Application of integral operators to the theory of partial differential equations with singular coefficients

1962 ◽  
Vol 10 (1) ◽  
pp. 323-340 ◽  
Author(s):  
Stefan Bergman ◽  
Ranko Bojanić
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Integral Operators ◽  
Partial Differential ◽  
Singular Coefficients

The Hardy Space H1 on Manifolds and Submanifolds

1972 ◽  
Vol 24 (5) ◽  
pp. 915-925 ◽  
Author(s):  
Robert S. Strichartz
Keyword(s):  
Partial Differential Equations ◽  
Hardy Space ◽  
Euclidean Space ◽  
Differential Equations ◽  
Singular Integral ◽  
Integral Operators ◽  
Singular Integral Operators ◽  
Riesz Transforms ◽  
Partial Differential ◽  
Integrable Functions

It is well-known that the space L1(Rn) of integrable functions on Euclidean space fails to be preserved by singular integral operators. As a result the rather large Lp theory of partial differential equations also fails for p = 1. Since L1 is such a natural space, many substitute spaces have been considered. One of the most interesting of these is the space we will denote by H1(Rn) of integrable functions whose Riesz transforms are integrable.

On Some Relations Between Partial and Ordinary Differential Equations

1958 ◽  
Vol 10 ◽  
pp. 183-190 ◽  
Author(s):  
Erwin Kreyszig
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Complex Variable ◽  
Analytic Functions ◽  
Integral Operators ◽  
Basic Tool ◽  
Partial Differential ◽  
Image Position

The theory of solutions of partial differential equations (1.1) with analytic coefficients can be based upon the theory of analytic functions of a complex variable; the basic tool in this approach is integral operators which map the set of solutions of (1.1) onto the algebra of analytic functions. For certain classes of operators this mapping which is first defined in the small, can be continued to the large, cf. Bergman (3).

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

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.

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