A simple proof of an extrapolation theorem for Marcinkiewicz spaces

Abstract

It is well known that many operators in analysis, for example, the Hilbert operator or the conjugate function operator are bounded in the space Lp for p ∈ (1,∞), but they are unbounded in L1 or L∞. In Yano’s paper [1] (also see [2]), the extrapolation theorems were first used to define the “extremal” space in the limit case, which led to the appearance of many papers [3]–[5] generalizing the Yano and Zygmund theorems. In the present paper, we consider another approach to the proof of extrapolation theorems for quasilinear operators in Marcinkiewicz spaces. In this approach, the geometric properties of these spaces are taken into account. A similar approach was first demonstrated for the Lorentz spaces in [6]. Comparing our results with the results of the above papers and, in particular, with [3], we note that, first, we consider operators in the scale of Marcinkiewicz spaces rather than in the scale of Lebesgue spaces, and second, which is more important, from the character of growth of the operator norm as the exponent tends to the critical exponent, we write out the exact Marcinkiewicz space, which is the extrapolation space. The theorems given below include both the classical Zygmund theorem (see, e.g., Example 1 at the end of the paper) and recent results about extrapolation of operators near L∞ (see [7]). We assume that (Ω,Σ, μ) is a space equipped with a σ-finite measure, S(μ) is the set of measurable functions on Ω, and χ(D) is the characteristic function of the set D; moreover, we assume that the measure is continuous. For f : Ω → R, we let f∗ denote its permutation in nonascending order. We assume that X is a symmetric space [8] and ψ(X, s) = {‖χ(D)|X‖ : μ(D) = s}