Abstract
In this paper we introduce the atomic Hardy space \(\mathcal {H}^1((0,\infty ),\gamma _\alpha )\) associated with the non-doubling probability measure \(d\gamma _\alpha (x)=\frac{2x^{2\alpha +1}}{\Gamma (\alpha +1)}e^{-x^2}dx\) on \((0,\infty )\), for \({\alpha >-\frac{1}{2}}\). We obtain characterizations of \(\mathcal {H}^1((0,\infty ),\gamma _\alpha )\) by using two local maximal functions. We also prove that the truncated maximal function defined through the heat semigroup generated by the Laguerre differential operator is bounded from \(\mathcal {H}^1((0,\infty ),\gamma _\alpha )\) into \(L^1((0,\infty ),\gamma _\alpha )\).
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References
Betancor, J.J., Dalmasso, E., Quijano, P., Scotto, R.: Harmonic analysis operators associated with Laguerre polynomial expansions on variable Lebesgue spaces, (2022). arXiv:2202.11137
Betancor, J.J., Dziubański, J., Torrea, J.L.: On Hardy spaces associated with Bessel operators. J. Anal. Math. 107, 195–219 (2009)
Bownik, M.: Boundedness of operators on Hardy spaces via atomic decompositions. Proc. Am. Math. Soc. 133(12), 3535–3542 (2005)
Carbonaro, A., Mauceri, G., Meda, S.: \(H^1\) and BMO for certain locally doubling metric measure spaces of finite measure. Colloq. Math. 118(1), 13–41 (2010)
Cha, L., Liu, H.: BMO-boundedness of maximal operators and \(g\)-functions associated with Laguerre expansions. J. Funct. Spaces Appl. 21, 923874 (2012)
Cha, L., Liu, H.: \(BMO\) spaces for laguerre expansions. Taiwanese J. Math. 16(6), 2153–2186 (2012)
Coifman, R.R., Weiss, G.: Analyse harmonique non-commutative sur certains espaces homogènes. Lecture Notes in Mathematics, vol. 242. Springer-Verlag, Berlin-New York (1971)
Dziubański, J.: Hardy spaces for Laguerre expansions. Constr. Approx. 27(3), 269–287 (2008)
Gosselin, J., Stempak, K.: A weak-type estimate for fourier-bessel multipliers. Proc. Am. Math. Soc. 106(3), 655–662 (1989)
Gutiérrez, C.E., Incognito, A., Torrea, J.L.: Riesz transforms, \(g\)-functions, and multipliers for the Laguerre semigroup. Houston J. Math. 27(3), 579–592 (2001)
Haimo, D.T.: Integral equations associated with Hankel convolutions. Trans. Am. Math. Soc. 116, 330–375 (1965)
Harboure, E., Torrea, J.L., Viviani, B.: On the search for weighted inequalities for operators related to the Ornstein-Uhlenbeck semigroup. Math. Ann. 318(2), 341–353 (2000)
Hirschman, I.I., Jr.: Variation diminishing Hankel transforms. J. Analyse Math. 8(61), 307–336 (1960)
Lebedev, N.N., “Special functions and their applications”. Dover Publications Inc, New York,: Revised edition, translated from the Russian and edited by Richard A. Unabridged and corrected republication, Silverman (1972)
Maas, J., van Neerven, J., Portal, P.: Conical square functions and non-tangential maximal functions with respect to the Gaussian measure. Publ. Mat. 55(2), 313–341 (2011)
Macías, R.A., Segovia, C.: A decomposition into atoms of distributions on spaces of homogeneous type. Adv. Math. 33(3), 271–309 (1979)
Mauceri, G., Meda, S.: \({\rm BMO}\) and \(H^1\) for the Ornstein-Uhlenbeck operator. J. Funct. Anal. 252(1), 278–313 (2007)
Mauceri, G., Meda, S., Sjögren, P.: A maximal function characterization of the Hardy space for the Gauss measure. Proc. Am. Math. Soc. 141(5), 1679–1692 (2013)
Muckenhoupt, B.: Conjugate functions for Laguerre expansions. Trans. Am. Math. Soc. 147, 403–418 (1970)
Portal, P.: Maximal and quadratic Gaussian Hardy spaces. Rev. Mat. Iberoam. 30(1), 79–108 (2014)
Sasso, E.: Spectral multipliers of Laplace transform type for the Laguerre operator. Bull. Austral. Math. Soc. 69(2), 255–266 (2004)
Stein, E.M.: Interpolation of linear operators. Trans. Amer. Math. Soc. 83, 482–492 (1956)
Stein, E.M.: “Topics in harmonic analysis related to the Littlewood-Paley theory”. Annals of Mathematics Studies, No. 63. Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, (1970)
Tolsa, X.: BMO, \(H^1\), and Calderón-Zygmund operators for non doubling measures. Math. Ann. 319(1), 89–149 (2001)
Urbina-Romero, W.: “Gaussian harmonic analysis”. Springer Monographs in Mathematics. Springer, Cham, (2019). With a foreword by Sundaram Thangavelu
Yang, D., Yang, D.: Real-variable characterizations of Hardy spaces associated with Bessel operators. Anal. Appl. (Singap.) 9(3), 345–368 (2011)
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The first author is partially supported by PID2019-106093GB-I00 (Ministerio de Ciencia e Innovación, Spain). The second and fourth authors are partially supported by grants PICT-2019-2019-00389 (ANPCyT), PIP-11220200101916CO (CONICET) and CAI+D 2019-015 (UNL).
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Betancor, J.J., Dalmasso, E., Quijano, P. et al. Maximal function characterization of Hardy spaces related to Laguerre polynomial expansions. Collect. Math. (2024). https://doi.org/10.1007/s13348-024-00433-z
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DOI: https://doi.org/10.1007/s13348-024-00433-z