Linear functional equations and their solutions in Lorentz spaces

Assume that $$\varOmega \subset {\mathbb {R}}^k$$ is an open set, V a separable Banach space over field $${\mathbb {K}}\in \{{\mathbb {R}},{\mathbb {C}}\}$$ and $$f_1,\ldots ,f_N :\varOmega \rightarrow \varOmega $$ , $$g_1,\ldots g_N:\varOmega {K}}$$ $$h_0:\varOmega V$$ are given functions. We interested in the existence uniqueness of solutions $$\varphi linear functional equation =\sum _{k=1}^{N}g_k\cdot (\varphi \circ f_k)+h_0$$ Lorentz spaces. The investigation case $$L^1([0,1],{\mathbb {R}})$$ $$L^{1,1}([0,1],{\mathbb was motivated by Nikodem’s paper On $$\epsilon -invariant measures published Czechoslovak Math. J., 41(116)(4):565–569, 1991. have studied it Some classes operators involved equations Ann. Funct. Anal., 10(3):381–394, 2019. Inspired Matkowski’s Dissertationes Integrable from 1975, we Orlicz our Linear their generalized spaces Aequationes Mathematicae 2021 volume 95, number 6. Here, build on ideas paper, using connection between as detailed Fine behavior functions whose gradients authored Malý, Swanson, Ziemer Studia 190(1), 33–71 2009. However, do not only look at real-valued functions, but allow target As about spaces, important tool result Hajłasz, Change variables formula under minimal assumptions Colloq. 64(1), 93–101 (1993), concerning optimal change variable formula. It used for proving can apply Banach’s fixed point theorem.