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.

Weighted Lp Boundedness of Pseudodifferential Operators and Applications

2012 ◽  
Vol 55 (3) ◽  
pp. 555-570 ◽  
Author(s):  
Nicholas Michalowski ◽  
David J. Rule ◽  
Wolfgang Staubach
Keyword(s):  
Pseudodifferential Operator ◽  
Wide Class ◽  
Pseudodifferential Operators ◽  
Maximal Functions ◽  
Bounded Mean Oscillation ◽  
Weighted Norm ◽  
Weighted Norm Inequalities ◽  
Norm Inequalities ◽  
Sharp Function ◽  
Mean Oscillation

AbstractIn this paper we prove weighted norm inequalities with weights in the Ap classes, for pseudodifferential operators with symbols in the class that fall outside the scope of Calderón– Zygmund theory. This is accomplished by controlling the sharp function of the pseudodifferential operator by Hardy–Littlewood type maximal functions. Our weighted norm inequalities also yield Lp boundedness of commutators of functions of bounded mean oscillation with a wide class of operators in .

scholarly journals Regularized functional calculi, semigroups, and cosine functions for pseudodifferential operators

1997 ◽  
Vol 2 (1-2) ◽  
pp. 121-136 ◽  
Author(s):  
Ralph Delaubenfels ◽  
Yansong Lei
Keyword(s):  
Pseudodifferential Operator ◽  
Functional Calculus ◽  
Pseudodifferential Operators ◽  
Explicit Representation ◽  
Stieltjes Transform ◽  
Strongly Continuous Groups ◽  
Regularized Semigroups ◽  
Cosine Functions

LetiAj(1≤j≤n)be generators of commuting bounded strongly continuous groups,A≡(A1,A2,…,An). We show that, whenfhas sufficiently many polynomially bounded derivatives, then there existk,r>0such thatf(A)has a(1+|A|2)−r-regularizedBCk(f(Rn))functional calculus. This immediately produces regularized semigroups and cosine functions with an explicit representation; in particular, whenf(Rn)⫅R, then, for appropriatek,r,t↦(1−it)−ke−itf(A)(1+|A|2)−ris a Fourier-Stieltjes transform, and whenf(Rn)⫅[0,∞), thent↦(1+t)−ke−tf(A)(1+|A|2)−ris a Laplace-Stieltjes transform. WithA≡i(D1,…,Dn),f(A)is a pseudodifferential operator onLp(Rn)(1≤p<∞)orBUC(Rn).

Some results on hypoelliptic pseudo-differential operators

1970 ◽  
Vol 68 (3) ◽  
pp. 685-695
Author(s):  
Robert J. Elliott
Keyword(s):  
Differential Operator ◽  
Differential Operators ◽  
Normal Operator ◽  
A Priori ◽  
Adjoint Operator ◽  
Elliptic Operators ◽  
Pseudo Differential Operator ◽  
Principal Type ◽  
Subelliptic Operators ◽  
Pseudo Differential Operators

In this paper, by extending the results of Yoshikawa (8), we obtain local a priori inequalities for hypoelliptic pseudo-differential operators. Using these inequalities we then show how the results of Hormander ((3), Theorem 8·7·2), on the solvability of the adjoint operator of a principally normal operator can be extended to the adjoint operator of a hypoelliptic pseudo-differential operator. Finally, we consider a class of operators which satisfy more particular a priori inequalities and we show that these operators are hypoelliptic. This class of operators was studied by Egorov (1), and he shows them to be ‘of principal type’. They include elliptic operators and also the subelliptic operators of Hormander (4).

scholarly journals HILBERT MODULAR PSEUDODIFFERENTIAL OPERATORS OF MIXED WEIGHT

2003 ◽  
Vol 46 (2) ◽  
pp. 269-277 ◽  
Author(s):  
Min Ho Lee ◽  
Hyo Chul Myung
Keyword(s):  
Pseudodifferential Operator ◽  
Modular Forms ◽  
Pseudodifferential Operators ◽  
Discrete Subgroup ◽  
Subject Classification ◽  
Sufficient Condition ◽  
Hilbert Modular Forms ◽  
Necessary And Sufficient Condition ◽  
Hilbert Modular ◽  
Necessary And Sufficient

AbstractWe introduce an action of a discrete subgroup $\varGamma$ of $SL(2,\mathbb{R})^n$ on the space of pseudodifferential operators of $n$ variables, and construct a map from the space of Hilbert modular forms for $\varGamma$ to the space of pseudodifferential operators invariant under such a $\varGamma$-action, which is a lifting of the symbol map of pseudodifferential operators. We also obtain a necessary and sufficient condition for a certain type of pseudodifferential operator to be $\varGamma$-invariant.AMS 2000 Mathematics subject classification: Primary 11F41; 35S05

scholarly journals A Note on the Regularity of the Solutions to Two Variational Inequalities Involving a Pseudodifferential Operator

2007 ◽  
Vol 2007 ◽  
pp. 1-8
Author(s):  
Randolph G. Cooper
Keyword(s):  
Variational Inequalities ◽  
Pseudodifferential Operator ◽  
Pseudodifferential Operators ◽  
Regularity Of Solutions ◽  
Local Operators ◽  
Regularity Results

The regularity of solutions to variational inequalities involving local operators has been studied extensively. Less attention has been paid to those involving nonlocal pseudodifferential operators. We present two regularity results for such problems.

scholarly journals Partial parabolicity of the boundary-value problem for pseudodifferential equations in a layer

2019 ◽  
Keyword(s):  
Boundary Value Problem ◽  
Pseudodifferential Operator ◽  
Boundary Value Problems ◽  
Pseudodifferential Operators ◽  
Boundary Value ◽  
Pseudodifferential Equation ◽  
Parabolic Boundary ◽  
Pseudodifferential Equations ◽  
Spatial Variables ◽  
Parabolic Boundary Value Problems

A nonlocal boundary-value problem for evolutional pseudodifferential equations in an infinite layer is considered in this paper. The notion of the partially parabolic boundary-value problem is introduced when a solving function decreases exponentially only by the part of space variables. This concept generalizes the concept of a parabolic boundary value problem, which was previously studied by one of the authors of this paper (A. A. Makarov). Necessary and sufficient conditions for the pseudodifferential operator symbol are obtained in which partially parabolic boundary-value problems exist. It turned out that the real part of the symbol of a pseudodifferential operator should increase unboundedly powerfully in some of the spatial variables. In this case, a specific type of boundary conditions is indicated, which depend on a pseudodifferential equation and are also pseudodifferential operators. It is shown that for solutions of partially parabolic boundary-value problems, smoothness in some of the spatial variables increases. The disturbed (excitated) pseudodifferential equation with a symbol which depends on space and temporal variables is also investigated. It has been found for partially parabolic boundary-value problems what pseudodifferential operators are possible to be disturbed in the way that the input equation of this boundary-value problem would remain correct in Sobolev-Slobodetsky spaces. It is also shown that although the properties of increasing the smoothness of solutions in part of the variables for partially parabolic boundary value problems are similar to the property of solutions of partially hypoelliptic equations introduced by L. H\"{o}rmander, these examples show that the partial parabolic boundary value problem does not follow from partial hipoellipticity; and vice versa - an example of a partially parabolic boundary value problem for a differential equation that is not partially hypoelliptic is given.

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