Sobolev embeddings control the integrability of some power of a function by an
integral of the derivative of the function at a lower power. The limiting case
where the latter power is taken to be 1 due to Gagliardo and Nirenberg, is
inaccessible to classical methods of harmonic analysis and turns out to be a
functional version of the isoperimetric inequality. If one considers vector
fields instead of functions, one can hope that some redundancy in the
derivative would allow to obtain estimates with an integrand that does not
involve all the components of the derivative. Such sparse estimates have been
obtained for the deformation operator (M.J. Strauss) and for the Hodge complex
(Bourgain and Brezis). I have characterized the homogeneous autonomous linear
differential operators for which they hold as elliptic and canceling
differential operator. I will also present various further questions that have
been solved or remain as open problems.
Thackeray Hall, Room 704