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Locality principle

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A collective concept that combines a number of assertions related mainly to elliptic (in some cases to hypo-elliptic) equations (operators) and that follows from the pointwise character of the singularity of a fundamental solution for this class of equations. For example, an elliptic operator $L(D,x)$ with variable coefficients, written in the form

$$L(D,x)\equiv\sum_{|\alpha|\leq m}a_\alpha(x)D^\alpha,\quad x\in\mathbf R^n,$$

can be represented, in an appropriate sense, in a neighbourhood of a point $x_0$ as a sum

$$L(D,x)=\sum_{|\alpha|\leq m}a_\alpha(x_0)D^\alpha+L'(x),$$

where the first term is an operator with constant coefficients, and $L'(x)$ is "sufficiently small" in the given neighbourhood.

References

[1] N. Dunford, J.T. Schwartz, "Linear operators. Spectral theory" , 2 , Interscience (1963)
How to Cite This Entry:
Locality principle. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Locality_principle&oldid=16547
This article was adapted from an original article by A.A. Dezin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article