Traces and extensions of matrix‐weighted Besov spaces

Let V be a matrix weight on ℝ n +1 and let W be a matrix weight on ℝ n , satisfying, for example, the matrix A p condition. Define the trace, or restriction, operator Tr by Tr ( f )( x ′ )= f ( x ′ , 0), where x ′ ∈ℝ n and f is a function on ℝ n +1 . If α−1/ p > n (1/ p −1) + +(β− n )/ p , where β is the doubling exponent of W , then the trace operator is bounded fromB . p α q( V )intoB . p α − 1 / p , q( W )(matrix‐weighted Besov spaces) if and only if the weights V and W uniformly satisfy an estimate controlling the average of | | W 1 / p ( t ) y → | | p on any dyadic cube I ⊆ ℝ n by the average of | | V 1 / p ( t ) y → | | p on Q ( I )= I ×[0, ℓ( I )], for all y → . If V and W satisfy the converse inequality, then there exists a continuous linear map Ext :B ˙ p α − 1 / p , q( W ) → B ˙ p α q( V ) . If both inequalities hold, then Tr ○ Ext is the identity onB ˙ p α − 1 / p , q( W ) .