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Weighted inequalities for one-sided maximal functions in Orlicz spaces

Pedro Ortega Salvador — 1998

Studia Mathematica

Let M g + be the maximal operator defined by M g + ( x ) = s u p h > 0 ( ʃ x x + h | | g ) / ( ʃ x x + h g ) , 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 Δ 2 . We characterize the pairs of positive functions (u,ω) such that the weak type inequality u ( x | M g + ( x ) > λ ) C / ( Φ ( λ ) ) ʃ Φ ( | | ) ω holds for every ⨍ in the Orlicz space L Φ ( ω ) . We also characterize the positive functions ω such that the integral inequality ʃ Φ ( | M g + | ) ω ʃ Φ ( | | ) ω holds for every L Φ ( ω ) . Our results include some already obtained for functions in L p and yield as consequences...

Convergence of the averages and finiteness of ergodic power funtions in weighted L spaces.

Pedro Ortega Salvador — 1991

Publicacions Matemàtiques

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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