Integrable Systems Seminar
Thursday, April 25, 2002, 4:00pm, 114 Physics
Svetlana Roudenko (MSU)
Theory of function spaces with matrix weights
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
{\it matrix weight setting}. In particular, we introduce matrix-weighted
continuous 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 $A_Q$ are reducing operators for $W$. We prove the norm
equivalences $\Vert \vec{f} \,\Vert_{\dot{B}^{\alpha q}_p(W)}
\approx \Vert \{ \vec{s}_Q \}_Q \Vert_{\dot{b}^{\alpha q}_p(W)}
\approx \Vert \{ \vec{s}_Q \}_Q \Vert_{\dot{b}^{\alpha q}_p(\{A_Q\})}$,
where $\{ \vec{s}_Q \}_Q$ is a vector-valued sequence of the $\ffi$-transform
coefficients of $\vec{f}$, under any of the three conditions on the weight $W$:
(i) the matrix $A_p$ condition;
(ii) for large $p$ under just the doubling condition;
(iii) the doubling condition with no restriction on $p$ if $W$ is diagonal.
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 operator is bounded on
$\dot{B}^{\alpha q}_p(W)$ under any of the three conditions on $W$.
This leads to the boundedness of Calder\'on-Zygmund operators (CZOs) on
$\dot{B}^{\alpha q}_p(W)$, in particular, {\it the Hilbert transform}. We apply
these results to wavelets to show that the above norm equivalence holds if the
$\ffi$-transform coefficients are replaced with the wavelet coefficients.
Finally, we construct dual spaces by the averaging method of reducing operators.
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