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Extension of $$C^{m, \omega}$$-smooth functions by linear operators. (English) Zbl 1173.46014
Let $$C^{m,\omega}(\mathbb{R}^n)$$ be the Banach space of functions on $$\mathbb{R}^n$$ whose $$m$$th derivatives have modulus of continuity majorised by a nondecreasing concave function $$\omega:\mathbb{R}_+\to \mathbb{R}_+$$ at zero. For $$E\subset\mathbb{R}^n$$, let $$C^{m, \omega}(E)$$ be the space of all restrictions to $$E$$ of functions in $$C^{m,\omega}(\mathbb{R}^n)$$. The main result is as follows.
Theorem. There exists a linear extension operator $$T:C^{m,\omega}(E)\to C^{m,\omega} (\mathbb{R}^n)$$ whose norm is bounded by a constant depending only on $$n,m$$.
The author also proves a generalization of this result related to the notion of Whitney convexity. This, in turn, allows to simplify the proof of his extremely difficult result from [Ann. Math. 166, No. 3, 779–835 (2007; Zbl 1161.46013)], on $$C^m$$ extension by linear operators.

##### MSC:
 46E15 Banach spaces of continuous, differentiable or analytic functions 49K24 Optimal control problems with differential inclusions (nec./ suff.) (MSC2000) 52A35 Helly-type theorems and geometric transversal theory
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