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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Matrix-weighted Besov spaces


Author: Svetlana Roudenko
Journal: Trans. Amer. Math. Soc. 355 (2003), 273-314
MSC (2000): Primary 42B25, 42B35, 47B37, 47B38
DOI: https://doi.org/10.1090/S0002-9947-02-03096-9
Published electronically: August 7, 2002
MathSciNet review: 1928089
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Abstract: Nazarov, Treil and Volberg defined matrix $A_p$ weights and extended the theory of weighted norm inequalities on $L^p$ to the case of vector-valued functions. We develop some aspects of Littlewood-Paley function space theory in the matrix weight setting. In particular, we introduce matrix- weighted homogeneous Besov spaces $\dot{B}^{\alpha q}_p(W)$and matrix-weighted sequence Besov spaces $\dot{b}^{\alpha q}_p(W)$, as well as $\dot{b}^{\alpha q}_p(\{A_Q\})$, where the $A_Q$ are reducing operators for $W$. Under any of three different conditions on the weight $W$, we prove the norm equivalences $\Vert \vec{f} \,\Vert_{\dot{B}^{\alpha q}_p(W)} \approx \Vert \{ \vec{s}_Q \}... ... q}_p(W)} \approx \Vert \{ \vec{s}_Q \}_Q \Vert_{\dot{b}^{\alpha q}_p(\{A_Q\})}$, where $\{ \vec{s}_Q \}_Q$ is the vector-valued sequence of $\varphi$-transform coefficients of $\vec{f}$. In the process, we note and use an alternate, more explicit characterization of the matrix $A_p$ class. Furthermore, we introduce a weighted version of almost diagonality and prove that an almost diagonal matrix is bounded on $\dot{b}^{\alpha q}_p(W)$ if $W$ is doubling. We also obtain the boundedness of almost diagonal operators on $\dot{B}^{\alpha q}_p(W)$ under any of the three conditions on $W$. This leads to the boundedness of convolution and non-convolution type Calderón-Zygmund operators (CZOs) on $\dot{B}^{\alpha q}_p(W)$, in particular, the Hilbert transform. We apply these results to wavelets to show that the above norm equivalence holds if the $\varphi$-transform coefficients are replaced by the wavelet coefficients. Finally, we construct inhomogeneous matrix-weighted Besov spaces ${B}^{\alpha q}_p(W)$ and show that results corresponding to those above are true also for the inhomogeneous case.


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Additional Information

Svetlana Roudenko
Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 48824
Address at time of publication: Department of Mathematics, Duke University, Box 90320, Durham, North Carolina 27708
Email: svetlana@math.msu.edu

DOI: https://doi.org/10.1090/S0002-9947-02-03096-9
Keywords: Besov spaces, matrix weights, $\varphi$-transform, $A_p$ condition, doubling measure, reducing operators, almost diagonal operators, Calder\'on-Zygmund operators, Hilbert transform, wavelets
Received by editor(s): March 15, 2002
Published electronically: August 7, 2002
Article copyright: © Copyright 2002 American Mathematical Society