Bourgain has made so many contributions to analysis. I will describe just a few of my favorites in harmonic analysis, and mention a few in number theory. First to cover my bases, I will name what I think he is best known for in harmonic analysis and are so significant that they do not fit your criteria:
- Progress on Stein's restriction conjecture and on the Kakeya conjecture.
- Introducing discrete restriction theory and the decoupling method with Ciprian Demeter.
I also include closely related problems to the above such as $\Lambda(p)$ sets, Roth's theorem, the study of solutions to certain PDEs(especially the Schroedinger equation), problems in additive combinatorics, etc.
Bourgain also made numerous contributions to number theory aside from his work with Demeter and Guth. These include:
- Disproving Montgomery's conjecture via the Kakeya problem
- The affine sieve with Gamburd and Sarnak and related work on expansion with Kontorovich
- Sarnak's conjecture on Mobius orthogonality
- Quantitative versions of Oppenheimer's conjecture and related works (e.g. with Lindenstrauss, Michel and Venkatesh)
- Statistics of eigenfunctions and of lattice points on spheres with Rudnick and Sarnak
- Additive combinatorics - He did so much here that I am not sure where to begin, maybe Mordell's exponential sums revisited?
Here are three of Bourgain's results in harmonic analysis that I think were big when he proved them but are not as well known now:
- $L^p$-boundedness of the circular maximal function
- Dimension-free bounds for maximal functions over convex sets
- Pointwise ergodic theorems for arithmetic sets
The first two may not be known outside of harmonic analysis. The last one is known to ergodic theorists though they seem less interested in it than harmonic analysts. In succession, I will justify why I think of these works were important.
Bourgain proved that the circular maximal function is bounded on $L^p(\mathbb{R}^2)$ for $p>2$. The circular maximal function is unbounded on $L^2$. That suggests that purely Fourier analytic methods are insufficient to solve the problem. This is contrast to higher dimensions where Fourier analysis suffices which was done by Stein about 10 years prior to Bourgain's result. Bourgain's result was considered quite an achievement at the time. To this date, no one has adequately explained Bourgain's proof to me. (There are other, "better" proofs nowadays.) More importantly I think Bourgain's solution foreshadowed the direction he would take harmonic analysis towards. Bourgain's proof combined Fourier analysis of the operator with incidence geometry and combinatorics. Later Bourgain used this perspective on the restriction problem mentioned above. These approaches currently dominate approaches to related problems in harmonic analysis. (Bourgain even joked during his talk at Stein's 80th birthday conference that harmonic analysts need to stop doing Fourier analysis and start doing combinatorics.)
Bourgain's work on dimension free inequalities began roughly at the same time as his work on the circular maximal function. I believe that this was also considered a big problem at the time. Here, Bourgain generalized a result of Stein and Strömberg for balls to convex sets with a bound on their geometry. His expertise of and intuition from functional analysis is on display in this work. Various lemmas in these works are still of use today. In particular he uses a discretized form of the classical Littlewood--Paley inequality to derive certain bounds which are much more intuitive than Stein's g-functions. (Bourgain did not introduce these LP decompositions - I would like to know who the first to do so was.) Recently these works were adapted to variational operators and discrete operators by Bourgain--Mirek--Stein--Wrobel. For instance, one may readily prove results for variational operators by combining Bourgain's approach for maximal functions with Jones--Seeger--Wright's work on variations.
Bourgain's work on pointwise ergodic theorems for arithmetic sets solved a then-outstanding problem about sparse averages in ergodic theory. (I think the question was posed by Furstenberg.) Bourgain combined methods from harmonic analysis and number theory to attack this problem. In these works, Bourgain introduced discrete operators, the circle method, transference principles and variational operators to harmonic analysts. This is a field that Bourgain initiated which is still active. One important contribution here is the paradigm that Bourgain demonstrated which is that the circle method is analogous to Littlewood--Paley theory in a sense. This paradigm was later used by Bourgain's seminal work on discrete restriction theory and periodic non-linear Schroedinger equation, and recently superseded by decoupling for the same problem.
