Abstract
We define the vector-valued, matrix-weighted function spaces \(\dot{F}^{\alpha q}_p(W)\) (homogeneous) and \(F^{\alpha q}_p(W)\) (inhomogeneous) on \({\mathbb {R}}^n\), for \(\alpha \in {\mathbb {R}}\), \(0<p<\infty \), \(0<q \le \infty \), with the matrix weight W belonging to the \(A_p\) class. For \(1<p<\infty \), we show that \(L^p(W) = \dot{F}^{0 2}_p(W)\), and, for \(k \in {\mathbb {N}}\), that \(F^{k 2}_p(W)\) coincides with the matrix-weighted Sobolev space \(L^p_k(W)\), thereby obtaining Littlewood-Paley characterizations of \(L^p(W)\) and \(L^p_k (W)\). We show that a vector-valued function belongs to \(\dot{F}^{\alpha q}_p(W)\) if and only if its wavelet or \(\varphi \)-transform coefficients belong to an associated sequence space \(\dot{f}^{\alpha q}_p(W)\). We also characterize these spaces in terms of reducing operators associated to W.
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Acknowledgements
We thank Fedor Nazarov, who provided us with the formulation and proof of Theorem 3.7. S.R. was partially supported by the NSF-DMS CAREER grant # 1151618/1929029.
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Communicated by Loukas Grafakos.
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Frazier, M., Roudenko, S. Littlewood–Paley theory for matrix-weighted function spaces. Math. Ann. 380, 487–537 (2021). https://doi.org/10.1007/s00208-020-02088-0
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DOI: https://doi.org/10.1007/s00208-020-02088-0