Embeddings for A-weakly differentiable functions on domains

We prove that the inhomogeneous estimate of vector fields on balls in R-n (integral(B)vertical bar D(k-1)u vertical bar(n/(n-1))dx) ((n-1)/n )<= c (integral(B )vertical bar Au vertical bar + vertical bar u vertical bar dx) for all u is an element of C-infinity ((B) over bar, R-N) holds if and only if the linear, constant coefficient differential Sobolev inequalities operator A of order k has finite dimensional null-space (FDN). Jones extension This generalizes the Gagliardo-Nirenberg-Sobolev inequality on domains and provides the local version of the analogous homogeneous embedding in full-space (integral(Rn)vertical bar D(k-1)u vertical bar(n/(n-1))dx)((n-1)/n )<= c integral(Rn)vertical bar Au vertical bar dx for all u is an element of C-c(infinity)(R-n, R-N), proved by Van Schaftingen precisely for elliptic and cancelling (EC) operators, building on fundamental L-1-estimates from the works of Bourgain and Brezis. We prove that FDN strictly implies EC and discuss the contrast between homogeneous and inhomogeneous estimates on both algebraic and analytic level. (C) 2019 Elsevier Inc. All rights reserved.