Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-23T06:30:34.304Z Has data issue: false hasContentIssue false
  • English
  • Français

Some Properties of Triebel–Lizorkin and Besov Spaces Associated with Zygmund Dilations

Published online by Cambridge University Press:  20 November 2018

Fanghui Liao  and
Zongguang Liu
Fanghui Liao
Affiliation:
School of Mathematics & Computational Science, Xiangtan University, Xiangtan 411105, P. R. China e-mail: [email protected]
Zongguang Liu
Affiliation:
Department of Mathematics, China University of Mining & Technology, Beijing 100083, P. R. China e-mail: [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, using Calderón’s reproducing formula and almost orthogonality estimates, we prove the lifting property and the embedding theorem of the Triebel–Lizorkin and Besov spaces associated with Zygmund dilations.

Keywords

42B2042B35Triebel–Lizorkin and Besov spacesRiesz potentialCalderón’s reproducing formulaalmost orthogonality estimateZygmund dilationembedding theorem
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 59 , Issue 4 , 01 December 2016 , pp. 834 - 848
Copyright
Copyright © Canadian Mathematical Society 2016

References

[1] Frazier, M. and Jawerth, B., A discrete transform and decomposition of distribution spaces. J. Funct. Anal. 93(1990), no. 1, 34–170. http://dx.doi.Org/10.1016/0022-1236(90)90137-A Google Scholar
[2] Fefferman, R. and Pipher, J., Multiparameter operators and sharp weighted inequalities. Amer. J. Math. 11(1997), no. 2, 337–369. http://dx.doi.Org/10.1353/ajm.1997.0011 Google Scholar
[3] Gatto, A. E., Product rule and chain rule estimates for fractional derivatives in spaces that satisfy the doubling condition. J. Funct. Anal. 188(2002), no. 1, 27–37. http://dx.doi.Org/10.1OO6/jfan.2OO1.3836 Google Scholar
[4] Gatto, A. E., Segovia, C., and Vagi, S., On fractional differentiation and integration on spaces of homogeneous type. Rev. Mat. Iberoamericana 12(1996), no. 1,111-145. http://dx.doi.Org/10.4171/RMI/196 Google Scholar
[5] Gatto, A. E. and Vagi, S., On Sobolev spaces of fractional order and e-families of operators on spaces of homogeneous type. Studia Math. 133(1999), no. 1,19-27.Google Scholar
[6] Han, Y., The embedding theorem for the Besov and Triebel-Lizorkin spaces on spaces of homogeneous type. Proc. Amer. Math. Soc. 123(1995), no. 7, 2181–2189. http://dx.doi.Org/10.1090/S0002-9939-1995-1249880-9 Google Scholar
[7] Han, Y. and Lu, G., Endpoint estimates for singular integral operators and Hardy spaces associated with Zygmund dilations. Trends in Partial Differential Equations, ALM 10, igh Education Press and International Press, Beijing-Boston, 2009, pp. 99–191.Google Scholar
[8] Liao, F. and Liu, Z., Multi-parameter Triebel-Lizorkin and Besov spaces assaciated with Zygmund dilation. Taiwanese J. Math. 6(2013), no. 6, 2019–2037. http://dx.doi.Org/10.1165O/tjm.17.2013.3243 Google Scholar
[9] Ricci, F. and Stein, E. M., Multiparameter singular integrals and maximal functions. Ann. Inst. Fourier(Grenoble) 42(1992), no. 3, 637–670. http://dx.doi.Org/10.58O2/aif.1304 Google Scholar
[10] Triebel, H., Theory of function spaces. Monographs in Mathematics, 78, Birkhauser-Verlag, Basel, 1983. http://dx.doi.Org/10.1007/978-3-0346-0416-1 Google Scholar
[11] Yang, D., Riesz potentials in Besov and Triebel-Lizorkin spaces over spaces of homogeneous type. Potential Anal. 19(2003), no. 2, 193–210. http://dx.doi.Org/10.1023/A:1023217617339 Google Scholar
[12] Yang, D., Embedding theorems of Besov and Triebel-Lizorkin spaces on spaces of homogeneous type. Sci. China Ser. A 46(2003), no. 2, 187–199. http://dx.doi.Org/10.1360/03ys9020 Google Scholar
[13] Yang, D., Besov and Triebel-Lizorkin spaces related to singular integrals with flag kernel. Rev. Mat. Complut. 22(2009), no. 1, 253–302. http://dx.doi.Org/10.5209/revj:EMA.2009.v22.n1.16358 Google Scholar