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Intrinsic Ultracontractivity of Feynman-Kac Semigroups for Symmetric Jump Processes
Consider the symmetric non-local Dirichlet form (D,\D(D)) given by with
\D(D) the closure of the set of functions on with compact
support under the norm , where and is a nonnegative symmetric measurable function on
. Suppose that there is a Hunt process on
corresponding to (D,\D(D)), and that (L,\D(L)) is its infinitesimal
generator. We study the intrinsic ultracontractivity for the Feynman-Kac
semigroup generated by , where is a
non-negative locally bounded measurable function such that Lebesgue measure of
the set is finite for every . By using
intrinsic super Poincar\'{e} inequalities and establishing an explicit lower
bound estimate for the ground state, we present general criteria for the
intrinsic ultracontractivity of . In particular, if
J(x,y)\asymp|x-y|^{-d-\alpha}\I_{\{|x-y|\le
1\}}+e^{-|x-y|^\gamma}\I_{\{|x-y|> 1\}} for some and
, and the potential function for some
, then is intrinsically ultracontractive if and
only if . When , we have the following explicit estimates
for the ground state where are
constants. We stress that, our method efficiently applies to the Hunt process
with finite range jumps, and some irregular potential
function such that .Comment: 31 page
Perturbations of Functional Inequalities for L\'evy Type Dirichlet Forms
Perturbations of super Poincar\'e and weak Poincar\'e inequalities for L\'evy
type Dirichlet forms are studied. When the range of jumps is finite our results
are natural extensions to the corresponding ones derived earlier for diffusion
processes; and we show that the study for the situation with infinite range of
jumps is essentially different. Some examples are presented to illustrate the
optimality of our results
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