Quantitative Differentiation: A General Formulation

Let 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{emp