Weighted ergodic theorems for mean ergodic 𝐿₁-contractions

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It is shown that any bounded weight sequence which is good for all probability preserving transformations (a universally good weight) is also a good weight for any L 1 L_{1} -contraction with mean ergodic (ME) modulus, and for any positive contraction of L p L_{p} with 1 > p > ∞ 1 > p >\infty . We extend the return times theorem by proving that if S S is a Dunford-Schwartz operator (not necessarily positive) on a Lebesgue space, then for any g g bounded measurable { S n g ( ω ) } \{S^{n} g(\omega )\} is a universally good weight for a.e. ω . \omega . We prove that if a bounded sequence has "Fourier coefficents", then its weighted averages for any L 1 L_{1} -contraction with mean ergodic modulus converge in L 1 L_{1} -norm. In order to produce weights, good for weighted ergodic theorems for L 1 L_{1} -contractions with quasi-ME modulus (i.e., so that the modulus has a positive fixed point supported on its conservative part), we show that the modulus of the tensor product of L 1 L_{1} -contractions is the product of their moduli, and that the tensor product of positive quasi-ME L 1 L_{1} -contractions is quasi-ME.