Quantitative Differentiation: A General Formulation

Abstract

AbstractLet dx denote Lebesgue measure on \documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath, amssymb} \pagestyle{empty} \begin{document} \begin{align*}\mathbb{R}^n\end{align*} \end{document}. With respect to the measure \documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath, amssymb} \pagestyle{empty} \begin{document} \begin{align*}\mathcal C=r^{-1}\, dr\times dx\end{align*} \end{document} on the collection of balls \documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath, amssymb} \pagestyle{empty} \begin{document} \begin{align*}B_r(x)\subset \mathbb{R}^n\end{align*} \end{document}, the subcollection of balls \documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath, amssymb} \pagestyle{empty} \begin{document} \begin{align*}B_r(x)\subset B_1(0)\end{align*} \end{document} has infinite measure. Let \documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath, amssymb} \pagestyle{empty} \begin{document} \begin{align*}f:B_1(0)\to {\mathbb R}\end{align*} \end{document} have bounded gradient, \documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath, amssymb} \pagestyle{empty} \begin{document} \begin{align*}|\nabla f |\leq 1\end{align*} \end{document}. For any \documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath, amssymb} \pagestyle{empty} \begin{document} \begin{align*}B_r(x)\subset B_1(0)\end{align*} \end{document} there is a natural scale‐invariant quantity that measures the deviation of \documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath, amssymb} \pagestyle{empty} \begin{document} \begin{align*}f\,|\, B_r(x)\end{align*} \end{document} from being an affine linear function. The most basic case of quantitative differentiation (due to Peter Jones) asserts that for all \documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath, amssymb} \pagestyle{empty} \begin{document} \begin{align*}\epsilon > 0\end{align*} \end{document}, the measure of the collection of balls on which the deviation from linearity is \documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath, amssymb} \pagestyle{empty} \begin{document} \begin{align*}\geq \epsilon\end{align*} \end{document} is finite and controlled by \documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath, amssymb} \pagestyle{empty} \begin{document} \begin{align*}\epsilon\end{align*} \end{document}, independent of the particular function \documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath, amssymb} \pagestyle{empty} \begin{document} \begin{align*}f\end{align*} \end{document}. The purpose of this paper is to explain the sense in which this model case is actually a particular instance of a general phenomenon that is present in many different geometric/analytic contexts. Essentially, in each case that fits the framework, to prove the relevant quantitative differentiation theorem, it suffices to verify that the family of relative defects is monotone and coercive. We indicate one recent application to theoretical computer science and others to partial regularity theory in geometric analysis and nonlinear PDEs. © 2012 Wiley Periodicals, Inc.