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 Cα, 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

Read more about this topic:  Pseudo-differential Operator

Famous quotes containing the words definition of and/or definition:

    “It is very hard to give a just definition of love. The most we can say of it is this: that in the soul, it is a desire to rule; in the spirit, it is a sympathy; and in the body, it is but a hidden and subtle desire to possess—after many mysteries—what one loves.”
    —François, Duc De La Rochefoucauld (1613–1680)

    “... we all know the wag’s definition of a philanthropist: a man whose charity increases directly as the square of the distance.”
    —George Eliot [Mary Ann (or Marian)