On limiting trace inequalities for vectorial differential operators
(2021) Indiana University Mathematics Journal — Vol. 70, n° 5, p. 2133-2176 (2021)
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- Abstract
- We establish that trace inequalities ∥Dk−1u∥Ln−sn−1(Rn,dμ)≤c∥μ∥n−1n−sL1,n−s(Rn)∥A[D]u∥L1(Rn,dLn) hold for vector fields u∈C∞(Rn;RN) if and only if the k-th order homogeneous linear differential operator A[D] on Rn 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∥L1,λ denotes the (1,λ)-Morrey norm of the measure μ, so that such traces can be taken, for example, with respect to the Hausdorff measure Hn−s restricted to fractals of codimension 0<s<1. The above 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 moreover establish a multiplicative version of the above inequality, which implies (A-)strict continuity of the associated trace operators on BVA.
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Gmeineder, F., Raita, B., & Van Schaftingen, J. (2021). On limiting trace inequalities for vectorial differential operators. Indiana University Mathematics Journal, 70(5), 2133-2176. https://doi.org/10.1512/iumj.2021.70.8682 (Original work published 2021)