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Transactions of the American Mathematical Society

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Harmonic functions on semidirect extensions of type $ H$ nilpotent groups


Author: Ewa Damek
Journal: Trans. Amer. Math. Soc. 290 (1985), 375-384
MSC: Primary 43A80; Secondary 22E27, 22E30, 31C12
DOI: https://doi.org/10.1090/S0002-9947-1985-0787971-5
MathSciNet review: 787971
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Abstract: Let $ S = NA$ be a semidirect extension of a Heisenberg type nilpotent group $ N$ by the one-parameter group of dilations, equipped with the Riemannian structure, which generalizes this of the symmetric space. Let $ {\{ {P_a}(y)\} _{a > 0}}$ be a Poisson kernel on $ N$ with respect to the Laplace-Beltrami operator. Then every bounded harmonic function $ F$ on $ S$ is a Poisson integral $ F(yb) = f \ast {P_b}(y)$ of a function $ f \in {L^\infty }(N)$. Moreover the harmonic measures $ \mu _a^b$ defined by $ {P_b} = {P_a} \ast \mu _a^b,b > a$, are radial and have smooth densities. This seems to be of interest also in the case of a symmetric space of rank $ 1$.


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  • [1] J. Cygan, A tangential convergence for bounded harmonic functions on a rank one symmetric space, Trans. Amer. Math. Soc. 265 (1981), 405-418. MR 610957 (83h:43010)
  • [2] E. Damek, A Poisson kernel on type $ H$ nilpotent groups, Colloq. Math. (to appear). MR 924068 (89d:22006)
  • [3] -, The geometry of a semi-direct extension of a type $ H$ nilpotent group, Colloq. Math. (to appear).
  • [4] A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher transcendental functions, McGraw-Hill, New York, 1953.
  • [5] G. B. Folland, Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat. 13 (1975), 161-207. MR 0494315 (58:13215)
  • [6] H. Furstenberg, A Poisson formula for semi-simple Lie groups, Ann. of Math. (2) 77 (1963), 335-386. MR 0146298 (26:3820)
  • [7] A. Hulanicki and F. Ricci, A Tauberian theorem and tangential convergence of bounded harmonic functions on balls in $ {C^n}$, Invent. Math. 62 (1980), 325-331. MR 595591 (82e:32008)
  • [8] A. Kaplan, Fundamental solution for a class of hypoelliptic $ PDE$ generated by composition of quadratic forms, Trans. Amer. Math. Soc. 258 (1980), 147-153. MR 554324 (81c:58059)
  • [9] A. Korányi, Harmonic functions on symmetric spaces, Symmetric Spaces, Marcel Dekker, New York, 1972. MR 0407541 (53:11314)
  • [10] T. Przebinda, A tangential convergence for bounded harmonic functions on a rank one symmetric space (preprint).
  • [11] F. Ricci, Harmonic analysis on generalized Heisenberg groups (preprint).
  • [12] C. Riehm, The automorphism group of a composition of quadratic forms, Trans. Amer. Math. Soc. 269 (1982), 403-414. MR 637698 (82m:10038)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1985-0787971-5
Keywords: Laplace-Beltrami operator, type $ H$ nilpotent groups
Article copyright: © Copyright 1985 American Mathematical Society

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