Other Mathematics Commons^™

Extension Theorems On Matrix Weighted Sobolev Spaces, Christopher Ryan Loga

Doctoral Dissertations

Let D a subset of Rⁿ [R n] be a domain with Lipschitz boundary and 1 ≤ p < ∞ [1 less than or equal to p less than infinity]. Suppose for each x in Rⁿ that W(x) is an m x m [m by m] positive definite matrix which satisfies the matrix A_p [A p] condition. For k = 0, 1, 2, 3;... define the matrix weighted, vector valued, Sobolev space [L p k of D,W] with

[the weighted L p k norm of vector valued f over D to the p power equals the sum over all alpha with order less than k of the integral over D of the the pth power …