Bourgain-Brezis estimates for the solutions of the divergence equation

This thesis focuses on two results by J. Bourgain and H. Brezis regarding endpoint Sobolev regularity of solutions to PDEs; the proofs of these results are quite convoluted and rely heavily on techniques from Harmonic Analysis, such as Littlewood-Paley decompositions. The first theorem concerns the equation div(Y)=f on the d-dimensional torus. Usual regularity theory guarantees that if f belongs to L^d then there exists a solution in W^{1,d}, but in general this isn't in L^∞, since the Sobolev embedding fails in the endpoint case. Bourgain and Brezis proved that it is possible to find a vector field in W^{1,d}∩L^∞ that solves the equation and has norm controlled by that of f. While for d=2 this result follows from a duality argument, the case d>2 is much harder to prove and needs an ad hoc construction: the core step is finding scalar fields with only one derivative bounded, and this is done by decomposing the function as a sum of Fourier projections, using antiderivatives to define a function with a better control on the derivative and then define the scalar field as a series using this new function and localizations of Fourier frequencies. The second theorem is a refinement of the previous one: it states that given f in W^{1,d} it is possible to find Y in W^{1,d}∩L^∞ with d-1 derivatives controlled, instead of only one as before. This result is harder to prove since the bounding function cannot be defined using antiderivatives, which would give a control in only one direction; then the construction passes through a sup-convolution and similar arguments as the previous proof, with additional care in differentiating high and low frequencies since they have very different behaviors. My goal is to present and shed some light on these proofs, highlighting the many nuances of this construction and the similarities among the two, and focusing on why similar but less refined constructions fail to produce the correct regularity.