Parametrix

In mathematics, and specifically the field of partial differential equations (PDEs), a parametrix is a generalization of the notion of a fundamental solution of a PDE. A fundamental solution for a differential operator "P"("D") with constant coefficients is a distribution "u" on R^"n" such that

:$P(D)u\; =\; delta$

in the weak sense, where δ is the Dirac delta distribution. A parametrix for "P"("D") is a distribution "u" such that

:$P(D)u\; =\; delta\; +\; omega$

where ω is some C^∞ function with compact support. The parametrix is a useful concept in the study of elliptic differential operators and, more generally, of hypoelliptic pseudodifferential operators, since for such operators over appropriate domains a parametrix can be shown to exist, and be a smooth function away from the origin.

More generally, if "L" is any pseudodifferential operator, then another pseudodifferential operator "L"⁺ is called a parametrix for "L" if the operators:$Lcirc\; L^+\; -\; I,quad\; L^+circ\; L\; -I$are both compact on a suitable Sobolev space. A common further requirement is that they must both be operators of order −1 in the sense that they admit continuous extension to an operator from each Sobolev space "H"^"s" to "H"^"s"+1.

References

*|first=L.|last= Hörmander|authorlink=Lars Hörmander|title=The analysis of linear partial differential operators I|series= Grundl. Math. Wissenschaft. |volume= 256 |publisher= Springer |year=1983|ISBN=3-540-12104-8 .
*.
*.
*
*