Composition operator on the space of functions Triebel-Lizorkin and bounded variation type (Q2913257)

The Bourdaud-Kateb class \(U^1_p\) is the collection of all Lipschitz continuous functions \(f: {\mathbb R} \to {\mathbb R}\) such that NEWLINE\[NEWLINE A_p (f):= \left (\sup _{t>0} \int \limits _{-\infty }^\infty \sup _{| h| \leq t}| f' (x+h) - f'(x)| ^p\, dx\right)^{1/p} <\infty . NEWLINE\]NEWLINE The author studies the operator \(T_f: g \mapsto f \circ g\) on the space \(L_p ({\mathbb R}^n) \cap {\mathcal V}_p ({\mathbb R}^n)\), where \({\mathcal V}_p ({\mathbb R}^n)\) is defined as follows. A~function \(g: {\mathbb R}^n \to {\mathbb R}\) belongs to \({\mathcal V}_p ({\mathbb R}^n)\) if the expression NEWLINE\[NEWLINE \| g \| _{{\mathcal V}_p ({\mathbb R}^n)}:= \sum _{j=1}^n \left (\int \limits _{{\mathbb R}^{n-1}} \| g_{x_j'} \| _{BV^1_p ({\mathbb R})} \, dx_j'\right)^{1/p} NEWLINE\]NEWLINE is finite, where \(BV^1_p ({\mathbb R})\) is the Wiener class of primitives of functions with bounded \(p\)-variation and NEWLINE\[NEWLINE g_{x_j'} (y):= g(x_1, \dots , x_{j-1}, y, x_{j+1}, \dots ,x_n) , \quad y \in {\mathbb R}, \quad x \in {\mathbb R}^n. NEWLINE\]NEWLINE Let \(0 <p,q<\infty \) and \(0< s < 1+1/p\). For \(f\) belonging to the Bourdaud-Kateb class \(U^1_p\) and satisfying \(f(0)=0\), the author proves the inequality NEWLINE\[NEWLINE \| f \circ g\| _{F^s_{p,q}({\mathbb R}^n)} \leq c\Bigl (\| f \| _{L_\infty ({\mathbb R})} + A_p (f)\Bigr) \, \Bigl (\| g \| _{L_p ({\mathbb R})} + \| g \| _{{\mathcal V}_p ({\mathbb R})}\Bigr) NEWLINE\]NEWLINE with \(c\) independent of \(f\) and \(g\).