Abstract
Let 0 < p ≤ 1 and w in the Muckenhoupt class A1. Recently, by using the weighted atomic decomposition and molecular characterization, Lee, Lin and Yang[11] established that the Riesz transforms Rj, j = 1,2, …,n, are bounded on Hpw(Rn). In this note we extend this to the general case of weight w in the Muckenhoupt class A∞ through molecular characterization. One difficulty, which has not been taken care in [11], consists in passing from atoms to all functions in Hpw(Rn). Furthermore, the Hpw-boundedness of θ-Calderón-Zygmund operators are also given through molecular characterization and atomic decomposition.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bownik, M., Boundedness of Operators on Hardy Spaces via Atomic Decompositions, Proc. Amer. Math. Soc., 133 (2005), 3535–3542.
Bownik, M., Li, B., Yang, D. and Zhou, Y., Weighted Anisotropic Hardy Spaces and Their Applications in Boundedness of Sublinear Operators, Indiana Univ. Math. J., 57:7(2008), 3065–3100.
Ding, Y., Lee, M. Y. and Lin, C. C., Fractional Integrals on Weighted Hardy Spaces, J. Math. Anal. Appl., 282(2003), 356–368.
Fefferman, C. and Stein, E. M., H p Spaces of Several Variables, Acta Math., 129 (1972), 137–193.
García-Cuerva, J., Weighted H p Spaces, Dissertations Math., 162 (1979), 1–63.
García-Cuerva, J. and Rubio de Francia, J. L., Weighted norm Inequalities and Related Topics, North- Holland Math. Stud. 116, 1985.
Journé, J. L., Calderón-Zygmund Operators, Pseudo-Differential Operators and the Cauchy Integral of Calderón, Lecture notes in Math. 994, Springer-Verlag, Berlin, 1983.
Ky, L. D., New Hardy Spaces of Musielak-Orlicz type and boundedness of Sublinear Operators, submitted, arXiv:1103.3757
Lee, M. Y., Different Approaches to the H p Boundedness of Riesz Transforms, Commun. Math. Anal., 1:2 (2006), 101–108.
Lee, M. Y. and Lin, C. C., The Molecular Characterization of Weighted Hardy Spaces, J. Funct. Anal., 188 (2002), 442–460.
Lee, M. Y. Lin, C. C. and Yang, W. C., Hpw-boundedness of Riesz Transforms, J. Math. Anal. Appl., 301(2005), 394–400.
Lu, S. Z., Four Lectures on Real Hp Spaces, World Scientific Publishing Co. Pte. Ltd, 1995.
Meda, S., Sjögren, P. and Vallarino, M., On the H1-L1 Boundedness of Operators, Proc. Amer. Math., Soc. 136 (2008), 2921–2931.
Meyer, Y. and Coifman, R., Wavelets, Calderón-Zygmund and Multilinear Operators, Advanced Mathematics. Cambridge University Press, 1997.
Quek, T. and Yang, D., Calderón-Zygmund-type Operators on Weighted weak Hardy Spaces over Rn, Acta Math. Sin. (Engl. Ser.), 16 (2000), 141–160.
Strömberg, J. O. and Torchinsky, A., Weighted Hardy Spaces, Lecture Notes in Mathematics, Vol. 1381, Springer-Verlag, Berlin/New York, 1989.
Taibleson, M. H. and Weiss, G., The Molecular Characterization of Certain Hardy Spaces; Astérisque., 77 (1980), 67–149.
Yabuta, K., Generalizations of Calderón-Zygmund Operators, Studia Math., 82 (1985) 17–31.
Yang, D. and Zhou, Y., A Boundedness Criterion via Aboms for Linear Operators in Hardy Spaces, Constr. Approx., 29(2009), 207–218.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Ky, L.D. A note on Hpw-boundedness of Riesz transforms and θ -Calderón-Zygmund operators through molecular characterization. Anal. Theory Appl. 27, 251–264 (2011). https://doi.org/10.1007/s10496-011-0251-z
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10496-011-0251-z