# Controlled bilipschitz extension

A map $f\colon X\to Y$ is $L$-bilipschitz if $L^{-1} |a-b| \le |f(a)-f(b)| \le L |a-b|$ for all $a,b\in X$. This definition makes sense if X and Y are general metric spaces, but let’s suppose they are subsets on the plane $\mathbb R^2$.

Definition 1. A set $A\subset \mathbb R^2$ has the BL extension property if any bilipschitz map $f\colon A\to\mathbb R^2$ can be extended to a bilipschitz map $F\colon \mathbb R^2\to\mathbb R^2$. (Extension means that $F$ is required to agree with $f$ on $A$.)

Lines and circles have the BL extension property. This was proved in early 1980s independently by Tukia, Jerison and Kenig, and Latfullin.

Definition 2. A set $A\subset \mathbb R^2$ has the controlled BL extension property if there exists a constant $C$ such that any $L$-bilipschitz map $f\colon A\to\mathbb R^2$ can be extended to a $C L$-bilipschitz map $F\colon \mathbb R^2\to\mathbb R^2$.

Clearly, Definition 2 asks for more than Definition 1. I can prove that a line has the controlled BL extension property, even with a modest constant such as $C=2000$. (Incidentally, one cannot take $C=1$.) I still can’t prove the controlled BL extension property for a circle.

Update: extension from line is done in this paper.