Estimates for the Derivatives of the Poisson Kernel on Nilpotent Meta-Abelian Groups |
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Authors: | Richard Penney Roman Urban |
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Institution: | 1. Department of Mathematics, Purdue University, 150 N. University St, West Lafayette, IN, 47907, USA 2. Institute of Mathematics, Wroclaw University, Plac Grunwaldzki 2/4, 50-384, Wroclaw, Poland
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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 ? k , 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?∈?? k , and for each a?∈?? k , 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 ? k . 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 ? k . |
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