Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-27T22:34:32.880Z Has data issue: false hasContentIssue false
  • English
  • Français

Anisotropic Hardy-Lorentz Spaces with Variable Exponents

Published online by Cambridge University Press:  20 November 2018

Víctor Almeida ,
Jorge J. Betancor  and
Lourdes Rodríguez-Mesa
Víctor Almeida
Affiliation:
Departamento de Análisis Matemático, Universidad de la Laguna,Campus de Anchieta, Avda., Astrofísico Francisco Sánchez, s/n, 38271, La Laguna (Sta. Cruz de Tenerife, Spain e-mail: valmeida@ull.es, jbetanco@ull.es, lrguez@ull.es
Jorge J. Betancor
Affiliation:
Departamento de Análisis Matemático, Universidad de la Laguna,Campus de Anchieta, Avda., Astrofísico Francisco Sánchez, s/n, 38271, La Laguna (Sta. Cruz de Tenerife, Spain e-mail: valmeida@ull.es, jbetanco@ull.es, lrguez@ull.es
Lourdes Rodríguez-Mesa
Affiliation:
Departamento de Análisis Matemático, Universidad de la Laguna,Campus de Anchieta, Avda., Astrofísico Francisco Sánchez, s/n, 38271, La Laguna (Sta. Cruz de Tenerife, Spain e-mail: valmeida@ull.es, jbetanco@ull.es, lrguez@ull.es
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 we introduce Hardy-Lorentz spaces with variable exponents associated with dilations in ${{\mathbb{R}}^{n}}$. We establish maximal characterizations and atomic decompositions for our variable exponent anisotropic Hardy-Lorentz spaces.

Keywords

42B3042B2542B35variable exponent Hardy spaceHardy-Lorentz spaceanisotropic Hardy spacemaximal functionatomic decomposition
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 69 , Issue 6 , 01 December 2017 , pp. 1219 - 1273
Copyright
Copyright © Canadian Mathematical Society 2016

References

[1] Abu-Shammala, W. and Torchinsky, A., TheHardy-Lorentz spaces Hp,q (). Studia Math. 182(2007), no. 3, 283–294. http://dx.doi.Org/10.4064/sm182-3-7 Google Scholar
[2] Almeida, A. and Caetano, A. M., Generalized Hardy spaces. Acta Math. Sin. (Engl. Ser.) 26(2010), no. 9,1673–1692.http://dx.doi.org/10.1007/s10114-010-8647-9 Google Scholar
[3] Bennett, C. and Sharpley, R., Interpolation of operators. Pure and Applied Mathematics, 129, Academic Press, Inc., Boston, MA, 1988.Google Scholar
[4] Bownik, M., Anisotropic Hardy spaces and wavelets. Mem. Amer. Math. Soc. 164(2003), no. 781.http://dx.doi.Org/10.1090/memo/0781 Google 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/S0002-9939-05-07892-5 Google Scholar
[6] Bownik, M., Li, B., Yang, D., and Zhou, Y., Weighted anisotropic Hardy spaces and their applications in boundedness ofsublinear operators. Indiana Univ. Math. J. 57(2008), no. 7, 3065–3100.http://dx.doi.org/10.1512/iumj.2008.57.3414 Google Scholar
[7] Bui, T. A. and Duong, X. T., Hardy spaces associated to the discrete Laplacians on graphs and boundedness of singular integrals. Trans. Amer. Math. Soc. 366(2014), no. 7, 3451–3485.http://dx.doi.org/10.1090/S0002-9947-2014-05915-1 Google Scholar
[8] Carro, M. J., Raposo, J. A., and Soria, J., Recent developments in the theory ofLorentz spaces and weighted inequalities. Mem. Amer. Math. Soc. 187(2007), no. 877.http://dx.doi.Org/10.1090/memo/0877 Google Scholar
[9] Chang, D.-C., Krantz, S. G., and Stein, E. M., Hp theory on a smooth domain in RN and elliptic boundary value problems. J. Funct. Anal. 114(1993), no. 2, 286–347.http://dx.doi.Org/10.1006/jfan.1993.1069 Google Scholar
[10] Coifman, R. R., A real variable characterization of Hp. Studia Math. 51(1974), 269–274.Google Scholar
[11] Coifman, R. R. and Weiss, G., Analyse harmonique non-commutative sur certains espaces homogénes. Lecture Notes in Mathematics, 242, Springer-Verlag, Berlin-New York, 1971.Google Scholar
[12] Coifman, R. R., Extensions of Hardy spaces and their use in analysis. Bull. Amer. Math. Soc. 83(1977), no. 4, 569–645.http://dx.doi.org/10.1090/S0002-9904-1977-14325-5 Google Scholar
[13] Cruz-Uribe, D., Diening, L., and Fiorenza, A., A new proof of the boundedness of maximal operators on variable Lebesgue spaces. Boll. Unione Mat. Ital. (9) 2(2009), no. 1,151–173.Google Scholar
[14] Cruz-Uribe, D., Fiorenza, A., Martell, J. M., and Perez, C., The boundedness of classical operators on variable Lp spaces. Ann. Acad. Sci. Fenn. Math. 31(2006), no. 1, 239–264.Google Scholar
[15] Cruz-Uribe, D. and Wang, L.-A. D., Variable Hardy spaces. Indiana Univ. Math. J. 63(2014), no. 2, 447–493. http://dx.doi.org/10.1512/iumj.2014.63.5232 Google Scholar
[16] Cruz-Uribe, D., Extrapolation and weighted norm inequalities in the variable Lebesgue spaces. Trans. Amer. Math. Soc. 369(2017), no. 2,1205–1235.Google Scholar
[17] Cruz-Uribe, D. V. and Fiorenza, A., Variable Lebesgue spaces. In: Applied and numerical harmonic analysis, Birkhäuser/Springer, Heidelberg, 2013. http://dx.doi.Org/10.1007/978-3-0348-0548-3 Google Scholar
[18] Cruz-Uribe, D. V., Martell, J. M., and Perez, C., Weights, extrapolation and the theory of Rubio de Francia. Operator Theory: Advances and Applications, 215, Birkhäuser/Springer Basel AG, Basel, 2011.http://dx.doi.Org/10.1007/978-3-0348-0072-3 Google Scholar
[19] Diening, L., Maximal function on generalized Lebesgue spaces L(·). Math. Inequal. Appl. 7(2004), no. 2, 245–253.http://dx.doi.org/10.7153/mia-07-27 Google Scholar
[20] Diening, L., Harjulehto, P., Hästö, P., and Ruzicka, M., Lebesgue and Sobolev spaces with variable exponents. Lecture Notes in Mathematics, 2017 Springer, Heidelberg, 2011.http://dx.doi.org/10.1007/978-3-642-18363-8 Google Scholar
[21] Dierolf, P., Some locally convex spaces of regular distributions. Studia Math. 77(1984), no. 4, 393–412.Google Scholar
[22] Duong, X. T. and Li, J., Hardy spaces associated to operators satisfying Davies-Gaffney estimates and bounded holomorphic functional calculus. J. Funct. Anal. 264(2013), no. 6,1409–1437.http://dx.doi.Org/10.1016/j.jfa.2O13.01.006 Google Scholar
[23] Duong, X. T. and Yan, L., Duality of Hardy and BMO spaces associated with operators with heat kernel bounds. J. Amer. Math. Soc. 18(2005), no. 4, 943–973. http://dx.doi.Org/10.1090/S0894-0347-05-00496-0 Google Scholar
[24] Ephremidze, L., Kokilashvili, V., and Samko, S., Fractional, maximal and singular operators in variable exponent Lorentz spaces. Fract. Calc. Appl. Anal. 11(2008), no. 4, 407–420.Google Scholar
[25] Fefferman, C., Riviére, N. M., and Sagher, Y., Interpolation between Hp spaces: the real method. Trans. Amer. Math. Soc. 191(1974), 75–81.http://dx.doi.Org/10.2307/1996982 Google Scholar
[26] Fefferman, C. and Stein, E. M., Hp spaces of several variables. Acta Math. 129(1972), no. 3-4, 137–193.http://dx.doi.org/10.1007/BF02392215 Google Scholar
[27] Fefferman, R. and Soria, F., The space Weak H1. Studia Math. 85(1986), no. 1,1–16.Google Scholar
[28] Garcia-Cuerva, J., Weighted Hp spaces. Dissertationes Math. (Rozprawy Mat.) 162(1979).Google Scholar
[29] Gogatishvili, A. and Kopaliani, T., Extensions of Rubio de Francia's extrapolation theorem in variable Lebesgue space and application. arxiv:1407.5216 Google Scholar
[30] Grafakos, L., Classical and modern Fourier analysis. Pearson Education, Inc., Upper Saddle River, NJ, 2004.Google Scholar
[31] Grafakos, L. and He, D., Weak Hardy spaces. To appear in a special volume in honor of Professor Shanzhen Lu, Higher Education Press of China. Will be republished by International Press in US. Google Scholar
[32] He, D., Square function characterization of weak Hardy spaces. J. Fourier Anal. Appl. 20(2014), no. 5, 1083–1110.http://dx.doi.org/10.1007/s00041-014-9346-1 Google Scholar
[33] Hofmann, S., Lu, G., Mitrea, D., Mitrea, M., and Yan, L., Hardy spaces associated to non-negative self-adjoint operators satisfying Davies-Gaffney estimates. Mem. Amer. Math. Soc. 214(2011), no. 1007.http://dx.doi.org/10.1090/S0065-9266-2011-00624-6 Google Scholar
[34] Hofmann, S. and Mayboroda, S., Hardy and BMO spaces associated to divergence form elliptic operators. Math. Ann. 344(2009), no. 1, 37–116.http://dx.doi.Org/10.1007/s00208-008-0295-3 Google Scholar
[35] G.|Hu, Littlewood-Paley characterization of weighted anisotropic Hardy spaces. Taiwanese J. Math. 17(2013), no. 2, 675–700.http://dx.doi.Org/10.1165O/tjm.17.2013.1466 Google Scholar
[36] Kempka, H. and Vybfral, J., Lorentz spaces with variable exponents. Math. Nachr. 287(2014), no. 8-9, 938–954.http://dx.doi.org/10.1002/mana.2012002 78 Google Scholar
[37] Kinnunen, J., A stability result on Muckenhoupt-s weights. Publ. Mat. 42(1998), no. 1,153–163.http://dx.doi.Org/10.5565/PUBLMAT_42198_08 Google Scholar
[38] Latter, R. H., A characterization of Hp (Rn ) in terms of atoms. Studia Math. 62(1978), no. 1, 93–101.Google Scholar
[39] Lindenstrauss, J. and Tzafriri, L., Classical Banach spaces. II. Function spaces. Ergebnisse der Mathematik und ihrer Grenzgebiete, 97, Springer-Verlag, Berlin-New York, 1979.Google Scholar
[40] Liu, J., Yang, D., and Yuan, W., Anisotropic Hardy-Lorentz spaces and their applications. Sci. China Math. 59(2016), no. 9, 1669–1720.Google Scholar
[41] Lorentz, G. G., Some new functional spaces. Ann. of Math. (2) 51(1950), 37–55. http://dx.doi.Org/10.2307/1969496 Google Scholar
[42] Lorentz, G. G., On the theory of spaces Λ. Pacific J. Math. 1(1951), 411–429.Google Scholar
[43] Lu, S. Z., Four lectures on real Ht spaces. World Scientific Publishing Co., Inc., River Edge, NJ, 1995.http://dx.doi.Org/10.1142/9789812831194 Google Scholar
[44] Macias, R. A. and Segovia, C.,A decomposition into atoms of distributions on spaces of homogeneous type. Adv. in Math. 33(1979), no. 3, 271–309.http://dx.doi.Org/10.1016/0001-8708(79)90013-6 Google Scholar
[45] 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/S0002-9939-08-09365-9 Google Scholar
[46] Meyer, Y., Wavelets and operators. Cambridge Studies in Advanced Mathematics, 37, Cambridge University Press, Cambridge, 1992.Google Scholar
[47] Meyer, Y., Taibleson, M. H., and Weiss, G., Some functional analytic properties of the spaces Bq generated by blocks. Indiana Univ. Math. J. 34(1985), no. 3, 493–515.http://dx.doi.org/10.1512/iumj.1985.34.34028 Google Scholar
[48] Nakai, E. and Sawano, Y., Hardy spaces with variable exponents and generalized Campanato spaces. J. Funct. Anal. 262(2012), no. 9,3665–3748. http://dx.doi.Org/10.1016/j.jfa.2O12.01.004 Google Scholar
[49] Nekvinda, A., A note on maximal operator on l(pn) and Lp(x) (). J. Funct. Spaces Appl. 5(2007), no. 1, 49–88. http://dx.doi.Org/10.1155/2007/294367 Google Scholar
[50] Ricci, F. and Verdera, J., Duality in spaces of finite linear combinations of atoms. Trans. Amer. Math. Soc. 363(2011), no. 3, 1311–1323.http://dx.doi.org/10.1090/S0002-9947-2010-05036-6 Google Scholar
[51] Samko, S., On a progress in the theory ofLebesgue spaces with variable exponent: maximal and singular operators. Integral Transforms Spec. Funct. 16(2005), no. 5-6, 461–482.http://dx.doi.Org/10.1080/10652460412331320322 Google Scholar
[52] Sawano, Y., Atomic decompositions of Hardy spaces with variable exponents and its application to bounded linear operators. Integral Equations Operator Theory 77(2013), no. 1,123–148.http://dx.doi.org/10.1007/s00020-013-2073-1 Google Scholar
[53] Schwartz, L., Théorie des distributions. Publications de l'Institut de Mathématique de l'Université de Strasbourg, no. IX-X, Hermann, Paris, 1966.Google Scholar
[54] Strichartz, R. S., The Hardy space Hl on manifolds and submanifolds. Canad. J. Math. 24(1972), 915–925.http://dx.doi.Org/10.4153/CJM-1972-091-5 Google Scholar
[55] Strömberg, J.-O. and Torchinsky, A., Weighted Hardy spaces. Lecture Notes in Mathematics, 1381, Springer-Verlag, Berlin, 1989.http://dx.doi.Org/10.1007/BFb0091154 Google Scholar
[56] Szmydt, Z., On regular temperate distributions. Studia Math. 44(1972), 309–314.Google Scholar
[57] Szmydt, Z., Characterization of regular tempered distributions. Ann. Polon. Math. 41(1983), no. 3,255–258.Google Scholar
[58] Tolsa, X., The space Hl for nondoubling measures in terms of a grand maximal operator. Trans.Amer. Math. Soc. 355(2003), no. 1, 315–348.http://dx.doi.org/10.1090/S0002-9947-02-03131-8 Google Scholar
[59] Wang, L.-A. D., Multiplier theorems on anisotropic Hardy spaces. Ph.D. Thesis, University of Oregon, 2012.Google Scholar
[60] Yan, L., Classes of Hardy spaces associated with operators, duality theorem and applications. Trans. Amer. Math. Soc. 360(2008), no. 8, 4383–4408.Google Scholar
[61] Yang, D. and Zhou, Y., A boundedness criterion via atoms for linear operators in Hardy spaces. Constr. Approx. 29(2009), no. 2, 207–218. http://dx.doi.org/10.1007/s00365-008-9015-1 Google Scholar
[62] Zhuo, C., Sawano, Y., and Yang, D., Hardy spaces with variable exponents in RD-spaces and applications. Dissertationes Math. 520(2016), 1–74.http://dx.doi.Org/10.4064/dm744-9-2015 Google Scholar
[63] Zhuo, C., Yang, D., and Liang, Y.|, Intrinsic square function characterizations of Hardy spaces with variable exponents. Bull. Malays. Math. Sci. Soc. 39(2016), no. 4,1541–1577. http://dx.doi.org/10.1007/s40840-015-0266-2 Google Scholar