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Register a new userOn $L^p$ Liouville theorems for Dirichlet forms
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We study harmonic functions for general Dirichlet forms. First we review consequences of Fukushimas ergodic theorem for the harmonic functions in the domain of the $ L^{p} $ generator. Secondly we prove analogues of Yaus and Karps Liouville theorems for weakly harmonic functions. Both say that weakly harmonic functions which satisfy certain $ L^{p} $ growth criteria must be constant. As consequence we give an integral criterion for recurrence.
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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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