Abstract
We give a characterization of weighted Hardy spaces H p (w), valid for a rather large collection of wavelets, 0 <p ≤ 1,and weights w in the Muckenhoupt class A ∞ We improve the previously known results and adopt a systematic point of view based upon the theory of vector-valued Calderón-Zygmund operators. Some consequences of this characterization are also given, like the criterion for a wavelet to give an unconditional basis and a criterion for membership into the space from the size of the wavelet coefficients.
Similar content being viewed by others
References
Andersen, K.F. and John, R.T. Weighted inequalities for vector-valued maximal functions and singular integrals,Studia Math.,69, 19–31, (1980).
Benedek, A., Calderón, A.P., and Panzone, R. Convolution operators on Banach space valued functions,Proc. Nat. Acad. Sci. USA,48, 356–365, (1962).
Bochkarev, S.V. Existence of bases in the space of analytic functions and some properties of the Franklin system,Mat. Sb.,98, 3–18, (1974).
Carleson, L. An explicit unconditional basis inH 1,Bull. Sci. Math.,104, 405–116, (1980).
Chang, A. and Ciesielski, Z. Spline characterization ofH 1,Studia Math.,75, 183–192, (1983).
Ciesielski, Z. Properties of the orthonormal Franklin system,Studia Math.,23, 141–157, (1963).
Ciesielski, Z. Properties of the orthonormal Franklin system II,Studia Math.,27, 289–323, (1966).
Coifman, R. and Fefferman, C. Weighted norm inequalities for maximal functions and singular integrals,Studia Math.,51, 241–250, (1974).
García-Cuerva, J. WeightedH p spaces,Dissertationes Math.,162, (1979).
García-Cuerva, J. and Kazarian, K.S. Calderón-Zygmund operators and unconditional bases of weighted Hardy spaces,Studia Math.,109(3), 255–276, (1994).
García-Cuerva, J. and Kazarian, K.S. Spline wavelet bases of weightedL p spaces, 1 ≤p < ∞,Proc. Am. Math. Soc.,123(2), 433–139, (1995).
García-Cuerva, J. and Rubio de Francia, J.L. Weighted norm inequalities and related topics,North-Holland Math. Stud.,116, (1985).
Hernández, E. and Weiss, G.A First Course on Wavelets, CRC Press LLC, Boca Raton, FL, 1996.
Maurey, B. Isomorphismes entre espacesH 1,Acta Math.,145, 79–120, (1980).
Meyer, Y. Wavelets and operators,Cambrigde studies in mathematics,37, Cambrigde University Press, Cambrigde, 1992.
Muckenhoupt, B. Weighted norm inequalities for the Hardy maximal function,Trans. Am. Math. Soc.,165, 207–226, (1972).
Rubio de Francia, J.L., Ruiz, F.J., and Torrea, J.L. Calderón-Zygmund theory for operator-valued kernels,Adv. Math.,62, 7–18, (1986).
Sjölin, P. and Strömberg, J.O. Basis properties of Hardy spaces,Ark. Mat.,21, 111–125, (1983).
Strömberg, J.O. A modified Franklin system and higher order spline systems on ℝn as unconditional bases for Hardy spaces, in,Proc. Conf. in Honor of Antoni Zygmund, Beckner, W., Calderón, A.P., Fefferman, R., and Jones, P.W., Eds., Wadsworth, 475–193, 1981.
Strömberg, J.O. and Torchinsky, A. Weighted Hardy Spaces,Lecture Notes in Math.,1381, Springer, 1989.
Wojtaszczyk, P. The Franklin system is an unconditional basis inH 1,Ark. Mat.,20, 293–300, (1982).
Wu, S. A wavelet characterization for weighted Hardy spaces,Rev. Mat. Iberoamericana,8(3), 329–349, (1992).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
García-Cuerva, J., Martell, J.M. Wavelet characterization of weighted spaces. J Geom Anal 11, 241–264 (2001). https://doi.org/10.1007/BF02921965
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02921965