Schwartz's core sentence

The Schwartz kernel theorem (or set of core ) is an important mathematical statement in the field of distribution theory that a branch of functional analysis is. It was proven by the mathematician Laurent Schwartz in 1952. However, this statement is not called the core theorem because of its importance, but because it is a statement about integral kernels . These integral kernels dealt with here are called Schwartz kernels .

introduction

Each function can be an integral operator by ${\displaystyle K\in C(X_{1}\times X_{2})}$ ${\displaystyle A:C_{c}(X_{2})\to C(X_{1})}$

${\displaystyle A(\phi )(x_{1})=\int _{X_{2}}K(x_{1},x_{2})\phi (x_{2})\mathrm {d} x_{2}}$

define. The symbol denotes the continuous functions with a compact carrier . In addition, the identity applies ${\displaystyle C_{c}}$

${\displaystyle \langle A(\phi ),\psi \rangle =K(\psi \otimes \phi )}$

for all and , which is to be understood here as a -Scalar product and the tensor product of two functions through ${\displaystyle \psi \in C_{c}^{\infty }(X_{1})}$ ${\displaystyle \phi \in C_{c}^{\infty }(X_{2})}$ ${\displaystyle K(\psi \otimes \phi )}$ ${\displaystyle L^{2}}$

${\displaystyle (\psi \otimes \phi )(x_{1},x_{2}):=\psi (x_{1})\phi (x_{2}).}$

is defined. In the following, this idea will be expanded to include distribution theory . So be so and . In addition, there can be a distribution again. ${\displaystyle K\in {\mathcal {D}}'(X_{1}\times X_{2})}$ ${\displaystyle \phi \in C_{c}^{\infty }(X_{2})}$ ${\displaystyle A(\phi )}$

Schwartz's core sentence

Each distribution defines a linear mapping , which is the identity ${\displaystyle K\in {\mathcal {D}}'(X_{1}\times X_{2})}$ ${\displaystyle A:C_{c}^{\infty }(X_{2})\to {\mathcal {D}}'(X_{1})}$

${\displaystyle \langle A(\phi ),\psi \rangle =K(\psi \otimes \phi )}$

is sufficient and is continuous with regard to the weak - * - topology . That is, if there is a null sequence, then there is also a null sequence in ${\displaystyle \phi _{j}\to 0}$ ${\displaystyle A(\phi _{j})\to 0}$ ${\displaystyle {\mathcal {D}}'(X_{1}).}$

Conversely, there is exactly one distribution for every linear mapping , so that applies. ${\displaystyle A:C_{c}^{\infty }(X_{2})\to {\mathcal {D}}'(X_{1})}$ ${\displaystyle K}$ ${\displaystyle \langle A(\phi ),\psi \rangle =K(\psi \otimes \phi )}$

This distribution is called the Schwartz core. ${\displaystyle K}$

literature

• Lars Hörmander : The Analysis of Linear Partial Differential Operators. Volume 1: Distribution Theory and Fourier Analysis. Second edition. Springer-Verlag, Berlin et al. 1990, ISBN 3-540-52345-6 ( basic teaching of mathematical sciences 256).