Only a start
What I have written here is meant to be the first paragraph, a lay-person's introduction. If I or someone else gets around to extending the article, what I envisage is to add sections bellow with formal definitions of the symbol, singular support composition formula and examples. Billlion 08:51, 10 Sep 2004 (UTC)
The definition references formula (1), yet no such formula can be found in this article. I suspect that formula one is supposed to be some definition of ordinary differential operators... ie - where "p" is a polynomial. Also - should the language "we" be used? Is this proper for an encyclopedia entry? I'm working on a thesis related to this topic, and perhaps when I have more free time, and a good history written I'll take a swing at a rewrite...My thanks to all for the contributions. — Preceding unsigned comment added by 220.127.116.11 (talk) 17:06, 6 June 2014 (UTC)
First time more detailed content
When I found the article, it was a 'stub'. No examples, no definitions, no motivation. Probably, one has to change the structure of the article. My idea was to start with a motivation and present a formal definition afterwards. Maybe its better the other way round.
Please correct my "Simple Bad English", I'm not a native speaker...
A. Slateff 18.104.22.168 15:33, 22 January 2006 (UTC)
Well done A Slateff! I'll have a look over the English as a native speaker. Billlion 20:30, 22 January 2006 (UTC)
Actually, we forgot… non-Archimedean analysis
Current definition is restricted to ℝn, but this is not the only possible space. ΨDOs are very important in, say, p-adic analysis, because there is no differential operators. Incnis Mrsi (talk) 11:42, 10 May 2012 (UTC)
Over Generalization in lead?
Well maybe I am being a bit rusty, but I think we are over generalizing a bit in the lead. Can someone fact check me? Not every Calderon-Zygmund singular integral operator is a pseudo-differential operator. To see a counter example consider a more or less standard multiplier, say for example the multiplier corresponding to the Hilbert Transform. Such a multiplier is not typically considered as a pseudo differential operator because the symbol would lack of differentiablity. If I am not greatly mistaken the inclusion is really the other direction, many of the standard symbol classes give rise to CZ operators? Thenub314 (talk) 16:28, 21 April 2014 (UTC)
Dear Thenub314, why did you remove the section about "motivation", that was written by me in 2006, has been overlooked by several experts and has been robust for 8 years? Why do you state, it was "misleading"? In fact, the study of resolvents/inverses of elliptic operators was the main historical origin for inventing them! This also is the motivation given in Hoermander, vol3. I would strongly suggest to reestablish the "motivation"-section.
Moreover, there exist more general definitions of pseudo-differential operators, eg. with distributional or operator valued symbols, abstract C-star-algebraic extensions, and so on. IMHO, smoothness of the symbol is only a technical assumption, which is convenient for easily establishing a "first theory" as a start. Pseudo-differential operators with non-smooth symbols are common technical tools when dealing with nonlinear PDEs, cf. eg. the books of M.E. Taylor. Some of these classes include the Hilbert-transform operator too. ASlateff 22.214.171.124 (talk) 17:57, 10 July 2014 (UTC)
Today, I reestablished the "motivation" section. Thenub314's focus on the Atiyah-Singer-index theorem is by far too narrow a view, IMHO.
As for the existence of the integrals: In general, they are oscillatory integrals. They can be given a meaning of weak integrals as well. IMHO, a "motivation" need not be rigorous, but is intended to present some intuitive "ideas". This is IMHO, what wikipedia should be about. Wikipedia should make clear the main ideas, but not go too much into technical/rigorous detail. Wikipedia is neither a book nor a thesis.
The main idea why I included the suspicious "double integral" in the motivation is: There exist more general definitions of pseudo-differential operators which use amplitudes instead of symbols. Cf. eg. the (German) book of Niels Jacob on PDEs or Shubin's book. Point coordinates on the one hand, and frequency-co-variables on the other, have a different geometric meaning; this motivates writing the double integral. In fact, invariant definitions of pseudo-differential operators on manifolds use the co-normal-bundle of the diagonal with a different meaning of base and fibre coordinates, see eg. the lecture notes of R.B. Melrose or Hoermander's book vols 3,4.
IMHO, the symbolic calculus really is about the operator kernel. Applying a pseudo-differential operator to a "function" uses the "kernel theorem". Then, extending the operator to act on generalized functions uses eg. duality or continuity.
As for CZ-operators: In general, these two operator classes overlap. In an article about pseudo-differential operators, however, I would not focus too much on CZ-operators, except in a section about "history". May I suggest to you to write a new wikipedia-entry about CZ-operators and to provide a link to it? ASlateff 126.96.36.199 (talk) 18:42, 10 July 2014 (UTC)