We study some basic properties of weak Orlicz spaces and their applications to harmonic analysis.We first discuss the absolute continuity of the quasi-norm and its normality,then prove the boundedness of several maximal operators.We also establish a kind of Marcinkiewicz-type interpolation theorem between weak Orlicz spaces.As applications,the weak type analogues of several classical inequalities in harmonic analysis is obtained.
Let x =(xn)n1 be a martingale on a noncommutative probability space(M,τ) and(wn)n1 a sequence of positive numbers such that Wn=∑nk=1wk→∞asn→∞.We prove that x =(xn)n1 converges bilaterally almost uniformly(b.a.u.) if and only if the weighted average(σn(x))n1 of x converges b.a.u.to the same limit under some condition,where σn(x) is given by σn(x)=W1n∑nk=1wkxk,n=1,2,...Furthermore,we prove that x=(xn)n1 converges in Lp(M) if and only if(σn(x))n1 converges in Lp(M),where 1p<∞.We also get a criterion of uniform integrability for a family in L1(M).
Under appropriate conditions on Young's functions Φ1 and Φ2,we give necessary and sufficient conditions in order that weighted integral inequalities hold for Doob's maximal operator M on martingale Orlicz setting.When Φ1 = tp and Φ2 = tq,the inequalities revert to the ones of strong or weak(p,q)-type on martingale space.
Let x =(xn) n ≥1 be a martingale on a noncommutative probability space(M,τ) and(wn) n ≥1 a sequence of positive numbers such that Wn = ∑nk=1 wk →∞ as n →∞.We prove that x =(xn) n≥1 converges in E(M) if and only if(σn(x))n≥1 converges in E(M),where E(M) is a noncommutative rearrangement invariant Banach function space with the Fatou property and σ n(x) is given by k=1 If in addition,E(M) has absolutely continuous norm,then,(σ n(x)) n ≥1 converges in E(M) if and only if x =(x n) n ≥1 is uniformly integrable and its limit in measure topology x ∞∈ E(M).
Let B be a Banach space, Φ1 , Φ2 be two generalized convex Φ-functions and Ψ 1 , Ψ 2 the Young complementary functions of Φ1 , Φ2 respectively with ∫t t 0 ψ2 (s) s ds ≤ c 0 ψ1 (c 0 t) (t > t 0 ) for some constants c 0 > 0 and t 0 > 0, where ψ1 and ψ2 are the left-continuous derivative functions of Ψ 1 and Ψ 2 , respectively. We claim that: (i) If B is isomorphic to a p-uniformly smooth space (or q-uniformly convex space, respectively), then there exists a constant c > 0 such that for any B-valued martingale f = (f n ) n ≥ 0 , ‖f*‖Φ1 ≤ c‖S (p) (f ) ‖Φ2 (or ‖S (q) (f )‖Φ1 ≤ c‖f*‖Φ2 , respectively), where f and S (p) (f ) are the maximal function and the p-variation function of f respec- tively; (ii) If B is a UMD space, T v f is the martingale transform of f with respect to v = (v n ) n ≥ 0 (v*≤ 1), then ‖(T v f )*‖Φ1 ≤ c ‖f *‖Φ2 .
