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Register a new userSharp Harnack inequalities for a family of hypoelliptic diffusions
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P.W.Y. Lee
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We prove sharp Harnack inequalities for a family of Kolmogorov-Fokker-Planck type hypoelliptic diffusions.
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We consider nonnegative solutions $u:Omegalongrightarrow mathbb{R}$ of second order hypoelliptic equations begin{equation*} mathscr{L} u(x) =sum_{i,j=1}^n partial_{x_i} left(a_{ij}(x)partial_{x_j} u(x) right) + sum_{i=1}^n b_i(x) partial_{x_i} u(x) =0, end{equation*} where $Omega$ is a bounded open subset of $mathbb{R}^{n}$ and $x$ denotes the point of $Omega$. For any fixed $x_0 in Omega$, we prove a Harnack inequality of this type $$sup_K u le C_K u(x_0)qquad forall u mbox{ s.t. } mathscr{L} u=0, ugeq 0,$$ where $K$ is any compact subset of the interior of the $mathscr{L}$-propagation set of $x_0$ and the constant $C_K$ does not depend on $u$.
We give some a priori estimates of type sup*inf for Yamabe and prescribed scalar curvature type equations on Riemannian manifolds of dimension >2. The product sup*inf is caracteristic of those equations, like the usual Harnack inequalities for non negative harmonic functions. First, we have a lower bound for sup*inf for some classes of PDE on compact manifolds (like prescribed scalar cuvature). We also have an upper bound for the same product but on any Riemannian manifold not necessarily compact. An application of those result is an uniqueness solution for some PDE.
We prove matrix and scalar differential Harnack inequalities for linear parabolic equations on Riemannian and Kahler manifolds.
This work examines a class of switching jump diffusion processes. The main effort is devoted to proving the maximum principle and obtaining the Harnack inequalities. Compared with the diffusions and switching diffusions, the associated operators for switching jump diffusions are non-local, resulting in more difficulty in treating such systems. Our study is carried out by taking into consideration of the interplay of stochastic processes and the associated systems of integro-differential equations.
The sharp trace inequality of Jose Escobar is extended to traces for the fractional Laplacian on R^n and a complete characterization of cases of equality is discussed. The proof proceeds via Fourier transform and uses Liebs sharp form of the Hardy-Littlewood-Sobolev inequality.
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