Abstract
Let S be a semi direct product \(S=N\rtimes A\) where N is a connected and simply connected, non-abelian, nilpotent meta-abelian Lie group and A is isomorphic with ℝk, k > 1. We consider a class of second order left-invariant differential operators on S of the form \(\mathcal{L}_\alpha=L^a+\Delta_\alpha,\) where α ∈ ℝk, and for each a ∈ ℝk, L a is left-invariant second order differential operator on N and \(\Delta_\alpha=\Delta-\langle\alpha,\nabla\rangle,\) where Δ is the usual Laplacian on ℝk. Using some probabilistic techniques (skew-product formulas for diffusions on S and N respectively, the concept of the derivative of a measure, etc.) we obtain an upper bound for the derivatives of the Poisson kernel for \(\mathcal{L}_\alpha.\) During the course of the proof we also get an upper estimate for the derivatives of the transition probabilities of the evolution on N generated by L σ(t), where σ is a continuous function from [0, ∞ ) to ℝk.
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Penney, R., Urban, R. Estimates for the Derivatives of the Poisson Kernel on Nilpotent Meta-Abelian Groups. Potential Anal 41, 187–214 (2014). https://doi.org/10.1007/s11118-013-9368-3
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DOI: https://doi.org/10.1007/s11118-013-9368-3
Keywords
- Poisson kernel
- Evolution kernel
- Harmonic functions
- Left invariant differential operators
- Elliptic and parabolic equations
- Evolution equation
- Meta-Abelian nilpotent Lie groups
- Solvable Lie groups
- Homogeneous groups
- Higher rank NA groups
- Brownian motion
- Exponential functionals of Brownian motion