## A characterization of weighted Carleson measure spaces.(English)Zbl 1406.42031

Summary: Using Frazier and Jawerth’s $$\phi$$-transform, we characterize weighted generalized Carleson measure spaces $$\dot{C}MO^{\alpha,q}_{p,w}$$ for a weight $$w$$ and show that the definition of this space is well-defined by a Plancherel-Pôlya inequality. Note that $$\dot{C}MO^{0,2}_{1,w}$$ is the weighted $$BMO$$ space.

### MSC:

 42B35 Function spaces arising in harmonic analysis
Full Text:

### References:

 [1] K. F. Andersen and R. T. John, Weighted inequalities for vector-valued maximal functions and singular integrals, Studia Math. 69 (1980/81), no. 1, 19–31. · Zbl 0448.42016 [2] M. Bownik and K.-P. Ho, Atomic and molecular decompositions of anisotropic Triebel-Lizorkin spaces, Trans. Amer. Math. Soc. 358 (2006), no. 4, 1469–1510. · Zbl 1083.42016 [3] H.-Q. Bui, M. Paluszyński and M. H. Taibleson, A maximal function characterization of weighted Besov-Lipschitz and Triebel-Lizorkin spaces, Studia Math. 119 (1996), no. 3, 219–246. · Zbl 0861.42009 [4] ——–, Characterization of the Besov-Lipschitz and Triebel-Lizorkin spaces: The case $$q < 1$$, J. Fourier Anal. Appl. 3 (1997), Special Issue, 837–846. · Zbl 0897.42010 [5] H.-Q. Bui and M. H. Taibleson, The characterization of the Triebel-Lizorkin spaces for $$p = ∞$$, J. Fourier Anal. Appl. 6 (2000), no. 5, 537–550. · Zbl 0972.42015 [6] M. Frazier and B. Jawerth, A discrete transform and decompositions of distribution spaces, J. Funct. Anal. 93 (1990), no. 1, 34–170. · Zbl 0716.46031 [7] M. Izuki and Y. Sawano, Wavelet bases in the weighted Besov and Triebel-Lizorkin spaces with $$A^{\operatorname{loc}}_p$$-weights, J. Approx. Theory. 161 (2009), no. 2, 656–673. · Zbl 1183.42037 [8] G. Kyriazis, P. Petrushev and Y. Xu, Decomposition of weighted Triebel-Lizorkin and Besov spaces on the ball, Proc. Lond. Math. Soc. (3) 97 (2008), no. 2, 477–513. · Zbl 1153.41004 [9] M.-Y. Lee, C.-C. Lin and Y.-C. Lin, A wavelet characterization for the dual of weighted Hardy spaces, Proc. Amer. Math. Soc. 137 (2009), no. 12, 4219–4225. · Zbl 1181.42020 [10] C.-C. Lin and K. Wang, Generalized Carleson measure spaces and their applications, Abstr. Appl. Anal. 2012 (2012), Art. ID 879073, 26 pp. · Zbl 1252.42024 [11] 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; St. Petersburg Math. J. 8 (1997), no. 5, 721–824. · Zbl 0873.42011 [12] S. Roudenko, Matrix-weighted Besov spaces, Trans. Amer. Math. Soc. 355 (2003), no. 1, 273–314. · Zbl 1010.42011 [13] A. Torchinsky, Real-variable Methods in Harmonic Analysis, Pure and Applied Mathematics 123, Academic Press, Orlando, FL, 1986. · Zbl 0621.42001 [14] H. Triebel, Theory of Function Spaces, Monographs in Mathematics 78, Birkhäuser Verlag, Basel, 1983.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.