A subelliptic Bourgain-Brezis inequality

We prove an approximation lemma on (stratified) homogeneous groups that allows one to approximate a function in the non-isotropic Sobolev space $\dot{NL}^{1,Q}$ by $L^{\infty}$ functions, generalizing a result of Bourgain-Brezis \cite{MR2293957}. We then use this to obtain a Gagliardo-Nirenberg inequality for $\bar{\partial}_b$ on the Heisenberg group $\mathbb{H}^n$.Comment: 44 page

Bourgain-Brezis Estimates on Symmetric Spaces of Non-compact Type

Let M be a globally Riemannian symmetric space. We prove a duality estimate between pairings of vector fields with divergence zero and and in L^1 with vector fields in a critical Sobolev space on M. As a consequence we get a sharp Calderon-Zygmund estimate for solutions to Poisson's equation on M, where the right side data is manufactured from divergence free vector fields which are in L^1. Such a result was proved earlier by Jean Bourgain and Haim Brezis on Euclidean space.Comment: Final version of the paper to appear in J. Functional Analysi

Applications of Bourgain-Brezis inequalities to Fluid Mechanics and Magnetism

We apply the borderline Sobolev inequalities of Bourgain-Brezis to the vorticity equation and Navier-Stokes equation in 2D. We take the initial vorticity to be in the space of functions of Bounded variation(BV). We obtain the subsequent vorticity to be in the space of functions of bounded variation, uniformly for small time, and the velocity vector to be uniformly bounded for small time. Such a conclusion cannot follow for initial vorticity taken to be just a measure or in L^1 from the Lamb-Oseen vortex example. Secondly we apply an improved Strichartz inequality obtained earlier by the first and third authors to the Maxwell equations of Electromagnetism. In particular we estimate the size of the magnetic field vector in terms of the gradient of the current density vector. The main point is that in this inequality only the L^1 norm in space appears for the gradient of the current density vector. Such a result is only possible because of a vanishing divergence inhomogeneity in the wave equation for the Magnetic field vector stemming from the Maxwell equations. A key ingredient in the proof of the improved Strichartz inequality is the Bourgain-Brezis borderline Sobolev inequalities.Comment: References added to the work of M. Ben-Artzi and also Haim Brezis in ARMA. Introduction revise