On limiting trace inequalities for vectorial differential operators (original) (raw)

2021, Indiana University Mathematics Journal

We establish that trace inequalities for vector fields u ∈ C ∞ c (R n , R N) D k−1 u L n−s n−1 (dµ) c µ n−1 n−s L 1,n−s A[D]u L 1 (dL n) (*) hold if and only if the k-th order homogeneous linear differential operator A[D] on R n is elliptic and cancelling, provided that s < 1, and give partial results for s = 1, where stronger conditions on A[D] are necessary. Here, µ L 1,λ denotes the Morrey norm of µ so that such traces can be taken, for example, with respect to H n−s-measure restricted to fractals of codimension s < 1. The class of inequalities (*) give a systematic generalisation of Adams' 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 BV A. the supremum ranging over all open balls B ⊂ R n ; r(B) denotes the radius of the ball B. Measures µ satisfying µ L 1,λ (R n) < ∞ are sometimes called λ-Ahlfors regular. Note that, even for s = 0, the inequality (1.2) does not extend to p = 1. This can Key words and phrases. Trace embeddings, overdetermined elliptic operators, elliptic and cancelling operators, C-elliptic operators, Triebel-Lizorkin spaces, functions of bounded variation, functions of bounded deformation, BV A-spaces, strict convergence, Sobolev spaces.