Quantitative Differentiation: A General Formulation



Let dx denote Lebesgue measure on equation image. With respect to the measure equation image on the collection of balls equation image, the subcollection of balls equation image has infinite measure. Let equation image have bounded gradient, equation image. For any equation image there is a natural scale-invariant quantity that measures the deviation of equation image from being an affine linear function. The most basic case of quantitative differentiation (due to Peter Jones) asserts that for all equation image, the measure of the collection of balls on which the deviation from linearity is equation image is finite and controlled by equation image, independent of the particular function equation image. 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.