This paper is mainly devoted to establishing an atomic decomposition of a predictable martingale Hardy space with variable exponents defined on probability spaces. More precisely, let (Ω,F, ℙ) be a probability space and p(·): Ω →(0,∞) be a F-measurable function such that 0 < {\inf _{x \in \Omega }}p(x) \leqslant {\sup _{x \in \Omega }}p(x) < \infty . It is proved that a predictable martingale Hardy space Pp(·) has an atomic decomposition by some key observations and new techniques. As an application, we obtain the boundedness of fractional integrals on the predictable martingale Hardy space with variable exponents when the stochastic basis is regular., Zhiwei Hao., and Obsahuje seznam literatury
We prove and discuss some new (Hp,Lp)-type inequalities of weighted maximal operators of Vilenkin-Nörlund means with non-increasing coefficients {q_{k}:k\geqslant 0}. These results are the best possible in a special sense. As applications, some well-known as well as new results are pointed out in the theory of strong convergence of such Vilenkin-Nörlund means. To fulfil our main aims we also prove some new estimates of independent interest for the kernels of these summability results. In the special cases of general Nörlund means tn with non-increasing coefficients analogous results can be obtained for Fejér and Cesàro means by choosing the generating sequence {q_{k}:k\geqslant 0} in an appropriate way., István Blahota, Lars-Erik Persson, Giorgi Tephnadze., and Obsahuje seznam literatury