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Traces and extensions of matrix‐weighted Besov spaces

Traces and extensions of matrix‐weighted Besov spaces Let V be a matrix weight on ℝn+1 and let W be a matrix weight on ℝn, satisfying, for example, the matrix Ap condition. Define the trace, or restriction, operator Tr by Tr (f)(x′)=f(x′, 0), where x′∈ℝn and f is a function on ℝn+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 from B.pαq(V) into B.pα−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 ||W1/p(t)y→||p on any dyadic cube I ⊆ ℝn by the average of ||V1/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˙pα−1/p,q(W)→B˙pαq(V). If both inequalities hold, then Tr ○ Ext is the identity on B˙pα−1/p,q(W). http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

Traces and extensions of matrix‐weighted Besov spaces

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References (23)

Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/bdm108
Publisher site
See Article on Publisher Site

Abstract

Let V be a matrix weight on ℝn+1 and let W be a matrix weight on ℝn, satisfying, for example, the matrix Ap condition. Define the trace, or restriction, operator Tr by Tr (f)(x′)=f(x′, 0), where x′∈ℝn and f is a function on ℝn+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 from B.pαq(V) into B.pα−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 ||W1/p(t)y→||p on any dyadic cube I ⊆ ℝn by the average of ||V1/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˙pα−1/p,q(W)→B˙pαq(V). If both inequalities hold, then Tr ○ Ext is the identity on B˙pα−1/p,q(W).

Journal

Bulletin of the London Mathematical SocietyWiley

Published: Apr 1, 2008

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