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.
Birth-death processes with killing
The purpose of this note is to point out that Karlin and McGregor's integral representation for the transition probabilities of a birth-death process on a semi-infinite lattice with an absorbing bottom state remains valid if one allows the possibility of absorption into the bottom state from any other state. Conditions for uniqueness of the minimal transition function are also given.
Some inequalities of Hardy-Littlewood type
AbstractКОНЮШКОВ [3] установи л, ч то если при некоторомτ>0 последовательностьn−τan убывает, то одно из четырех широко извес тных неравенств Харди и Литтлвуда можно обр атить. Мы доказываем, ч то при некоторых близких ус ловиях остальные три нераве нства Харди и Литтлву да также можно обратить. Это утверждение мы по лучаем как частный сл учай более общих результатов. Показано также, что пр и некоторых условиях типа слабой регулярности найден ные достаточные условия являются и необходим ыми.