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