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Does $C^{k,s-k}$ function with lipschitz lower order derivatives give a certain bound on the...

Your desired conclusion is the hypothesis of (a version of) the Whitney extension theorem: it implies that there exists $F \in C^{k, s-k}(\mathbb R)$ such that $F^{(j)}|_{\bar{\Omega}} = f^{(j)}$ for …
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