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Register a new userThe norm of linear extension operators for $C^{m-1,1}(mathbb{R}^n)$
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Fix integers $m geq 2$, $n geq 1$. Let $C^{m-1,1}(mathbb{R}^n)$ be the space of $(m-1)$-times differentiable functions $F : mathbb{R}^n rightarrow mathbb{R}$ whose $(m-1)$st order partial derivatives are Lipschitz continuous, equipped with a standard seminorm. Given $E subseteq mathbb{R}^n$, let $C^{m-1,1}(E)$ be the trace space of all restrictions $F|_E$ of functions $F$ in $C^{m-1,1}(mathbb{R}^n)$, equipped with the standard quotient (trace) seminorm. We prove that there exists a bounded linear operator $T : C^{m-1,1}(E) rightarrow C^{m-1,1}(mathbb{R}^n)$ satisfying $Tf|_E = f$ for all $f in C^{m-1,1}(E)$, with operator norm at most $exp( gamma D^k)$, where $D := binom{m+n-1}{n}$ is the number of multiindices of length $n$ and order at most $m-1$, and $gamma,k > 0$ are absolute constants (independent of $m,n,E$). Our results improve on the previous construction of linear extension operators with norm at most $ exp(gamma D^k 2^D)$.
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