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Register a new userPrecise estimates for the subelliptic heat kernel on H-type groups
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Authors
Nathaniel Eldredge
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We establish precise upper and lower bounds for the subelliptic heat kernel on nilpotent Lie groups $G$ of H-type. Specifically, we show that there exist positive constants $C_1$, $C_2$ and a polynomial correction function $Q_t$ on $G$ such that $$C_1 Q_t e^{-frac{d^2}{4t}} le p_t le C_2 Q_t e^{-frac{d^2}{4t}}$$ where $p_t$ is the heat kernel, and $d$ the Carnot-Caratheodory distance on $G$. We also obtain similar bounds on the norm of its subelliptic gradient $| abla p_t|$. Along the way, we record explicit formulas for the distance function $d$ and the subriemannian geodesics of H-type groups.
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The aim of this note is twofold. The first one is to find conditions on the asymptotic sequence which ensures differentiation of a general asymptotic expansion with respect to it. Our method results from the classical one but generalizes it. As an application, our second aim is to give sharp asymptotic estimates at infinity of the heat kernel on H-type groups by the method of differentiation provided we have the result of the isotropic Heisenberg groups.
We obtain Sobolev inequalities for the Schrodinger operator -Delta-V, where V has critical behaviour V(x)=((N-2)/2)^2|x|^{-2} near the origin. We apply these inequalities to obtain pointwise estimates on the associated heat kernel, improving upon earlier results.
We study the long-time behavior of the Cesaro means of fundamental solutions for fractional evolution equations corresponding to random time changes in the Brownian motion and other Markov processes. We consider both stable subordinators leading to equations with the Caputo-Djrbashian fractional derivative and more general cases corresponding to differential-convolution operators, in particular, distributed order derivatives.
We discuss some aspects of the theory of subelliptic estimates.
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