Abstract
Let p∈(0,1] and s≥[n(1/p−1)], where [n(1/p−1)] denotes the maximal integer no more than n(1/p−1). In this paper, the authors prove that a linear operator T extends to a bounded linear operator from the Hardy space H p(ℝn) to some quasi-Banach space ℬ if and only if T maps all (p,2,s)-atoms into uniformly bounded elements of ℬ.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aoki, T.: Locally bounded linear topological spaces. Proc. Imp. Acad. Tokyo 18, 588–594 (1942)
Bownik, M.: Boundedness of operators on Hardy spaces via atomic decompositions. Proc. Am. Math. Soc. 133, 3535–3542 (2005)
Calderón, A.P.: An atomic decomposition of distributions in parabolic H p spaces. Adv. Math. 25, 216–225 (1977)
Coifman, R.R.: A real variable characterization of H p. Studia Math. 51, 269–274 (1974)
Coifman, R.R., Lions, P.-L., Meyer, Y., Semmes, S.: Compensated compactness and Hardy spaces. J. Math. Pures Appl. 72(9), 247–286 (1993)
Fefferman, C., Stein, E.M.: H p spaces of several variables. Acta Math. 129, 137–193 (1972)
Folland, G.B., Stein, E.M.: Hardy Spaces on Homogeneous Groups. Princeton University Press, Princeton (1982)
Frazier, M., Jawerth, B., Weiss, G.: Littlewood-Paley Theory and the Study of Function Spaces. CBMS Regional Conference Series in Mathematics, vol. 79. American Mathematical Society, Providence (1991)
García-Cuerva, J., Rubio de Francia, J.L.: Weighted Norm Inequalities and Related Topics. North-Holland, Amsterdam (1985)
Grafakos, L.: Classical and Modern Fourier Analysis. Prentice Hall, Upper Saddle River (2004)
Hu, G., Yang, D., Zhou, Y.: Boundedness of singular integrals in Hardy spaces on spaces of homogeneous type. Taiwan. J. Math. (2008, to appear)
Latter, R.H.: A characterization of H p(ℝn) in terms of atoms. Studia Math. 62, 93–101 (1978)
Meda, S., Sjögren, P., Vallarino, M.: On the H 1–L 1 boundedness of operators. Proc. Am. Math. Soc. (2008, to appear)
Meyer, Y., Taibleson, M., Weiss, G.: Some functional analytic properties of the spaces B q generated by blocks. Indiana Univ. Math. J. 34, 493–515 (1985)
Meyer, Y., Coifman, R.R.: Wavelets. Calderón–Zygmund and Multilinear Operators. Cambridge University Press, Cambridge (1997)
Müller, S.: Hardy space methods for nonlinear partial differential equations. Tatra Mt. Math. Publ. 4, 159–168 (1994)
Semmes, S.: A primer on Hardy spaces, and some remarks on a theorem of Evans and Müller. Commun. Partial Differ. Equ. 19, 277–319 (1994)
Stein, E.M.: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton University Press, Princeton (1993)
Stein, E.M., Weiss, G.: On the theory of harmonic functions of several variables. I. The theory of H p-spaces. Acta Math. 103, 25–62 (1960)
Taibleson, M.H., Weiss, G.: The molecular characterization of certain Hardy spaces. Astérisque 77, 67–149 (1980)
Triebel, H.: Fractals and Spectra. Birkhäuser, Basel (1997)
Triebel, H.: The Structure of Functions. Birkhäuser, Basel (2001)
Yabuta, K.: A remark on the (H 1,L 1) boundedness. Bull. Fac. Sci. Ibaraki Univ. Ser. A 25, 19–21 (1993)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Pencho Petrushev.
Rights and permissions
About this article
Cite this article
Yang, D., Zhou, Y. A Boundedness Criterion via Atoms for Linear Operators in Hardy Spaces. Constr Approx 29, 207–218 (2009). https://doi.org/10.1007/s00365-008-9015-1
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00365-008-9015-1