Early successes of Schwartz distribution theory 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?
 A: 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.
A: 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.

