Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-28T14:24:08.604Z Has data issue: false hasContentIssue false
  • English
  • Français

Hardy spaces estimates for multilinear operators with homogeneous kernels

Published online by Cambridge University Press:  22 January 2016

Yong Ding  and
Shanzhen Lu
Yong Ding
Affiliation:
Department of Mathematics, Beijing Normal University, Beijing 100875, China, [email protected]
Shanzhen Lu
Affiliation:
Department of Mathematics, Beijing Normal University, Beijing 100875, China, [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper the authors prove that a class of multilinear operators formed by the singular integral or fractional integral operators with homogeneous kernels are bounded operators from the product spaces Lp1 × Lp2 × · · · × LpK (ℝn) to the Hardy spaces Hq (ℝn) and the weak Hardy space Hq,∞(ℝn), where the kernel functions Ωij satisfy only the Ls-Dini conditions. As an application of this result, we obtain the (Lp, Lq) boundedness for a class of commutator of the fractional integral with homogeneous kernels and BMO function.

Keywords

42B2547G10
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 170 , 2003 , pp. 117 - 133
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2003

References

[CZ] Calderón, A. and Zygmund, A., A note on the interpolation of Sublinear operators, Amer. J. Math., 78 (1956), 282–288.CrossRefGoogle Scholar
[Cha] Chanillo, S., A note on commutators, Indiana. Univ. Math. J., 31 (1982), 7–16.CrossRefGoogle Scholar
[CWW] Chanillo, S., Watson, D. and Wheeden, R. L., Some integral and maximal operators related to star-like, Studia Math., 107 (1993), 223–255.CrossRefGoogle Scholar
[Che] Chen, L. K., On a singular integral, Studia Math., 85 (1987), 61–72.CrossRefGoogle Scholar
[CG] Coifman, R. and Grafakos, L., Hardy spaces estimates for multilinear operators I, Rev. Math. Iber., 8 (1992), 45–68.CrossRefGoogle Scholar
[D] Ding, Y., Weak type bounds for a class of rough operators with power weights, Proc. Amer. Math. Soc., 125 (1997), 2939–2942.CrossRefGoogle Scholar
[DL1] Ding, Y. and Lu, S. Z., The Lp1 × Lp2 × … × Lpk boundedness for some rough operators, Jour. Math. Anal. Appl., 203 (1996), 166–186.CrossRefGoogle Scholar
[DL2] Ding, Y. and Lu, S. Z., Weighted norm inequalities for fractional integral operators with rough kernel, Canad. J. Math., 50 (1998), 29–39.CrossRefGoogle Scholar
[DL3] Ding, Y. and Lu, S. Z., Homogeneous fractional integrals on Hardy spaces, Tôhôku Math. J., 52 (2000), 153–162.Google Scholar
[DL4] Ding, Y. and Lu, S. Z., Hardy spaces estimates for a class of multilinear homoge neous operators, Science in China (A), 42 (1999), 1270–1278.CrossRefGoogle Scholar
[FS] Fefferman, R. and Soria, F., The Weak space H1 , Studia Math., 85 (1987), 1–16.CrossRefGoogle Scholar
[G] Grafakos, L., Hardy spaces estimates for multilinear operators II, Rev. Math. Iber., 8 (1992), 69–92.CrossRefGoogle Scholar
[KW] Kurtz, D. and Wheeden, R. L., Results on weighted norm inequalities for multipliers, Trans. Amer. Math. Soc., 255 (1979), 343–362.CrossRefGoogle Scholar
[Lu] Lu, S. Z., Four lectures on real Hp spaces, World Scientific Publishing Co. Pte. Ltd., 1995.CrossRefGoogle Scholar
[Mi] Miyachi, A., Hardy spaces estimates for the product of singular integrals, Canad. J. Math., 52 (2000), 281–311.CrossRefGoogle Scholar
[MW] Muckenhoupt, B. and Wheeden, R. L., Weighted norm inequalities for singular and fractional integrals, Trans. Amer. Math. Soc., 161 (1971), 249–258.CrossRefGoogle Scholar
[S] Stein, E. M., Harmonic Analysis: Real-Variable Methods, Orthogonality, and Ocillatory Integrals, Princeton Univ. Press, Princeton, N.J., 1993.CrossRefGoogle Scholar
[T] Torchinsky, A., Real-Variable Methods in Harmonic Analysis, Academic Press, 1986.Google Scholar