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Register a new userFunctional inequalities for some generalised Mehler semigroups
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We consider generalised Mehler semigroups and, assuming the existence of an associated invariant measure $sigma$, we prove functional integral inequalities with respect to $sigma$, such as logarithmic Sobolev and Poincar{e} type. Consequently, some integrability properties of exponential functions with respect to $sigma$ are deduced.
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We prove a sharp Hardy inequality for fractional integrals for functions that are supported on a general domain. The constant is the same as the one for the half-space and hence our result settles a recent conjecture of Bogdan and Dyda.
We investigate selfadjoint $C_0$-semigroups on Euclidean domains satisfying Gaussian upper bounds. Major examples are semigroups generated by second order uniformly elliptic operators with Kato potentials and magnetic fields. We study the long time behaviour of the $L_infty$ operator norm of the semigroup. As an application we prove a new $L_infty$-bound for the torsion function of a Euclidean domain that is close to optimal.
Let $f in M_+(mathbb{R}_+)$, the class of nonnegative, Lebesgure-measurable functions on $mathbb{R}_+=(0, infty)$. We deal with integral operators of the form [ (T_Kf)(x)=int_{mathbb{R}_+}K(x,y)f(y), dy, quad x in mathbb{R}_+, ] with $K in M_+(mathbb{R}_+^2)$.
Using the Laplace derivative a Perron type integral, the Laplace integral, is defined. Moreover, it is shown that this integral includes Perron integral and to show that the inclusion is proper, an example of a function is constructed, which is Laplace integrable but not Perron integrable. Properties of integrals such as fundamental theorem of calculus, Hakes theorem, integration by parts, convergence theorems, mean value theorems, the integral remainder form of Taylors theorem with an estimation of the remainder, are established. It turns out that concerning the Alexiewiczs norm, the space of all Laplace integrable functions is incomplete and contains the set of all polynomials densely. Applications are shown to Poisson integral, a system of generalised ordinary differential equations and higher-order generalised ordinary differential equation.
The use of Lyapunov conditions for proving functional inequalities was initiated in [5]. It was shown in [4, 30] that there is an equivalence between a Poincar{e} inequality, the existence of some Lyapunov function and the exponential integrability of hitting times. In the present paper, we close the scheme of the interplay between Lyapunov conditions and functional inequalities by $bullet$ showing that strong functional inequalities are equivalent to Lyapunov type conditions; $bullet$ showing that these Lyapunov conditions are characterized by the finiteness of generalized exponential moments of hitting times. We also give some complement concerning the link between Lyapunov conditions and in-tegrability property of the invariant probability measure and as such transportation inequalities , and we show that some unbounded Lyapunov conditions can lead to uniform ergodicity, and coming down from infinity property.
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