Hostname: page-component-5cf477f64f-rdph2 Total loading time: 0 Render date: 2025-04-05T17:59:02.039Z Has data issue: false hasContentIssue false
  • English
  • Français

A Multiplier Theorem on Anisotropic Hardy Spaces

Published online by Cambridge University Press:  20 November 2018

Li-an Daniel Wang
Li-an Daniel Wang*
Affiliation:
Department of Mathematics and Statistics, Sam Houston State University, Huntsville, TX, e-mail: ldw021@shsu.edu
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.

We present a multiplier theorem on anisotropic Hardy spaces. When $m$ satisfies the anisotropic, pointwise Mihlin condition, we obtain boundedness of the multiplier operator ${{T}_{m}}:H_{A}^{p}({{\mathbb{R}}^{n}})\,\to \,H_{A}^{p}({{\mathbb{R}}^{n}})$, for the range of $p$ that depends on the eccentricities of the dilation $A$ and the level of regularity of a multiplier symbol $m$. This extends the classical multiplier theorem of Taibleson and Weiss.

Keywords

42B30(42B2542B35)anisotropic Hardy spacemultiplierFourier transform
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 61 , Issue 2 , 01 June 2018 , pp. 390 - 404
Copyright
Copyright © Canadian Mathematical Society 2018

References

[1] Almeida, V., Betancor, J. J., and Rodriguez, L., Anisotropie Hardy-Lorentz Spaces with variable exponents. arxiv:1601.04487Google Scholar
[2] Baernstein, A. II and Sawyer, E., Embedding and multiplier theorems for Hp (R“). Mem. Amer. Math. Soc. 53(1985), no. 318. http://dx.doi.Org/10.1090/memo/031 8Google Scholar
[3] Benyi, Ä. and Bownik, M., Anisotropie classes of homogeneous pseudodifferential Symbols. Studia Math. 200 (2010), no. 1, 41–66. http://dx.doi.org/10.4064/sm200-1-3Google Scholar
[4] Bownik, M., Anisotropie Hardy Spaces and wavelets. Mem. Amer. Math. Soc. 164(2003), no. 781. http://dx.doi.Org/10.1090/memo/0781Google Scholar
[5] Bownik, M., Boundedness of Operators on Hardy Spaces via atomic decompositions. Proc. Amer. Math. Soc. 133 (2005), no. 12, 3535–3542. http://dx.doi.org/10.1090/S0002-9939-05-07892-5Google Scholar
[6] Bownik, M., Atomic and molecular decompositions of anisotropic Besov Spaces. Math. Z. 250 (2005), no. 3, 539–571. http://dx.doi.org/10.1007/s00209-005-0765-1Google Scholar
[7] Bownik, M. and Ho, K.-P., Atomic and molecular decompositions of anisotropic Triebel-Lizorkin Spaces. Trans. Amer. Math. Soc. 358(2006), no. 4, 1469–1510. http://dx.doi.org/10.1090/S0002-9947-05-03660-3Google Scholar
[8] 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 (2008), no. 7, 3065–3100. http://dx.doi.org/10.1512/iumj.2008.57.3414Google Scholar
[9] Bownik, M. and Wang, L.-A. D., Fourier transform of anisotropic Hardy Spaces. Proc. Amer. Math. Soc. 141 (2013), no. 7, 2299–2308. http://dx.doi.org/10.1090/S0002-9939-2013-11623-0Google Scholar
[10] Calderön, A. and Torchinsky, A., Parabolic maximal funetions associated with a distribution. Advances in Math. 16 (1975), 1–64. http://dx.doi.Org/10.101 6/0001 -8708(75)90099-7Google Scholar
[11] Dekel, S., Petrushev, P., and Weissblat, T., Hardy Spaces on R” with pointwise variable anisotropy. J. Fourier Anal. Appl. 17 (2011), no. 5, 1066–1107. http://dx.doi.org/10.1007/s00041-011-9176-3Google Scholar
[12] Ding, Y. and Lan, S., Some multiplier theorems for anisotropic Hardy Spaces. Anal. Theory Appl. 22 (2006), no. 4, 339–352. http://dx.doi.Org/10.1007/s10496-006-0339-zGoogle Scholar
[13] Fefferman, C. and Stein, E., HP Spaces ofseveral variables. Acta Math. 129(1972), no. 3-4, 137–193. http://dx.doi.Org/10.1007/BF02392215Google Scholar
[14] Lemarie-Rieusset, P., Projecteurs invariants, matrices de dilatation, ondelettes et analyses multi-resolutions. Rev. Mat. Iberoamericana 10 (1994), no. 2, 283–347. http://dx.doi.Org/10.41 71/RMI/1 53Google Scholar
[15] Meda, S., Sjögren, P., and Vallarino, M., On the Hl-Ll boundedness of Operators. Proc. Amer. Math. Soc. 136 (2008), no. 8, 2921–2931. http://dx.doi.org/10.1090/S0002-9939-08-09365-9Google Scholar
[16] Peetre, J., On Spaces of Triebel-Lizorkin type. Ark. Mat. 13 (1975), 123–130. http://dx.doi.org/10.1007/BF02386201Google Scholar
[17] Taibleson, M. and Weiss, G., The molecular characterization ofcertain Hardy Spaces. In: Representation theorems for Hardy Spaces, Asterisque, 77, Soc. Math. France, Paris, 1980, pp. 67–149.Google Scholar