Boundedness properties of maximal operators on Lorentz spaces

2019/05/08 by Kosz, Dariusz
#42B25 #46E30 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.1905.03232

Abstract

We study mapping properties of the centered Hardy--Littlewood maximal operator M acting on Lorentz spaces. Given p ∈ (1,∞) and a metric measure space \mathfrakX we let Ω^p_{\rm HL}(\mathfrakX) ⊂ [0,1]² be the set of all pairs ((1)/(q),(1)/(r)) such that M is bounded from L^p,q(\mathfrakX) to L^p,r(\mathfrakX). For each fixed p all possible shapes of Ω^p_{\rm HL}(\mathfrakX) are characterized. Namely, we show that the boundary of Ω^p_{\rm HL}(\mathfrakX) either is empty or takes the form \ δ\ × [0, lim_{u → δ} F(u)] ∪ \(u, F(u)) : u ∈ (δ, 1] \, where δ∈ [0,1] and F \colon [δ, 1] → [0,1] is concave, non-decreasing, and satisfying F(u) ≤ u. Conversely, for each such F we find \mathfrakX such that M is bounded from L^p,q(\mathfrakX) to L^p,r(\mathfrakX) if and only if the point ((1)/(q), (1)/(r)) lies on or under the graph of F, that is, (1)/(q) ≥ δ and (1)/(r) ≤ F((1)/(q)).

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