A sharp form of the Marcinkiewicz Interpolation Theorem for Orlicz spaces

2017/11/25 by Kerman, Ron, Rawat, Rama, Singh, Rajesh K. · 1 citation
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.1711.09278

Abstract

An extension of Marcinkiewicz Interpolation Theorem, allowing intermediate spaces of Orlicz type, is proved. This generalization yields a necessary and sufficient condition so that every quasilinear operator, which maps the set, S(X,μ), of all μ-measurable simple functions on σ- finite measure space (X,μ) into M(Y,ν), the class of ν-measurable functions on σ- finite measure space (Y,ν), and satisfies endpoint estimates of type: 1 < p< ∞, 1 ≤ r < ∞, λ ν( \lbrace y ∈ Y : |(Tf)(y)| gt; λ\rbrace )^(1)/(p) ≤ C_p,r ( ∫_\mathbbR₊ μ( \lbrace x ∈ X : |(f)(x)| gt; t \rbrace )^(r)/(p) t^r-1dt )^(1)/(r), for all f ∈ S(X,μ) and λ∈ \mathbbR₊; is bounded from an Orlicz space into another.

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• The Hardy averaging operator between Orlicz spaces and a new gauge functional characterizing a given Orlicz space

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