Semigroup inequalities, stochastic domination, Hardy's inequality, and strong ergodicity
For the classical L p -spaces of signed measures on N, we devise a framework in which bounds for a sub-Markovian semigroup of interest can be obtained, up to a constant factor, from bounds for another tractable semigroup that dominates stochastically the first one. The main tools are the Hardy inequality, the definition of related auxiliary L p spaces suited to take advantage of the domination, and the proof that the norms are equivalent to the classical ones if the reference measure is quasi-geometrically decreasing. We illustrate the results using birth-death and single-birth processes.
NOTE ON THE CONVERSES OF INEQUALITIES OF HARDY AND LITTLEWOOD
The inequalities of Hardy–Littlewood type play very important role in many theorems concerning convergence or summability of orthogonal series. In the applications many times their converses are also very useful. Both the original inequalities and their converses have been generalized in several directions by many authors, among others Leindler (see [5], [6], [7], [8]), Mulholland (see [10]), Chen Yung-Ming (see [1]), etc. The aim of this paper is to give further generalization of the inequalities of converse type replacing the power functions by more general ones.
General expression for the density effect for the ionization loss of charged particles
The equations for the density-effect correction $\ensuremath{\delta}(X)$ for the ionization loss as given in a previous paper have been extended to the case where $\ensuremath{\delta}({X}_{0})\ensuremath{\equiv}{\ensuremath{\delta}}_{0}g0$. Here ${X}_{0}$ is the lower limit of the range ${X}_{0}lXl{X}_{1}$, where a particularly simple expression for $\ensuremath{\delta}(X)$ applies; $X={log}_{10}(\frac{p}{{m}_{0}c})$, where $p$ is the momentum and ${m}_{0}$ is the mass of the incident particle. The case ${\ensuremath{\delta}}_{0}g0$ applies directly to the calculation of $\ensuremath{\delta}(X)$ for conductors (metals).