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 R ₀ in differentiating the two-function composition operator splits as R ₀=R ₁+R ₂, where R ₁≔fo(G+g)−foG and R ₂≔Fo(G+g)−FoG−(F′oG)·g. Then R ₂ is the remainder for the Nemytskii operator. Thus, this paper concentrates on R ₁. 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 Fréchet differentiability W _Φ×L ^s↦L ^q for an arbitrary Φ-variation space W _Φ, e.g. any p-variation space W _p. 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.