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This article reviews several definitions of the fractional Laplace operator (-Delta)^{alpha/2} (0<alpha<2) in R^d, also known as the Riesz fractional derivative operator, as an operator on Lebesgue… (More)

We investigate the classical eigenvalue problem that arises in hydrodynamics and is referred to as the sloshing problem. It describes free liquid oscillations in a liquid container W in R^3. We study… (More)

In this paper we study the supremum functional Mt=sup0≤s≤tXs, where Xt, t≥0, is a one-dimensional Levy process. Under very mild assumptions we provide a simple, uniform estimate of the cumulative… (More)

We prove a boundary Harnack inequality for jump-type Markov processes on metric measure state spaces, under comparability estimates of the jump kernel and Urysohn-type property of the domain of the… (More)

There exist only a few known examples of subordinators for which the transition probability density can be computed explicitly along side an expression for its L?evy measure and Laplace exponent.… (More)

Small-space and large-time estimates and asymptotic expansion of the distribution function and (the derivatives of) the density function of hitting times of points for symmetric Levy processes are… (More)

Abstract We prove a two-term Weyl-type asymptotic law, with error term O ( 1 n ) , for the eigenvalues of the operator ψ ( − Δ ) in an interval, with zero exterior condition, for complete Bernstein… (More)

We study supercontractivity and hypercontractivity of Markov semigroups obtained via ground state transformation of non-local Schrodinger operators based on generators of symmet- ric jump-paring Levy… (More)

Two-term Weyl-type asymptotic law for the eigenvalues of one-dimensional fractional Laplace operator (-d^2/dx^2)^(alpha/2) (0<alpha<2) in the interval (-1,1) is given: the n-th eigenvalue is equal to… (More)

We provide a large class of functions $f$ that are bell-shaped: the $n$-th derivative of $f$ changes its sign exactly $n$ times. This class is described by means of Stieltjes-type representation of… (More)