Usefulness of pseudodifferential and fourier integral operators in the study of the local solvability of linear partial differential equations

1974 ◽  
pp. 92-108
Author(s):  
F. Trèves
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Integral Operators ◽  
Local Solvability ◽  
Fourier Integral ◽  
Fourier Integral Operators ◽  
Partial Differential ◽  
Linear Partial Differential Equations

scholarly journals Regularity of Fourier integral operators with amplitudes in general Hörmander classes

2021 ◽  
Vol 11 (3) ◽  
Author(s):  
Alejandro J. Castro ◽  
Anders Israelsson ◽  
Wolfgang Staubach
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Sobolev Spaces ◽  
Integral Operators ◽  
Fourier Integral ◽  
Hyperbolic Partial Differential Equations ◽  
Fourier Integral Operators ◽  
Partial Differential

AbstractWe prove the global $$L^p$$ L p -boundedness of Fourier integral operators that model the parametrices for hyperbolic partial differential equations, with amplitudes in classical Hörmander classes $$S^{m}_{\rho , \delta }(\mathbb {R}^n)$$ S ρ , δ m ( R n ) for parameters $$0\le \rho \le 1$$ 0 ≤ ρ ≤ 1 , $$0\le \delta <1$$ 0 ≤ δ < 1 . We also consider the regularity of operators with amplitudes in the exotic class $$S^{m}_{0, \delta }(\mathbb {R}^n)$$ S 0 , δ m ( R n ) , $$0\le \delta < 1$$ 0 ≤ δ < 1 and the forbidden class $$S^{m}_{\rho , 1}(\mathbb {R}^n)$$ S ρ , 1 m ( R n ) , $$0\le \rho \le 1.$$ 0 ≤ ρ ≤ 1 . Furthermore we show that despite the failure of the $$L^2$$ L 2 -boundedness of operators with amplitudes in the forbidden class $$S^{0}_{1, 1}(\mathbb {R}^n)$$ S 1 , 1 0 ( R n ) , the operators in question are bounded on Sobolev spaces $$H^s(\mathbb {R}^n)$$ H s ( R n ) with $$s>0.$$ s > 0 . This result extends those of Y. Meyer and E. M. Stein to the setting of Fourier integral operators.

Integral Operators in the Theory of Linear Partial Differential Equations

1962 ◽  
Vol 46 (357) ◽  
pp. 256
Author(s):  
E. T. Copson ◽  
Stefan Bergman
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Integral Operators ◽  
Partial Differential ◽  
Linear Partial Differential Equations

Bounds for Solutions of a System of Linear Partial Differential Equations on Domains with Bergman-Silov Boundary

1966 ◽  
Vol 18 ◽  
pp. 1272-1280
Author(s):  
Josephine Mitchell
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Entire Functions ◽  
Integral Operators ◽  
Holomorphic Functions ◽  
Complex Variables ◽  
Partial Differential ◽  
Linear Partial Differential Equations ◽  
Systems Of Differential Equations ◽  
Image Position

The method of integral operators has been used by Bergman and others (4; 6; 7; 10; 12) to obtain many properties of solutions of linear partial differential equations. In the case of equations in two variables with entire coefficients various integral operators have been introduced which transform holomorphic functions of one complex variable into solutions of the equation. This approach has been extended to differential equations in more variables and systems of differential equations. Recently Bergman (6; 4) obtained an integral operator transforming certain combinations of holomorphic functions of two complex variables into functions of four real variables which are the real parts of solutions of the system1where z1, z1*, z2, z2* are independent complex variables and the functions Fj (J = 1, 2) are entire functions of the indicated variables.

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