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Transactions of the American Mathematical Society

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Matrix-weighted Besov spaces

Author: Svetlana Roudenko
Journal: Trans. Amer. Math. Soc. 355 (2003), 273-314
MSC (2000): Primary 42B25, 42B35, 47B37, 47B38
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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  • 1. M.CHRIST AND M.GOLDBERG, Vector $A_2$ weights and a Hardy-Littlewood maximal function, Trans. Amer. Math. Soc., 353, no. 5, (2001), 1995-2002.
  • 2. I.DAUBECHIES, Orthonormal bases of compactly supported wavelets, Comm. Pure Appl. Math. 41 (1988), 909-996. MR 90m:42039
  • 3. M.FRAZIER, B.JAWERTH AND G.WEISS, Littlewood-Paley Theory and Study of Function Spaces, CBMS Regional Conference Series in Mathematics 79, Amer. Math. Soc., Providence, RI (1991). MR 92m:42021
  • 4. M.FRAZIER AND B.JAWERTH, Decomposition of Besov Spaces, Indiana Univ. Math. J. 34 (1985), 777-799. MR 87b:46083
  • 5. M.FRAZIER AND B.JAWERTH, A discrete transform and decompositions of distribution spaces, J. Funct. Anal. 93 (1990), 34-170. MR 92a:46042
  • 6. M.FRAZIER, R.TORRES AND G.WEISS, The boundedness of Calderón-Zygmund operators on the spaces $\displaystyle\dot{F}^{\alpha q}_p(W)$, Rev. Mat. Iberoamericana 4, no.1, (1988), 41-72. MR 90k:42029
  • 7. D.GOLDBERG, A local version of real Hardy spaces, Duke Math. J. 46 (1979), 27-42. MR 80h:46052
  • 8. R.HUNT, B.MUCKENHOUPT AND R.WHEEDEN, Weighted norm inequalities for conjugate function and Hilbert transform, Trans. Amer. Math. Soc., 176 (1973), 227-251. MR 47:701
  • 9. P.G.LEMARIÉ AND Y.MEYER, Ondelettes et bases hilbertiennes, Rev. Mat. Iberoamericana 2 (1986), 1-18.
  • 10. Y.MEYER, Principe d'incertitude, bases Hilbertiennes et algèbres d'opérateurs, Séminaire Bourbaki 1985/86, Exposé 662, Astérisque, no. 145-146, Soc. Math. France, Paris, 1987, pp. 209-223. MR 88g:42012
  • 11. Y.MEYER, Wavelets and Operators, Cambridge University Press, 1992. MR 92f:42001
  • 12. F.NAZAROV AND S.TREIL, The hunt for a Bellman function: applications to estimates for singular integral operators and to other classical problems of harmonic analysis, Algebra i Analiz 8 (1996), no. 5, 32-162; English transl., St. Petersburg Math. J. 8 (1997), 721-824. MR 99d:42026
  • 13. S.ROUDENKO, The theory of function spaces with matrix weights, Ph.D. thesis, Michigan State Univ., 2002.
  • 14. E.STEIN, Harmonic Analysis, Princeton University Press, 1993. MR 95c:42002
  • 15. H.TRIEBEL, Theory of function spaces, Monographs in Math., vol. 78, Birkhäuser-Verlag, Basel, 1983. MR 86j:46026
  • 16. S.TREIL AND A.VOLBERG, Wavelets and the angle between past and future, J. Funct. Anal. 143 (1997), no.2, 269-308. MR 99k:42073
  • 17. A.VOLBERG, Matrix $A_p$ weights via $S$-functions, J. Amer. Math. Soc. 10 (1997), no.2, 445-466. MR 98a:42013

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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

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

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