Generalized Hardy-Cesàro operators between weighted spaces

2016/10/19 by Pedersen, Thomas Vils
#44A15 #47B34 #47B38 #47G10 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1610.05947

Abstract

We characterize those non-negative, measurable functions ψ on [0,1] and positive, continuous functions ω₁ and ω₂ on \mathbb R⁺ for which the generalized Hardy-Cesàro operator (U_ψf)(x)=∫₀¹ f(tx)ψ(t) dt defines a bounded operator U_ψ:L¹(ω₁)→ L¹(ω₂). Furthermore, we extend U_ψ to a bounded operator on M(ω₁) with range in L¹(ω₂)⊕\mathbb Cδ₀. Finally, we show that the zero operator is the only weakly compact generalized Hardy-Cesàro operator from L¹(ω₁) to L¹(ω₂).

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