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$$C^m$$ extension by linear operators. (English) Zbl 1161.46013
The author proves the existence of a linear bounded extension operator from the trace space $$C^m/S$$ into $$C^m$$; here $$S$$ is an arbitrary subset of $$\mathbb{R}^n$$ and $$C^m$$ is the space of $$m$$-times continuously differentiable functions on $$\mathbb{R}^n$$. This deep and profound fact is derived from a general result involving ideals in rings of $$m$$-jets. Namely, let $${\mathcal R}_x$$ be the ring of $$m$$-jets at $$x$$ for all functions of $$C^m$$ and $$I_m(F)$$ be the $$m$$-jet of $$F\in C^m$$. Given a compact set $$S\subset\mathbb{R}^n$$ and a family of ideals $$I(x)\in{\mathcal R}_x$$, $$x\in S$$, the set $${\mathcal T}_S:=\{F\in C^m:{\mathcal I}_x(F)\subset I(x)$$ for all $$x\in S\}$$ is an ideal in the ring $$C^m$$ and $$C^m/{\mathcal T}_S$$ is a Banach space.
Theorem. Let $$\pi:C^m\to C^m/{\mathcal T}_S$$ be the natural projection. There exists a linear right inverse to $$\pi$$ $$T:C^m\to C^m/{\mathcal T}_S$$ with norm bounded by $$c=c(m,n)$$. The above formulated extension result is recovered from here by choosing $$I(x):=\{{\mathcal I}_x(F):F(x)=0\}$$.

##### MSC:
 46E15 Banach spaces of continuous, differentiable or analytic functions 46E25 Rings and algebras of continuous, differentiable or analytic functions 46E10 Topological linear spaces of continuous, differentiable or analytic functions 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 26B35 Special properties of functions of several variables, HĂ¶lder conditions, etc.
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