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Keywords:
composition operator; homogeneous Besov space; homogeneous Triebel-Lizorkin space; realization
composition operator; homogeneous Besov space; homogeneous Triebel-Lizorkin space; realization
Summary:
We prove the acting by composition of nontrivial functions $f \colon \mathbb {R} \to \mathbb {R}$ (i.e., ${T_f \colon g\to f\circ g}$) on homogeneous Besov and Triebel-Lizorkin spaces realized as subspaces of ${\mathcal S}'(\mathbb {R}^n)$ in case $s=n/p<1+1/p$, and $q>1$ (Besov space) and $p>1$ (Triebel-Lizorkin space). These subspaces are dilation invariant and endowed with quasi-seminorms such that $\|g\|=0$ if and only if $g$ is constant.
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