Haim Brezis
Paris VI and Rutgers

New types of Sobolev-Nirenberg imbeddings, isoperimetric inequalities and elliptic estimates in $ L^1$

We present new estimates obtained jointly with J. Bourgain, and partially with P. Mironescu. Each one has a different flavour, but, in fact, they are closely related.

The first one asserts that

$\displaystyle \Big\vert\int_\Gamma f(s) \overset{\rightarrow}{t}(s) \Big\vert\leq C\vert\Gamma\vert \, \Vert \nabla
f\Vert _{L^3}\quad \forall f

where $ \Gamma\subset \Bbb R^3$ is a closed rectifiable curve, $ \vert\Gamma\vert$ denotes the length of $ \Gamma$, $ C$ is a universal constant and $ \overset{\rightarrow}{t}$ is the tangent to $ \Gamma$.

The second estimate concerns the classical system, in $ \Bbb R^3$,

$\displaystyle \aligned$   div $\displaystyle u &= 0\ $   curl $\displaystyle u &= f.\endaligned$

Our new estimate asserts that

$\displaystyle \Vert u \Vert _{L^{3/2}}\leq C\Vert f \Vert _{L^1}.

A third new estimate concerns the system

$\displaystyle \Delta u = f$$\displaystyle \text { in } \Bbb R^3,

where $ f$ is a divergence-free vector-field. Our new estimate asserts that

Such inequality is unusual because it is well-known that standard elliptic estimates fail in $ L^1$.