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What are the early successes of Schwartz distributions theory? What are the hard theorems that became simple and what open problems were solved with this new tool soon after Laurent Schwartz released his book in the fifties?

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Following the citation for the 1950 Fields medal I would argue that putting the Dirac delta function on a firm ground was the early success of the theory of distributions.$^\ast$

An extensive list of (later) applications is at Nice applications for Schwartz distributions

$^\ast$ Incidentally, this is Dyson's take on applications of the theory of distributions in physics [source]

The theory of generalized functions (alias distributions) is the only part of post-war pure mathematics which has turned out to have a genuine usefulness in physics. In fashionable field-theoretic circles a paper can now hardly be submitted for publication without at least a reference to Schwartz's two-volume Théorie des Distributions. In some recent papers one can find evidence that Schwartz's work has not only been quoted but has even been read. Applications of distribution theory are being found all over physics, in nonlinear mechanics and fluid dynamics as well as in field theory.

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    $\begingroup$ The firm ground for the Dirac delta function might look like nitpickery by mathematicians but for computations with distributions this is useful for practical applications. If you don't know what distributions are and how to work with them it can easily happen that you accidentily include 1=0 into your assumptions and then go on to prove amazing things. Having a precise definition helps you sort out what you can and cannot do with a Dirac delta and helps sort out which results are actually true. $\endgroup$
    – quarague
    Commented Jan 30, 2022 at 11:32
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    $\begingroup$ @quarague --- it might be of interest to give an example of a case where a naive use of the delta function gives a wrong result (I don't quite have an example ready myself). $\endgroup$
    – Carlo Beenakker
    Commented Jan 30, 2022 at 12:17
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    $\begingroup$ Here is a contrived example. Let $f$ be the function defined by $f(0)=0$ and $f(x)=1$ for $x \neq 0$. Now integrate $f$ as a test function against the Dirac delta distribution on the interval $[-1,1]$. If you do it naively, that is treat both as regular functions, the result is zero. If you do it properly the result is $1$. $\endgroup$
    – quarague
    Commented Jan 30, 2022 at 14:36
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    $\begingroup$ Here is an example where not using the Dirac function does not give the correct result. Let us try to solve the differential equation $y''+y = 0$ using the Fourier transform. We get $(1-\xi^2) \hat{y}(\xi) = 0$ which has no interesting function as a solution. It has however the two distributional solutions $\delta_{-1}$ and $\delta_1$ that account for the two classical solutions $e^{ix}$ and $e^{-ix}$ of the original equation. $\endgroup$
    – coudy
    Commented Jan 30, 2022 at 22:48
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    $\begingroup$ It seems to me that the "Dirac function" was already put on a firm ground with the advent of measure theory and Lebesgue integral, with the concept of Dirac measure. Also, it became pretty standard in probability theory after the work of Kolmogorov to embed positive integrable functions iin the space of measures by seeing them as densities. And this matches pretty closely the use of the "Dirac function" in physics as a distribution of masses. $\endgroup$
    – coudy
    Commented Jan 31, 2022 at 7:42
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This is a very broad topic. If you want a nice little book on the use of distributions in mathematical physics, I suggest this one by Demidov. In the Preface and in Chapter 1, Section 1, the author deals a bit with the history of the concept of function and the role there played by generalized functions (a particular case of which, in his terminology, are Schartz distributions).

The book proceeds by showing how some classical PDEs of mathematical physics admit a much more natural and explicative solution if considered in weak form, and how this is further generalized in the theory of distributions. I also quite like the final chapter on the (underestimated, in my opinion) concept of "generalized function according to Egorov".

So all in all I think this may be a useful read for your aim. But please notice that, as usual in the Russian school, quite a bit of work, including some key proofs, is left to the reader in the problems.

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