Chapter 3 - The superposition operator in Lebesgue spaces

Summary

By reformulating the general results of Chapter 2, one gets many results on the superposition operator in Lebesgue spaces. On the other hand, the theory in Lebesgue spaces is much richer than in general ideal spaces. The most interesting (and pleasant) fact is that one can give an acting condition for F, in terms of the generating function f, which is both necessary and sufficient. It follows, in particular, that F is always bounded and continuous, whenever F acts from some L_p into L_q (for 1 ≤ p ≤ ∞, 1 ≤ q < ∞ and Ω_d = Ø); the corresponding problems in the case Ω_c = Ø are more delicate.

Apart from continuity and boundedness conditions, we provide a concrete “recipe” to calculate the growth function of the superposition operator on balls in L_p. Moreover, criteria for absolute boundedness and uniform continuity are given, as well as two-sided estimates for the modulus of continuity of F.

As immediate consequences of some results of the preceding chapter, we get that F is weakly continuous from L_p into L_q if and only if f is affine in u. Further, it turns out that the Darbo or Lipschitz condition for F is equivalent to a Lipschitz condition for the function f with respect to u. Holder continuity of F is also briefly discussed.

Another pleasant fact concerns differentiability: in Lebesgue spaces one can give conditions for differentiability, asymptotic linearity, and higher differentiability which are both necessary and sufficient.