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Bulletin of the London Mathematical Society Advance Access originally published online on March 11, 2008
Bulletin of the London Mathematical Society 2008 40(2):181-192; doi:10.1112/blms/bdm108
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© 2008 London Mathematical Society

Traces and extensions of matrix-weighted Besov spaces

Michael Frazier

Mathematics Department
University of Tennessee
Knoxville, TN 37996
USA
frazier@math.utk.edu

Svetlana Roudenko

Department of Mathematics
Arizona State University
Tempe, AZ 85287-1804
USA

Received 26 February 2006. Revision received 27 July 2007.

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'isinRn and f is a function on Rn+1. If {alpha}–1/p>n (1/p–1)++(β–n)/p, where β is the doubling exponent of W, then the trace operator is bounded from Formula into Formula (matrix-weighted Besov spaces) if and only if the weights V and W uniformly satisfy an estimate controlling the average of Formula on any dyadic cube I {subseteq} Rn by the average of Formula on Q(I)=Ix[0, {ell}(I)], for all Formula . If V and W satisfy the converse inequality, then there exists a continuous linear map Formula . If both inequalities hold, then Tr {circ} Ext is the identity on Formula .

2000 Mathematics Subject Classification 42B35, 47B38, 42B25 (primary).

Second author is partially supported by the NSF grants DMS-0401602 and DMS-0531337.


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