Interpolation of operators of weak type (ϕ, ϕ)

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(ℝ n ) 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.