Pseudo-differential Operator - Definition of Pseudo-differential Operators

Definition of Pseudo-differential Operators

Here we view pseudo-differential operators as a generalization of differential operators. We extend formula (1) as follows. A pseudo-differential operator P(x,D) on Rn is an operator whose value on the function u(x) is the function of x:

$\quad P(x,D) u (x) = \frac{1}{(2 \pi)^n} \int_{\mathbb{R}^n} \int_{\mathbb{R}^n} e^{i (x - y) \xi} P(x,\xi) u(y) \, dy \, d\xi$

(2)

where the symbol P(x,ξ) in the integrand belongs to a certain symbol class. For instance, if P(x,ξ) is an infinitely differentiable function on Rn × Rn with the property

for all x,ξ ∈Rn, all multiindices α,β. some constants C_{α, β} and some real number m, then P belongs to the symbol class of Hörmander. The corresponding operator P(x,D) is called a pseudo-differential operator of order m and belongs to the class

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