Abstract
We characterize Lusin type and cotype for a Banach space in terms of the L p-boundedness of Littlewood-Paley g-functions associated with the Hermite and Laguerre expansions.
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J. J. Betancor, J. C. Fariña, M. T. Martínez and J. L. Torrea, Riesz transform and g-function associated with Bessel operators and their appropriate Banach spaces, Israel Journal of Mathematics 157 (2007), 259–282.
J. J. Betancor, J. C. Fariña, L. Rodríguez Mesa, A. Sanabria and J. L. Torrea, Transference between Laguerre and Hermite settings, Journal of Functional Analysis 254 (2008), 826–850.
A. C. Ruiz and E. Harboure, Weighted norm inequalities for heat-diffusion Laguerre’semigroups, Mathematische Zeitschrift 257 (2007), 329–354.
U. Dinger, Weak type (1, 1) estimates of the maximal function for the Laguerre semigroup in finite dimensions. Revista Matemática Iberoamericana 8 (1992), 93–120.
G. Garrigós, E. Harboure, T. Signes, J. L. Torrea and B. Viviani, A sharp weighted transplantation theorem for Laguerre function expansions, Journal of Functional Analysis 244 (2007), 247–276.
P. Graczyk, J.-J. Loeb, I. A. López P. A. Nowak and W. O. Urbina R., Higher order Riesz transforms, fractional derivatives, and Sobolev spaces for Laguerre expansions, Journal de Mathématiques Pures et Appliquées (9) 84 (2005), 375–405.
C. E. Gutiérrez, A. Incognito and J. L. Torrea, Riesz transforms, g-functions, and multipliers for the Laguerre semigroup, Houston Journal of Mathematics 27 (2001), 579–592.
E. Harboure, C. Segovia, J. L. Torrea and B. E. Viviani, Power weighted l p-inequalities for Laguerre Riesz Transforms, Arkiv för Matematik 46 (2008), 285–313.
E. Harboure, J. L. Torrea and B. E. Viviani, Riesz transforms for Laguerre expansions, Indiana University Mathematics Journal 55 (2006), 999–1014.
S. Kwapień, Isomorphic characterizations of inner product spaces by orthogonal series with vector valued coefficients, Studia Mathematica 44 (1972), 583–595, Collection of articles honoring the completion by Antoni Zygmund of 50 years of scientific activity, VI.
N. N. Lebedev, Special Functions and their Applications, Dover Publications Inc., New York, 1972. Revised edition, translated from the Russian and edited by Richard A. Silverman, unabridged and corrected republication.
R. Macías, C. Segovia and J. L. Torrea, Heat-diffusion maximal operators for Laguerre semigroups with negative parameters, Journal of Functional Analysis 229 (2005), 300–316.
T. Martínez, J. L. Torrea and Q. Xu, Vector-valued Littlewood-Paley-Stein theory for semigroups, Advances in Mathematics 203 (2006), 430–475.
B. Muckenhoupt, Hermite conjugate expansions, Transactions of the American Mathematical Society 139 (1969), 243–260.
B. Muckenhoupt, Conjugate functions for Laguerre expansions, Transactions of the American Mathematical Society 147 (1970), 403–418.
B. Muckenhoupt, Hardy’s inequality with weights, Studia Math. 44 (1972), 31–38. Collection of articles honoring the completion by Antoni Zygmund of 50 years of scientific activity, I.
A. Nowak, Heat-diffusion and Poisson integrals for Laguerre and special Hermite expansions on weighted L p spaces, Studia Mathematica 158 (2003), 239–268.
A. Nowak and K. Stempak, Riesz transforms and conjugacy for Laguerre function expansions of Hermite type, Journal of Functional Analysis 244 (2007), 399–443.
G. Pisier, Martingales with values in uniformly convex spaces, Israel Journal of Mathematics 20 (1975), 326–350.
G. Pisier, Probabilistic methods in the geometry of Banach spaces, in Probability and Analysis (Varenna, 1985), Lecture Notes in Mathematics, Vol. 1206, Springer, Berlin, 1986, pp. 167–241.
K. Stempak and J. L. Torrea, Poisson integrals and Riesz transforms for Hermite function expansions with weights, Journal of Functional Analysis 202 (2003), 443–472.
K. Stempak and J. L. Torrea, BMO results for operators associated to Hermite expansions, Illinois Journal of Mathematics 49 (2005), 1111–1131.
G. Szegő, Orthogonal Polynomials, fourth edn., American Mathematical Society, Colloquium Publications, Vol. XXIII. American Mathematical Society, Providence, RI, 1975.
S. Thangavelu, Lectures on Hermite and Laguerre expansions, Mathematical Notes, Vol. 42, Princeton University Press, Princeton, NJ, 1993. With a preface by Robert S. Strichartz.
Q. Xu, Littlewood-Paley theory for functions with values in uniformly convex spaces, Journal für die Reine und Angewandte Mathematik 504 (1998), 195–226.
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This paper is partially supported by MTM2004/05878, MTM2005-08350-C03-01 and MTM2008-06621-C02-01. The third and fourth authors are also partially supported by grant PI042004/067.
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Betancor, J.J., Fariña, J.C., Rodríguez-Mesa, L. et al. Lusin type and cotype for Laguerre g-functions. Isr. J. Math. 182, 1–30 (2011). https://doi.org/10.1007/s11856-011-0021-9
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DOI: https://doi.org/10.1007/s11856-011-0021-9