Locality principle

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.

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