Modular Estimates in Orlicz Spaces and Hammerstein Operator Equations

1. Let us recall the basic definitions of the theory of Orlicz spaces (see, for example, [1]). Henceforth (Ω,A, μ) is the triple consisting of a set Ω, σ-algebra A of its subsets and a σ-additive measure μ defined on A. It is assumed that μ is continuous on Ω (that is any set of positive measure can be divided into two sets of equal measures) and is finite: μ(Ω) < ∞. Let M : [0,+∞) → [0,+∞) be an N -function that is a convex function satisfying the conditions lim u→0 M(u) u = 0, lim u→∞ M(u) u = ∞.