arXiv subjects
Publications and source records attributed to Gérard Bourdaud.
We establish the Faà di Bruno formula, in the sense of almost everywhere equality, for derivatives of the composed function $f \circ g$, for all function $f : R \rightarrow R$ such that $f$ acts on $W^m_p(R^n)$ by composition, and all $g \in W^m_p(R^n)$.
We propose a survey on composition operators in classical Sobolev spaces. We mention results obtained in 2019, on the continuity of such operators.
We establish the most general Szasz type estimates for homogeneous Besov and Lizorkin-Triebel spaces, and their realizations.
We prove that all the composition operators $T_f(g):= f\circ g$, which take the Adams-Frazier space $W^{m}_{p}\cap \dot{W}^{1}_{mp}(\mathbb{R}^n)$ to itself, are continuous mappings from $W^{m}_{p}\cap \dot{W}^{1}_{mp}(\mathbb{R}^n)$ to itself, for every integer $m\geq 2$ and every real number $1\leq p<+\infty$. The same automatic continuity property holds for Sobolev spaces $W^m_p(\mathbb{R}^n)$ for $m\geq 2$ and $1\leq p<+\infty$.
We give intrinsic characterisations for the uniformly localized versions of the Besov spaces $B^{s}_{p,q}({\mathbb R}^n)$, where $p,q\in [1,+\infty]$, and of the Lizorkin-Triebel spaces $F^{s}_{p,q}({\mathbb R}^n)$, where $q\in [1,+\infty]$ and $p\in [1,+\infty[$, whatever be the real number $s>0$.