scholarly journals Propagation of singularities of solutions for pseudodifferential operators with multiple characteristics and their local solvability

1976 ◽  
Vol 52 (2) ◽  
pp. 49-51
Author(s):  
Sunao Ōuchi
Keyword(s):  
Pseudodifferential Operators ◽  
Local Solvability ◽  
Multiple Characteristics ◽  
Propagation Of Singularities

scholarly journals Local solvability and hypoellipticity for pseudodifferential operators of Egorov type with infinite degeneracy

1995 ◽  
Vol 139 ◽  
pp. 151-171 ◽  
Author(s):  
Yoshinori Morimoto
Keyword(s):  
Pseudodifferential Operator ◽  
Infinite Order ◽  
Pseudodifferential Operators ◽  
Adjoint Operator ◽  
Local Solvability ◽  
Subelliptic Operators ◽  
Subelliptic Operator

Let P be a pseudodifferential operator of the formwhere s, b ≥ 0 are even integers and is odd function with f′(t) > 0 (t ≠ 0). Here . We shall call P an operator of Egorov type because P with f(t) = tk, (k odd) is an important model of subelliptic operators studied by Egorov [1] and Hörmander [3], [4, Chapter 27]. Roughly speaking, any subelliptic operator can be reduced to this operator or Mizohata one after several steps of microlocalization arguments. In this paper we shall study the hypoellipticity of P and the local solvability of adjoint operator P* in the case where f(t) vanishes infinitely at the origin and moreover consider the case where ts and are replaced by functions with zero of infinite order.

scholarly journals MICROLOCAL ANALYSIS OF GENERALIZED FUNCTIONS: PSEUDODIFFERENTIAL TECHNIQUES AND PROPAGATION OF SINGULARITIES

2005 ◽  
Vol 48 (3) ◽  
pp. 603-629 ◽  
Author(s):  
Claudia Garetto ◽  
Günther Hörmann
Keyword(s):  
Pseudodifferential Operators ◽  
Hyperbolic Equations ◽  
Regularity Theory ◽  
Regularity Result ◽  
Generalized Functions ◽  
Generalized Solutions ◽  
Original Definition ◽  
Characteristic Sets ◽  
Propagation Of Singularities ◽  
Definition Of

AbstractWe characterize microlocal regularity, in the $\mathcal{G}^{\infty}$-sense, of Colombeau generalized functions by an appropriate extension of the classical notion of micro-ellipticity to pseudodifferential operators with slow-scale generalized symbols. Thus we obtain an alternative, yet equivalent, way of determining generalized wavefront sets that is analogous to the original definition of the wavefront set of distributions via intersections over characteristic sets. The new methods are then applied to regularity theory of generalized solutions of (pseudo)differential equations, where we extend the general non-characteristic regularity result for distributional solutions and consider propagation of $\mathcal{G}^{\infty}$-singularities for homogeneous first-order hyperbolic equations.

Stationary phase and microlocal analysis

2017 ◽  
Author(s):  
Christopher D. Sogge
Keyword(s):  
Stationary Phase ◽  
Wave Front ◽  
Pseudodifferential Operators ◽  
Microlocal Analysis ◽  
Equivalent Definition ◽  
Method Of Stationary Phase ◽  
Wave Front Sets ◽  
Propagation Of Singularities ◽  
Definition Of

This chapter discusses basic techniques from the theory of stationary phase. After giving an overview of the method of stationary phase, the chapter moves on to a discussion of pseudodifferential operators, by going over the basics from the calculus of pseudodifferential operators and their various microlocal properties, in the process obtaining an equivalent definition of wave front sets, before defining pseudodifferential operators on manifolds and going over some of their properties. The chapter then lays out the propagation of singularities as well as Egorov's theorem, which involves conjugating pseudodifferential operators. Finally, this chapter describes the Friedrichs quantization, and differentiates it from the Kohn-Nirenberg quantization presented earlier in the chapter.

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