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Interpolation of operators of weak type (ϕ, ϕ)

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Abstract

Considering the measurable and nonnegative functions ϕ on the half-axis [0, ∞) such that ϕ(0) = 0 and ϕ(t) → ∞ as t → ∞, we study the operators of weak type (ϕ, ϕ) that map the classes of ϕ-Lebesgue integrable functions to the space of Lebesgue measurable real functions on ℝn. We prove interpolation theorems for the subadditive operators of weak type (ϕ0, ϕ0) bounded in L ∞(ℝn) and subadditive operators of weak types (ϕ0, ϕ0) and (ϕ1, ϕ1) in L ϕ(ℝn) under some assumptions on the nonnegative and increasing functions ϕ(x) on [0, ∞). We also obtain some interpolation theorems for the linear operators of weak type (ϕ0, ϕ0) bounded from L ∞(ℝn) to BMO(ℝ n). For the restrictions of these operators to the set of characteristic functions of Lebesgue measurable sets, we establish some estimates for rearrangements of moduli of their values; deriving a consequence, we obtain a theorem on the boundedness of operators in rearrangement-invariant spaces.

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Authors and Affiliations

  1. Dnepropetrovsk Agricultural State University, Dnepropetrovsk, the Ukraine

    B. I. Peleshenko

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  1. B. I. Peleshenko
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Correspondence to B. I. Peleshenko.

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Original Russian Text Copyright © 2008 Peleshenko B. I.

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Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 49, No. 2, pp. 400–419, March–April, 2008.

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Peleshenko, B.I. Interpolation of operators of weak type (ϕ, ϕ). Sib Math J 49, 322–338 (2008). https://doi.org/10.1007/s11202-008-0032-x

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  • DOI: https://doi.org/10.1007/s11202-008-0032-x

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