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Introduction by J. Bony of pseudodifferential operators with limited regularity, proof of propagation of weak singularities for classes of nonlinear PDE. Let me put it this way.
Many natural objects in PDE theory are pseudodifferential operators. Just a few examples besides, obviously, differential operators :. And this list is quite incomplete. Now, pseudodifferential calculus is essentially a framework which shows the underlying common structure of all the previous examples, unifies their properties, and shows that many computations from different theories are just special cases of general theorems. To accommodate more and more interesting examples, the theory has been tweaked and enlarged several times, while keeping the same abstract structure.
Thus there does not exist a single calculus, but several calculi following similar guidelines.
In addition, the procedure associating a symbol to an operator, which is at the heart of pseudodifferential calculus gives a mathematical framework for the quantization procedure in physics. As others have mentioned, pseudodifferential operators arose from trying to establish existence and regularity theorems for a variable coefficient linear PDE using Fourier transform, extending known results for constant coefficient PDE's.
I believe the earliest work in this direction, mainly for hyperbolic PDE's, is by Leray and Petrovskii. Another important early paper is by Lax, "Asymptotic solutions of oscillatory initial value problems", DMJ Unlike the earlier work, I believe their focus was on elliptic PDE's. The extension to and use on manifolds was, I believe, done by Atiyah and Singer in their original work on the index theorem, as well as Seeley. Many applications of pseudo-differential operators, especially to boundary value problems for elliptic and hyperbolic equations, can be found in the book by F.
A different kind of pseudo-differential operators, typically with non-smooth e. Eidelman, S. Ivasyshen, and A. A slightly different motivation for fourier integral operators and pseudo-differential operators is given in the first chapter of this book - Fourier Integral Operators, chapter V. Guillemin: 25 Years of Fourier Integral Operators. In this chapter, V. Guillemin presents this subject from the conormal bundles point of view and then shows how pseudo-differential operators are a special case of operators coming out of the Fourier distributions.
Interesting, Guillemin also shows here that pseudo-differential operators can be used to fine tune the theory of conormal distributions themselves -- giving a natural but unexpected application of pseudo-differential operators. I the vol. I suggest this introduction of M. Wong, see 1. Here is a practical problem in signal processing which is solved with pseudodifferential operators.
The singularity of the kernel on the diagonal depends on the degree of the corresponding operator. Hanges , Almost Mizohata operators. Fourier-integral-operators, and A. Sorted by. Condition: UsedAcceptable.
This can be represented as. What if we want to compute the half derivative. That is,.
Of course any fractional derivative can be computed in this way. In signal processing we know that the full derivative performs a 90 degree phase shift on each frequency component on the data.
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Ricardo Andrade 5, 5 5 gold badges 35 35 silver badges 66 66 bronze badges. I'm not sure how exactly algebraic analysis influenced microlocal analysis and psido's, but you might find it interesting. I think that solving the heat equation on the circle using fourier series is perhaps the first "glimpse" of psido's that I can imagine, but this is not a particularly historically motivated remark. Paul Siegel Paul Siegel Pedro Lauridsen Ribeiro 3, 1 1 gold badge 18 18 silver badges 49 49 bronze badges.