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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)$.
We characterize positivity preserving, translation invariant, linear operators in $L^p(mathbb{R}^n)^m$, $p in [1,infty)$, $m,n in mathbb{N}$.
Let $ f $ be a real-valued function on a compact subset in $ mathbb{R}^n $. We show how to decide if $ f $ extends to a nonnegative and $ C^1 $ function on $ mathbb{R}^n $. There has been no known result for nonnegative $ C^m $ extension from a gener
We prove that a bi-Lipschitz image of a planar $BV$-extension domain is also a $BV$-extension domain, and that a bi-Lipschitz image of a planar $W^{1,1}$-extension domain is again a $W^{1,1}$-extension domain.
We construct a regular random projection of a metric space onto a closed doubling subset and use it to linearly extend Lipschitz and $C^1$ functions. This way we prove more directly a result by Lee and Naor and we generalize the $C^1$ extension theorem by Whitney to Banach spaces.
Let $S subset mathbb{R}^{n}$ be a~closed set such that for some $d in [0,n]$ and $varepsilon > 0$ the~$d$-Hausdorff content $mathcal{H}^{d}_{infty}(S cap Q(x,r)) geq varepsilon r^{d}$ for all cubes~$Q(x,r)$ centered in~$x in S$ with side length $2r i