Matrix-weighted Besov spaces

Roudenko, Svetlana

doi:10.1090/S0002-9947-02-03096-9

Matrix-weighted Besov spaces
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by Svetlana Roudenko PDF

Trans. Amer. Math. Soc. 355 (2003), 273-314 Request permission

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 $\Vert \vec {f} \Vert _{\dot {B}^{\alpha q}_p(W)} \approx \Vert \{ \vec {s}_Q \}_Q \Vert _{\dot {b}^{\alpha q}_p(W)} \approx \Vert \{ \vec {s}_Q \}_Q \Vert _{\dot {b}^{\alpha q}_p(\{A_Q\})}$, where $\{ \vec {s}_Q \}_Q$ is 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.

References

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.
Ingrid Daubechies, Orthonormal bases of compactly supported wavelets, Comm. Pure Appl. Math. 41 (1988), no. 7, 909–996. MR 951745, DOI 10.1002/cpa.3160410705
Michael Frazier, Björn Jawerth, and Guido Weiss, Littlewood-Paley theory and the study of function spaces, CBMS Regional Conference Series in Mathematics, vol. 79, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1991. MR 1107300, DOI 10.1090/cbms/079
Radu Bǎdescu, On a problem of Goursat, Gaz. Mat. 44 (1939), 571–577. MR 0000087
Michael Frazier and Björn Jawerth, A discrete transform and decompositions of distribution spaces, J. Funct. Anal. 93 (1990), no. 1, 34–170. MR 1070037, DOI 10.1016/0022-1236(90)90137-A
M. Frazier, R. Torres, and G. Weiss, The boundedness of Calderón-Zygmund operators on the spaces $\dot F^{\alpha ,q}_p$, Rev. Mat. Iberoamericana 4 (1988), no. 1, 41–72. MR 1009119, DOI 10.4171/RMI/63
David Goldberg, A local version of real Hardy spaces, Duke Math. J. 46 (1979), no. 1, 27–42. MR 523600
Richard Hunt, Benjamin Muckenhoupt, and Richard Wheeden, Weighted norm inequalities for the conjugate function and Hilbert transform, Trans. Amer. Math. Soc. 176 (1973), 227–251. MR 312139, DOI 10.1090/S0002-9947-1973-0312139-8
P.G.Lemarié and Y.Meyer, Ondelettes et bases hilbertiennes, Rev. Mat. Iberoamericana 2 (1986), 1-18.
Yves Meyer, Principe d’incertitude, bases hilbertiennes et algèbres d’opérateurs, Astérisque 145-146 (1987), 4, 209–223 (French). Séminaire Bourbaki, Vol. 1985/86. MR 880034
M. Samuélidès and L. Touzillier, Analyse harmonique, Collection La Chevêche. [La Chevêche Collection], Cépaduès Éditions, Toulouse, 1990 (French). MR 1093674
F. L. Nazarov and S. R. Treĭl′, 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 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 8 (1997), no. 5, 721–824. MR 1428988
S.Roudenko, The theory of function spaces with matrix weights, Ph.D. thesis, Michigan State Univ., 2002.
Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. MR 1232192
Hans Triebel, Theory of function spaces, Monographs in Mathematics, vol. 78, Birkhäuser Verlag, Basel, 1983. MR 781540, DOI 10.1007/978-3-0346-0416-1
S. Treil and A. Volberg, Wavelets and the angle between past and future, J. Funct. Anal. 143 (1997), no. 2, 269–308. MR 1428818, DOI 10.1006/jfan.1996.2986
A. Volberg, Matrix $A_p$ weights via $S$-functions, J. Amer. Math. Soc. 10 (1997), no. 2, 445–466. MR 1423034, DOI 10.1090/S0894-0347-97-00233-6