Hostname: page-component-76fb5796d-skm99 Total loading time: 0 Render date: 2024-04-27T14:54:29.050Z Has data issue: false hasContentIssue false
  • English
  • Français

Weak Type Estimates for Riesz-Laguerre Transforms

Published online by Cambridge University Press:  17 April 2009

Emanuela Sasso
Emanuela Sasso
Affiliation:
Dipartimento di Matematica, Università di Genova, via Dodecanesco 35, 16146 Genova, Italy e-mail: sasso@dima.unige.it
Rights & Permissions [Opens in a new window]

Extract

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 prove that the first order Riesz transforms associated to the Laguerre semigroup are weak-type (1, 1). We also present a counterexample showing that for the Riesz transforms of order three or higher the weak type (1, 1) estimate fails.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 75 , Issue 3 , June 2007 , pp. 397 - 408
Copyright
Copyright © Australian Mathematical Society 2007

References

lsqb;1]Coifman, R.R. and Weiss, G., Analyse harmonique non-commutative sur certains espaces homogènes Lecture Notes in Mathématiques 242 (Springer-Verlag, Berlin, New York, 1971).CrossRefGoogle Scholar
[2]Dinger, U., ‘Weak Type (1, 1) Estimates of the maximal function for the Laguerre semi-group in finite dimensions’, Rev. Mat. Iberoamericana 8 (1992), 93–118.CrossRefGoogle Scholar
[3]Garcìa-Cuerva, J., Mauceri, G., Sjögren, P. and Torrea, J.L., ‘Higher-Order Riesz operators for Ornstein-Uhlenbeck semigroups’, Potential Anal. 10 (1999), 379–407.CrossRefGoogle Scholar
[4]Graczyk, P., Loeb, J.J., Lopez, I., Nowak, A. and Urbina, W., ‘Higher order Riesz transforms, fractional derivatives, and Sobolev spaces for Laguerre expansions’, J. Math. Pures Appl. 84 (2005), 375–405.CrossRefGoogle Scholar
[5]Gutiérrez, C., Incognito, A. and Torrea, J.L., ‘Riesz Transforms, g-functions and multipliers for Laguerre semigroup’, Houston J. Math. 27 (2001), 579–592.Google Scholar
[6]Muckenhoupt, B., ‘Conjugate functions for Laguerre expansions’, Trans. Amer. Math. Soc. 139 (1969), 231–242.CrossRefGoogle Scholar
[7]Nowak, A., ‘On Riesz transforms for Laguerre expansions’, J. Funct. Anal. 215 (2004), 217–240.CrossRefGoogle Scholar
[8]Menárguez, T., Pérez, S. and Soria, F., ‘The Mehler maximal function: a geometric proof of the weak type 1’, J. London Math. Soc. (2) 61 (2000), 846–856.CrossRefGoogle Scholar
[9]Pérez, S. and Soria, F., ‘Operators associated with the Ornstein-Uhlenbeck semigroup’, J. London Math. Soc. 61 (2000), 857–871.Google Scholar
[10]Sasso, E., ‘Functional calculus for the Laguerre operator’, Math. Z. 249 (2005), 683–711.Google Scholar
[11]Sasso, E., ‘Maximal operators for the holomorphic Laguerre semigroup’, Math. Scand. 97 (2005), 235–265.CrossRefGoogle Scholar
[12]Sasso, E., ‘Spectral multipliers of Laplace transform type for the Laguerre operator’, Bull. Austral. Math. Soc. 69 (2004), 255–266.Google Scholar
[13]Szegö, G., Orthogonal polynomials, (Revised Edition) (Amer. Math. Soc., Providence, 1959).Google Scholar