AN EXTENSION THEOREM FOR MATRIX WEIGHTED SOBOLEV SPACES ON LIPSCHITZ DOMAINS

Let D be a subset of the n-dimensional real numbers considered as a domain with Lipschitz boundary and let p be a real number greater than or equal to 1. Suppose for each x in the n-dimensional real numbers that W(x) is an m by m positive definite matrix which satisfies the matrix A-p condition. For k = 0, 1, 2, 3,..., we can then define the matrix weighted, vector valued, Sobolev space L-k-p(D,W) with an associated norm. We show that for a vector valued function f in L-k-p(D,W) there exists an extension E(f) in the matrix weighted Sobolev space over the real numbers which is a linear operator in f such that E(f) is equal to f on D and there is an inequality bounding the norm of E(f) by the norm of f. This theorem generalizes a known result for scalar A-p weights.