This tag is for questions regarding to the Pseudo-differential operator, acting on a space of functions on a differentiable manifold, that can locally be described by definite rules using a certain function. It satisfies estimates for the derivatives analogous to the estimates for derivatives of polynomials, which are symbols of differential operators.
In mathematical analysis a pseudo-differential operator is an extension of the concept of differential operator. The role of pseudo-differential operators lies in the fact that there is a number of operations leading outside the class of differential operators but preserving the class of pseudo-differential operators. For example, the resolvent and complex powers of an elliptic differential operator on a compact manifold are classical pseudo-differential operators; they arise when reducing an elliptic boundary value problem to the boundary. They are used extensively in the theory of partial differential equations and quantum field theory. They are of similar importance to the theory of partial differential equations as Schwartz distributions, see also microlocal analysis.
Definition: Let $~X⊂\mathbb R^n~$ be open. A pseudodifferential operator is a Fourier integral operator of the form
$$A:C^∞_0(X)→D'(X)$$ $$Au(x)=\dfrac 1{(2π)^n}∫∫e^{i(x−y)θ}a(x,y,θ)u(y)dydθ$$ where the function a, called the symbol of the pseudodifferential operator $~A~$, belongs to the space $~S^m_{ρ,δ}(X×X×\mathbb R^n)~$ defined below.
If $~\mathcal F~$ denotes the Fourier transform a short hand notation for this definition is $~Au=\mathcal F^{−1}(a\mathcal Fu)~$, put in words: Fourier transform $~u~$, multiply with a and transform back.
For more details, see following the references
$1.~$ "Pseudodifferential operator"
$2.~$ "Introduction to pseudo-differential operators" by Michael Ruzhansky
$3.~$ "An introduction to pseudo-differential operators" by Jean-Marc Bouclet
$4.~$ "Pseudo-differential operator"
$5.~$ "A First Course on Pseudo-Differential Operators" by Nicolas Lerner
// copy from math.stackexchange.com \
