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• W2024456145 abstract "We prove a global implicit function theorem. In particular we show that any Lipschitz map $f:bR^ntimes bR^mtobR^n$ (with $n$-dim. image) can be precomposed with a bi-Lipschitz map $bar{g}:bR^ntimes bR^mto bR^ntimes bR^m$ such that $fcirc bar{g}$ will satisfy, when we restrict to a large portion of the domain $Esubset bR^ntimes bR^m$, that $fcirc bar{g}$ is bi-Lipschitz in the first coordinate, and constant in the second coordinate. Geometrically speaking, the map $bar{g}$ distorts $bR^{n+m}$ in a controlled manner, so that the fibers of $f$ are straightened out. Furthermore, our results stay valid when the target space is replaced by {bf any metric space}. A main point is that our results are quantitative: the size of the set $E$ on which behavior is good is a significant part of the discussion. Our estimates are motivated by examples such as Kaufman's 1979 construction of a $C^1$ map from $[0,1]^3$ onto $[0,1]^2$ with rank $leq 1$ everywhere. On route we prove an extension theorem which is of independent interest. We show that for any $Dgeq n$, any Lipschitz function $f:[0,1]^nto bR^D$ gives rise to a large (in an appropriate sense) subset $Esubset [0,1]^n$ such that $f|_E$ is bi-Lipschitz and may be extended to a bi-Lipschitz function defined on {bf all} of $bR^n$. The most interesting case is the case $D=n$. As a simple corollary, we show that $n$-dimensional Ahlfors-David regular spaces lying in $bR^{D}$ having big pieces of bi-Lipschitz images also have big pieces of big pieces of Lipschitz graphs in $bR^{D}$. This was previously known only for $Dgeq 2n+1$ by a result of G. David and S. Semmes." @default.
• W2024456145 created "2016-06-24" @default.
• W2024456145 creator A5014202794 @default.
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• W2024456145 date "2011-05-21" @default.
• W2024456145 modified "2023-09-27" @default.
• W2024456145 title "Hard Sard: Quantitative Implicit Function and Extension Theorems for Lipschitz Maps" @default.
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