Gaku Sadasue

On generalized martingale Morrey spaces we give necessary and sufficient conditions for the boundedness of generalized fractional integrals as martingale transforms.

Some new properties concerning BLO martingales are given. The BMO-BLO boundedness of martingale maximal functions and Bennett type characterizatipn of BLO martingales. are shown. Also, a non-negative BMO martingale that is not in BLO is constructed.

We introduce'generalized Campanato spaces L-p,L-phi on a probability Space (Omega, F, P), where p is an element of [1, infinity) and phi: (0,1] -> (0, infinity). If p = 1 and phi equivalent to 1, then L-p,L-phi = BMO. We give a characterization of the set of all pointwise multipliers on L-p,L-phi.

We prove the boundedness of the maximal operator for martingales on generalized Lebesgue spaces with variable exponent over probability spaces. We consider pointwise multipliers on martingale BMO as the variable exponent. (C) 2013 Elsevier B.V. All rights reserved.

We introduce generalized Morrey-Campanato spaces of martingales, which generalize both martingale Lipschitz spaces introduced by Weisz (1990) and martingale Morrey-Campanato spaces introduced in 2012. We also introduce generalized Morrey-Hardy and Campanato-Hardy spaces of martingales and study Burkholder-type equivalence. We give some results on the boundedness of fractional integrals of martingales on these spaces.

The purpose of this paper is to introduce five martingale Orlicz-Hardy spaces and to establish the atomic decomposition theorem. As applications we show the relation among five martingale Orlicz-Hardy spaces and the duality, namely, the dual of martingale Orlicz-Hardy spaces are generalized martingale Campanato spaces. Further, we prove a John-Nirenberg type inequality for generalized martingale Campanato spaces when the stochastic basis is regular.

Quasi-invariance of infinite product measures is studied when a locally compact second countable group acts on a standard Borel space. A characterization of l(2)-quasi-invariant infinite product measures is given. The group that leaves the measure class invariant is also studied. In the case where the group acts on itself by translations, our result extends previous ones obtained by Shepp (Ann. Math. Stat. 36:1107-1112, 1965) and by Hora (Math. Z. 206:169-192, 1991; J. Theor. Probab. 5:71-100, 1992) to all connected Lie groups.