On sequences of $C^{k,\delta }_{b}$ maps which converge in the uniform $C^{0}$-norm

Abstract:

We study maps $f\in C^{k,\delta }_{b}(U,Y)$ and give detailed estimates on $\|D^{k}f(x)\|,x\in U,$ in terms of $\|f\|$ and $\|f\|_{k,\delta }$. These estimates are used to prove a lemma by D. Henry for the case $k\geq 2$. Here $U\subset X$ is an open subset and $X$ and $Y$ are Banach spaces.