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By Revista De La, T. A. Gillespie, J. L. Torrea, Dedicated Professor and Roberto Macías


Abstract. A well known result by Rubio de Francia asserts that for every finite family of disjoint intervals {Ik} in R, and p in the range 2 ≤ p < ∞, there exists Cp> 0 such that ‖ X rkSI f ‖ p k L L k p ([0,1]) (R) ≤ Cp‖f‖L p (R), where the rk’s are the Rademacher functions. In this note we prove that, given a compact connected abelian group G with dual group Γ and p in the range 2 ≤ p < ∞, there is a constant Cp, independent of G and the particular ordering on Γ, such that for every sequence {Ik} of disjoint intervals in Γ, we have ‖ X k rkSI k f ‖ L p L p ([0,1]) (G) ≤ Cp‖f ‖ L p (G). We obtain the result by a transference approach that can be used for functions taking values in Banach spaces. 1

Year: 2014
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