Ono equation. The Benjamin–Ono equation has traveling wave solutions with algebraic decay and a nonlocal pseudo-differential operator, the Hilbert transform. When posed on R, the former makes Fourier collocation a poor discretization choice; the latter is challenging for any local method. We develop an RBF-FD approximation of the Hilbert transform, and discuss the challenges of implementing this and other pseudo-differential operators on unstructured grids. Numerical examples, simulation costs, convergence rates, and generalizations of this method are all discussed.

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