Let be the maximal operator defined by , where g is a positive locally integrable function on ℝ. Let Φ be an N-function such that both Φ and its complementary N-function satisfy . We characterize the pairs of positive functions (u,ω) such that the weak type inequality holds for every ⨍ in the Orlicz space . We also characterize the positive functions ω such that the integral inequality holds for every . Our results include some already obtained for functions in and yield as consequences...
Let (X, F, μ) be a finite measure space. Let T: X → X be a measure preserving transformation and let Af denote the average of Tf, k = 0, ..., n. Given a real positive function v on X, we prove that {Af} converges in the a.e. sense for every f in L(v dμ) if and only if inf v(Tx) > 0 a.e., and the same condition is equivalent to the finiteness of a related ergodic power function Pf for every f in L(v dμ). We apply this result to characterize, being T null-preserving, the finite measures ν for...
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