On Limiting Trace Inequalities for Vectorial Differential Operators

We establish that trace inequalities for vector fields u is an element of C-c(infinity) (R-n, R-N) (*) parallel to D(k-1)u parallel to(L(n-s)/(n-1)(d mu)) <= c parallel to mu parallel to((n-1)/(n-s))(L1,n-s) parallel to A[D]u parallel to(L1(dLn)) hold if and only if the k-th order homogeneous linear differ-ential operator A[D] on R-n is elliptic and cancelling, provided that s < 1, and we give partial results for s = 1, where stronger conditions on A[D] are necessary. Here, parallel to mu parallel to(L1,lambda) denotes the Morrey norm of mu so that such traces can be taken, for example, with respect to Hn-s-measure restricted to fractals of codimension s < 1. The class of inequalities (*) gives a systematic generalisation of Adams's trace inequalities to the limit case p = 1, and can be used to prove trace embeddings for functions of bounded A-variation, thereby comprising Sobolev functions and functions of bounded variation or deformation. We also prove a multiplicative version of (*), which implies strict continuity of the associated trace operators on BVA.