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Poisson kernels on semi-direct products of abelian groups

  • Richard Penney EMAIL logo and Roman Urban
Published/Copyright: December 30, 2016
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Mathematica Slovaca
From the journal Mathematica Slovaca Volume 66 Issue 6

Abstract

Let G be a semi direct product G = ℝd ⋊ ℝk. On G we consider a class of second order left-invariant differential operators of the form Lα=∑j=1de2λj(a)∂xj2+∑j=1k(∂aj2−2αj∂aj), where a ∈ ℝk and λ1,..., λd ∈ (ℝk )*. It is known that bounded 𝓛α-harmonic functions on G are precisely the “Poisson integrals” of L∞(ℝd) against the Poisson kernel να which is a smooth function on ℝd. We prove an upper bound of να and its derivatives.

MSC 2010: Primary 43A85; 31B05; 22E25; 22E30; 60J25; 60J60
Keywords: left invariant differential operators; Poisson kernel; time-dependent parabolic operators; Brownian motion; evolution kernel; diffusion process; solvable Lie groups

(Communicated by Gregor Dolinar)

Acknowledgement

The authors wishes to thank the anonymous referees for several remarks that improved the overall presentation of the results.

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Received: 2014-4-23
Accepted: 2015-1-16
Published Online: 2016-12-30
Published in Print: 2016-12-1

© 2016 Mathematical Institute Slovak Academy of Sciences

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https://doi.org/10.1515/ms-2016-0230
Keywords for this article
left invariant differential operators; Poisson kernel; time-dependent parabolic operators; Brownian motion; evolution kernel; diffusion process; solvable Lie groups

Articles in the same Issue

  2. A triple representation of lattice effect algebras
  4. Periods of morgan-voyce sequences and elliptic curves
  6. Multiplicative generalized derivations on ideals in semiprime rings
  8. A few remarks on Poincaré-Perron solutions and regularly varying solutions
  10. Applications of extremal theorem and radius equation for a class of analytic functions
  12. On the “bang-bang” principle for a class of Riemann-Liouville fractional semilinear evolution inclusions
  14. New criteria for global exponential stability of linear time-varying volterra difference equations
  16. On approximation of functions by some hump matrix means of Fourier series
  18. Pseudo-amenability and pseudo-contractibility for certain products of Banach algebras
  20. Poisson kernels on semi-direct products of abelian groups
  22. Locally convex projective limit cones
  24. On the properties (wL) and (wV)
  26. On the existence of solutions for quadratic integral equations in Orlicz spaces
  28. Existence and uniqueness of best proximity points under rational contractivity conditions
  30. Almost Weyl structures on null geometry in indefinite Kenmotsu manifolds
  32. On the internal approach to differential equations 3. Infinitesimal symmetries
  34. A Characterization of the discontinuity point set of strongly separately continuous functions on products
  36. Wick differential and Poisson equations associated to the 𝚀𝚆𝙽-Euler operator acting on generalized operators
  38. Multivariate EIV models
  40. On codes over 𝓡k, m and constructions for new binary self-dual codes
  42. Domination number of total graphs
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