Abstract
We consider singular integral operators of the form (a)Z 1L−1Z2, (b)Z 1Z2L−1, and (c)L −1Z1Z2, whereZ 1 andZ 2 are nonzero right-invariant vector fields, andL is theL 2-closure of a canonical Laplacian. The operators (a) are shown to be bounded onL p for allp∈(1, ∞) and of weak type (1, 1), whereas all of the operators in (b) and (c) are not of weak type (p, p) for anyp∈[1, ∞).
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References
Alexopoulos, G., Inégalités de Harnack paraboliques et transformées de Riesz sur les groupes de Lie résolubles à croissance polynômiale du volume,C. R. Acad. Sci. Paris Sér. I Math. 309 (1989), 661–662.
Bakry, D., Etude des transformées de Riesz dans les variétés riemanniennes à courbure de Ricci minorée,Séminaire de Probabilités XXI, 137–172, Lecture Notes in Math,1247, Springer-Verlag, Berlin-New York, 1987.
Burns, R. J. andRobinson, D. W., A priori inequalities on Lie groups,Preprint.
Fefferman, C. andSánchez-Calle, A. Fundamental solutions for second order subelliptic operators,Ann. of Math. 124 (1986), 247–272.
Hulanicki, A., On the spectrum of the Laplacian on the affine group of the real line,Studia Math. 54 (1976), 199–204.
Hulanicki, A., Subalgebra ofL 1(G) associated with Laplacian on a Lie group,Colloq. Math. 31 (1974), 269–297.
Khalil, I., Sur l'analyse harmonique du groupe affine de la droite,Studia Math. 51 (1974), 140–166.
Lohoué, N., Transformées de Riesz et fonctions de Littlewood-Paley sur les groupes non moyennables,C. R. Acad. Sci. Paris, Sér. I,306 (1988), 327–330.
Lohoué, N., Inégalités de Sobolev pour les formes différentielles sur une variété riemannienne,C. R. Acad. Sci. Paris, Sér. I,301 (1985), 277–280.
Lohoué, N. Comparaison des champs de vecteurs et des puissances du Laplacien sur une variété riemannienne à courbure non positive,J. Funct. Anal. 61 (1985), 164–201.
Magnus, W., Oberhettinger, F. andSoni, R. P.,Formulas and theorems for the special functions of mathematical physics, 3rd Edition, Springer-Verlag, Berlin-Heidelberg-New York, 1966.
Piers, J.-P.,Amenable locally compact groups, John Wiley and Sons, New York, 1984.
Reed, M. andSimon, B.,Methods of modern mathematical physics, I. Functional analysis, Academic Press, New York, 1972.
Saloff-Coste, L., Analyse sur les groupes de Lie à croissance polynomiale,Ark. Mat. 28 (1990), 315–331.
Stein, E. M.,Singular integrals and differentiability properties of functions, Princeton University Press, Princeton 1970.
Strichartz, R. S., Analysis of the Laplacian on the complete Riemannian manifold,J. Funct. Anal. 52 (1983), 48–79.
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Research supported by the Australian Research Council.
Research carried out as a National Research Fellow.
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Gaudry, G.I., Qian, T. & Sjögren, P. Singular integrals associated to the Laplacian on the affine groupax+b . Ark. Mat. 30, 259–281 (1992). https://doi.org/10.1007/BF02384874
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DOI: https://doi.org/10.1007/BF02384874