Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

An estimate for a first-order Riesz operator
on the affine group


Author: Peter Sjögren
Journal: Trans. Amer. Math. Soc. 351 (1999), 3301-3314
MSC (1991): Primary 43A80, 42B20; Secondary 22E30
DOI: https://doi.org/10.1090/S0002-9947-99-02222-9
Published electronically: March 29, 1999
MathSciNet review: 1475695
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: On the affine group of the line, which is a solvable Lie group of exponential growth, we consider a right-invariant Laplacian $\Delta$. For a certain right-invariant vector field $X$, we prove that the first-order Riesz operator $X\Delta^{-1/2}$ is of weak type (1, 1) with respect to the left Haar measure of the group. This operator is therefore also bounded on $L^p, \; 1<p\leq 2$. Locally, the operator is a standard singular integral. The main part of the proof therefore concerns the behaviour of the kernel of the operator at infinity and involves cancellation.


References [Enhancements On Off] (What's this?)

  • 1. G. Alexopoulos, An application of homogenization theory to harmonic analysis: Harnack inequalities and Riesz transforms on Lie groups of polynomial growth. Can. J. Math. 44 (1992), 691-727. MR 93j:22006
  • 2. J.-Ph. Anker, Sharp estimates for some functions of the Laplacian on noncompact symmetric spaces, Duke Math. J. 65 (1992), 257-297. MR 93b:43007
  • 3. D. Bakry, Etude des transformations de Riesz dans les variétés riemanniennes à courbure de Ricci minorée. In Séminaire de Probabilité XXI, J. Azéma et al. (eds), Lecture Notes in Mathematics 1247, Springer-Verlag 1987, 137-172. MR 89h:58208
  • 4. M.S. Birman and M.Z. Solomjak, Spectral theory of self-adjoint operators in Hilbert space. (Mathematics and its applications (Soviet series) Reidel, Dordrecht 1987. MR 93g:47001
  • 5. R. Burns, A.F.M. ter Elst and D.W. Robinson, $L_p$-regularity of subelliptic operators on Lie groups. J. Operator Theory 31 (1994), 165-187. MR 96a:22013
  • 6. Th. Coulhon and X.Th. Duong, Riesz transforms for $1 \leq p \leq 2$. To appear in Trans. Amer. Math. Soc. CMP 97:15
  • 7. R.E. Edwards and G.I. Gaudry, Littlewood-Paley and multiplier theory. Ergebnisse der Mathematik und ihrer Grenzgebiete 90, Springer-Verlag, Berlin Heidelberg New York 1977. MR 58:29760
  • 8. G.I. Gaudry, T. Qian and P. Sjögren, Singular integrals related to the Laplacian on the affine group $ax+b$. Ark. Mat. 30 (1992), 259-281.
  • 9. G. Gaudry and P. Sjögren, Singular integrals on Iwasawa $NA$ groups of rank $1$. J. Reine Angew. Math. 479 (1996), 39-66.MR 97k:22012
  • 10. -, Singular integrals on the complex affine group. Coll. Math. 75 (1998), 133-148. CMP 98:06
  • 11. -, Haar-like expansions and boundedness of a Riesz operator on a solvable group. To appear in Math. Zeitschr.
  • 12. A. Hulanicki, On the spectrum of the Laplacian on the affine group of the real line. Studia Math. 54 (1976), 199-204. MR 53:3195
  • 13. N. Lohoué, Comparaison des champs de vecteurs et des puissances du laplacien sur une variétés riemanniennes à courbure non positive. J. Funct. Anal. 61 (1985), 164-201. MR 86k:58117
  • 14. -, Transformées de Riesz et fonctions de Littlewood-Paley sur les groupes non moyennables. C. R. Acad. Sci. Paris Sér. I Math. 306 (1988), 327-330.MR 89b:43008
  • 15. N. Lohoué and N.Th. Varopoulos, Remarques sur les transformées de Riesz sur les groupes de Lie nilpotents. C. R. Acad. Sci. Sér. I Math. 301 (1985), 559-560. MR 87b:43008
  • 16. W. Magnus and F. Oberhettinger, Formulas and theorems for the special functions of mathematical physics. Chelsea Publishing Company, New York 1949. MR 10:532b
  • 17. L. Saloff-Coste, Analyse sur les groupes de Lie à croissance polynômiale. Ark. Mat. 28 (1990), 315-331. MR 92d:22014
  • 18. E.M. Stein, Singular integrals and differentiability properties of functions. Princeton University Press, Princeton 1970. MR 44:7280
  • 19. E.M. Stein and N.J. Weiss, On the convergence of Poisson integrals. Trans. Amer. Math. Soc. 140 (1969), 35-54. MR 39:3024
  • 20. J.-O. Strömberg, Weak type $L^1$ estimates for maximal functions on non-compact symmetric spaces and biinvariant convolutions. Unpublished manuscript, 1979.
  • 21. -, Weak type $L^1$ estimates for maximal functions on non-compact symmetric spaces. Ann. Math. 114 (1981), 115-126. MR 82k:43010

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 43A80, 42B20, 22E30

Retrieve articles in all journals with MSC (1991): 43A80, 42B20, 22E30


Additional Information

Peter Sjögren
Affiliation: Department of Mathematics Chalmers University of Technology and Göteborg University S-412 96 Göteborg Sweden
Email: peters@math.chalmers.se

DOI: https://doi.org/10.1090/S0002-9947-99-02222-9
Received by editor(s): December 15, 1996
Received by editor(s) in revised form: August 15, 1997
Published electronically: March 29, 1999
Article copyright: © Copyright 1999 American Mathematical Society

American Mathematical Society