Quantitative Differentiation: A General Formulation
Article first published online: 27 SEP 2012
Copyright © 2011 Wiley Periodicals, Inc.
Communications on Pure and Applied Mathematics
Special Issue: Second Special Issue Commemorating the 75th Anniversary of the Courant Institute
Volume 65, Issue 12, pages 1641–1670, December 2012
How to Cite
Cheeger, J. (2012), Quantitative Differentiation: A General Formulation. Comm. Pure Appl. Math., 65: 1641–1670. doi: 10.1002/cpa.21424
- Issue published online: 27 SEP 2012
- Article first published online: 27 SEP 2012
- Manuscript Revised: AUG 2011
Let dx denote Lebesgue measure on . With respect to the measure on the collection of balls , the subcollection of balls has infinite measure. Let have bounded gradient, . For any there is a natural scale-invariant quantity that measures the deviation of from being an affine linear function. The most basic case of quantitative differentiation (due to Peter Jones) asserts that for all , the measure of the collection of balls on which the deviation from linearity is is finite and controlled by , independent of the particular function . 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.