Differentiability of the composition and quantile operators for regulated and A. E. continuous functions

Abstract

If F is a function defined on the range of a function G, let (FoG)(x)=F(G(x)) for all x. Let (Ω, μ) be a finite measure space. The paper treats differentiability of the two-function composition operator f, g↦(F+f)o(G+g) into L q (Ω, μ). where g→0 in L s and 1≤q<s. The case where f=0, namely g↦Fo(G+g), for suitable F, G, is a special case of the so-called Nemytskii or superposition operator, which has been extensively studied, as in the book by J. Appell and P. P. Zabrejko, Nonlinear Superposition Operators, Cambridge University Press, 1990, Chapter 3. The remainder R0 in differentiating the two-function composition operator splits as R0=R1+R2, where R1≔fo(G+g)−foG and R2≔Fo(G+g)−FoG−(F′oG)·g. Then R2 is the remainder for the Nemytskii operator. Thus, this paper concentrates on R1. For suitable G, the question then is, for what f, and uniformly over what classes of f, is ‖tf○(G+g)-tf○g‖ q =o({t{+‖g‖ s ) as {t{+‖g‖ g →0, or equivalently ‖f○(G+g)-f○G‖ g =o(1) as ‖g‖ s →0. This is a question of continuity or equicontinuity of Nemytskii operators at points. Previously, for the most part, global continuity had been treated. The individual f are shown to be exactly those which are continuous almost everywhere, suitably measurable, and such that {f(x){/(1+{x{ s/q ) is bounded in x. Large classes of f, called “uniformly Riemann,” are given over which the differentiability is uniform. These give in particular Frechet differentiability WΦ×L s ↦L q for an arbitrary Φ-variation space WΦ, e.g. any p-variation space Wp. Very similar results are found for the quantile operator g↦(G+g)← for functions G and g from an interval J into ℝ, where H←(y)≔inf{x∈J:H(x)≥y}. Also, a theorem is given on composition of Banach-valued functions with supremum norms, where again f need not be differentiable.