Linear functional equations and their solutions in Lorentz spaces.

Assume that Ω ⊂ R k is an open set, V is a separable Banach space over a field K ∈ { R , C } and f 1 , … , f N : Ω → Ω , g 1 , … , g N : Ω → K , h 0 : Ω → V are given functions. We are interested in the existence and uniqueness of solutions φ : Ω → V of the linear functional equation φ = ∑ k = 1 N g k · (φ ∘ f k) + h 0 in Lorentz spaces. The investigation of the existence of the solutions in the case L 1 ([ 0 , 1 ] , R) , that is in the Lorentz space L 1 , 1 ([ 0 , 1 ] , R) , was motivated by Nikodem’s paper On ϵ -invariant measures and a functional equation published in Czechoslovak Math. J., 41(116)(4):565–569, 1991. We have studied it in Some classes of linear operators involved in functional equations published in Ann. Funct. Anal., 10(3):381–394, 2019. Inspired by Matkowski’s Dissertationes Math. paper Integrable solutions of functional equations from 1975, we studied the Orlicz case in our paper Linear functional equations and their solutions in generalized Orlicz spaces published in Aequationes Mathematicae in 2021 in volume 95, number 6. Here, we build on the ideas from that paper, using the connection between Lorentz and Orlicz spaces as detailed in the paper Fine behavior of functions whose gradients are in an Orlicz space authored by Malý, Swanson, and Ziemer and published in Studia Math. 190(1), 33–71 in 2009. However, we do not only look at real-valued functions, but allow separable Banach spaces as target spaces. As in the paper about Orlicz spaces, an important tool is a result by Hajłasz, Change of variables formula under minimal assumptions as published in Colloq. Math. 64(1), 93–101 (1993), concerning an optimal change of variable formula. It is used for proving that we can apply Banach’s fixed point theorem. [ABSTRACT FROM AUTHOR]