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Register a new userHarnack inequality for hypoelliptic second order partial differential operators
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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 $u$.
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We prove weighted $L^p$-Liouville theorems for a class of second order hypoelliptic partial differential operators $mathcal{L}$ on Lie groups $mathbb{G}$ whose underlying manifold is $n$-dimensional space. We show that a natural weight is the right-invariant measure $check{H}$ of $mathbb{G}$. We also prove Liouville-type theorems for $C^2$ subsolutions in $L^p(mathbb{G},check{H})$. We provide examples of operators to which our results apply, jointly with an application to the uniqueness for the Cauchy problem for the evolution operator $mathcal{L}-partial_t$.
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In this paper we characterize global regularity in the sense of Shubin of twisted partial differential operators of second order in dimension $2$. These operators form a class containing the twisted Laplacian, and in bi-unique correspondence with second order ordinary differential operators with polynomial coefficients and symbol of degree $2$. This correspondence is established by a transformation of Wigner type. In this way the global regularity of twisted partial differential operators turns out to be equivalent to global regularity and injectivity of the corresponding ordinary differential operators, which can be completely characterized in terms of the asymptotic behavior of the Weyl symbol. In conclusion we observe that we have obtained a new class of globally regular partial differential operators which is disjoint from the class of hypo-elliptic operators in the sense of Shubin.
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