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ON LITTLEWOOD-PALEY FUNCTIONS ASSOCIATED WITH BESSEL OPERATORS

Published online by Cambridge University Press:  01 January 2009

J. J. BETANCOR ,
J. C. FARIÑA  and
A. SANABRIA
J. J. BETANCOR
Affiliation:
Departamento de Análisis Matemático, Universidad de La Laguna, 38271 – La Laguna, Tenerife, Islas Canarias, España e-mail: jbetanco@ull.es
J. C. FARIÑA
Affiliation:
Departamento de Análisis Matemático, Universidad de La Laguna, 38271 – La Laguna, Tenerife, Islas Canarias, España e-mail: jcfarina@ull.es
A. SANABRIA
Affiliation:
Departamento de Análisis Matemático, Universidad de La Laguna, 38271 – La Laguna, Tenerife, Islas Canarias, España e-mail: asgarcia@ull.es
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Abstract

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In this paper, we study Lp-boundedness properties for higher order Littlewood-Paley g-functions in the Bessel setting. We use the Calderón-Zygmund theory in a homogeneous-type space (in the sense of Coifman and Weiss) ((0, ∞), d, γα), where d represents the usual metric on (0, ∞) and γα denotes the doubling measure on (0, ∞) with respect to d defined by dγα(x) = x2α+1dx, with α > −1/2.

Keywords

42C05 (primary)42C15 (secondary)
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 51 , Issue 1 , January 2009 , pp. 55 - 70
Copyright
Copyright © Glasgow Mathematical Journal Trust 2008

References

REFERENCES

1.Andersen, K. F. and Kerman, R. A., Weighted norm inequalities for generalized Hankel conjugate transformations, Stud. Math. 71 (1) (1981/82), 15–26.CrossRefGoogle Scholar
2.Betancor, J. J., Buraczewski, D., Fariña, J. C., Martínez, M. and Torrea, J. L., Riesz transforms related to Bessel operators, Proc. Roy. Soc. Edinb. 137A (2007), 701–725.CrossRefGoogle Scholar
3.Buraczewski, D., Martínez, M., Torrea, J. L. and Urban, R., On the Riesz transform associated to the ultraspherical polynomials, J. d'Analyse Math. 98 (2006), 113–143.CrossRefGoogle Scholar
4.Buraczewski, D., Martínez, M. and Torrea, J. L., Calderón-Zygmund operators associated to ultraspherical expansions, Can. J. Math. 59 (6) (2007), 1223–1244.CrossRefGoogle Scholar
5.Coifman, R. and Meyer, Y., Au dela des operateurs pseudo-differentielles. (Astérisque 57 (Société Mathématique de France, Paris, 1978).Google Scholar
6.Coifman, R. and Weiss, G., Analyse harmonique non-commutative sur certains espaces homogènes. Lecture Notes in Mathematics, Vol. 242 (Springer-Verlag, Berlin-New York, 1971).CrossRefGoogle Scholar
7.Diestel, J. and Ull, J., Vector measures, Mathematical Surveys, No. 15. (American Mathematical Society, Providence, RI, 1977).CrossRefGoogle Scholar
8.Duoandikoetxea, J., Fourier analysis. Graduate Studies in Mathematics, Vol. 29 (American Mathematical Society, Providence, RI, 2001).Google Scholar
9.García-Cuerva, J., Mauceri, G., Sjögren, P. and Torrea, J. L., Spectral multipliers for the Ornstein-Uhlenbeck semigroup, J. Anal. Math. 78 (1999), 281–305.CrossRefGoogle Scholar
10.Macías, R. and Segovia, C., A decomposition into atoms of distributions on spaces of homogeneous type, Adv. Math. 33 (3) (1979), 271–309.CrossRefGoogle Scholar
11.Macías, R. and Segovia, C., Lipschitz functions on spaces of homogeneous type, Adv. Math. 33 (3) (1979), 257–270.CrossRefGoogle Scholar
12.Macías, R., Segovia, C. and Torrea, J. L., Heat-diffusion maximal operators for Laguerre semigroups with negative parameters, J. Funct. Anal. 229 (2) (2005), 300–316.CrossRefGoogle Scholar
13.Muckenhoupt, B. and Stein, E., Classical expansions and their relations to conjugate harmonic functions, Trans. Amer. Math. Soc. 118 (1965), 17–92.CrossRefGoogle Scholar
14.Nowak, A. and Stempak, K., Riesz transforms for multi-dimensional Laguerre function expansions, Adv. Math. 215 (2007), 642–678.CrossRefGoogle Scholar
15.Rubio de Francia, J. L., Ruiz, F. J. and Torrea, J. L., Les opérateurs de Calderón-Zygmund vectoriels, C. R. Acad. Sci. Paris Serie I 297 (1983), 477–480.Google Scholar
16.Ruiz, F. J. and Torrea, J. L., Vector-valued Calderón-Zygmund theory and Carleson measures on spaces of homogeneous nature, Stud. Math. 88 (3) (1988), 221–243.CrossRefGoogle Scholar
17.Stein, E. M., Singular integrals and differentiability properties of functions. Princeton Mathematical Series, No. 30 (Princeton University Press, Princeton, NJ 1970).Google Scholar
18.Stein, E. M., Topics in harmonic analysis related to the Littlewood Paley theory, (Princeton University. Press, Princeton, NJ 1970).CrossRefGoogle Scholar
19.Stempak, K., The Littlewood-Paley theory for the Fourier-Bessel transform. Preprint no. 45 (Mathematical Institute of Wroclaw, Poland, 1985).Google Scholar
20.Stempak, K. and Torrea, J. L., Poisson integrals and Riesz transforms for Hermite function expansions with weights, J. Funct. Anal. 202 (2) (2003), 443–472.CrossRefGoogle Scholar
21.Stempak, K. and Torrea, J. L., On g-functions for Hermite function expansions, Acta Math. Hungar. 109 (1–2) (2005), 99–125.CrossRefGoogle Scholar
22.Stempak, K. and Torrea, J. L., BMO results for operators associated to Hermite expansions, Illinois J. Math. 49 (4) (2005), 1111–1131.CrossRefGoogle Scholar
23.Stempak, K. and Torrea, J. L., Higher Riesz transforms and imaginary powers associated to the harmonic oscillator, Acta Math. Hungar. 111 (1–2) (2006), 43–64.CrossRefGoogle Scholar
24.Watson, G. N., A treatise on the theory of Bessel functions, Cambridge University Press, (Cambridge, England; The Macmillan Company, New York, 1944).Google Scholar
25.Weinstein, A., Discontinuous integrals and generalized potential theory, Trans. Amer. Math. Soc. 63 (1948), 342–354.CrossRefGoogle Scholar