On the composition operators on Besov and Triebel–Lizorkin spaces with power weights
Volume 129 / 2022
Abstract
Let be a continuous function. Under some assumptions on G, s,\alpha ,p and q we prove that \{G(f):f\in A_{p,q}^{s}(\mathbb R^{n},|\cdot |^{\alpha })\}\subset A_{p,q}^{s}(\mathbb R^{n},|\cdot |^{\alpha }) implies that G is a linear function. Here A_{p,q}^{s}(\mathbb R^{n},|\cdot |^{\alpha }) stands either for the Besov space B_{p,q}^{s}(\mathbb R^{n},|\cdot |^{\alpha }) or for the Triebel–Lizorkin space F_{p,q}^{s}(\mathbb R^{n},|\cdot |^{\alpha }). These spaces unify and generalize many classical function spaces such as Sobolev spaces with power weights.