Quantitative Differentiation: A General Formulation

Abstract

Let dx denote Lebesgue measure on R n . With respect to the measure C D r 1 dr dx on the collection of balls B r .x/ R n , the subcollection of balls B r .x/ B 1 .0/ has infinite measure. Let f W B 1 .0/ ! R have bounded gradient, jrf j Ä 1. For any B r .x/ B 1 .0/ there is a natural scale-invariant quantity that measures the deviation of f j B r .x/ from being an affine linear function. The most basic case of quantitative differentiation (due to Peter Jones) asserts that for all > 0, the measure of the collection of balls on which the deviation from linearity is is finite and controlled by , independent of the particular function f . 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.

Volume Bounds for the Quantitative Singular Strata of Non Collapsed RCD Metric Measure Spaces

“…To this aim we need a lemma which, together with the almost rigidity result about metric cones proved in Theorem 1.12, will give us the sought result. In this lemma we use a technique reminding the general machinery of quantitative differentiation (see [Ch12]). We recall here the definition of conicality given in Definition 1.15.…”

“…In the case of our interest special configurations are the conical ones. We refer to [Ch12] for a general survey about quantitative differentiation and detailed list of references to the recent applications of this tools in the various contexts. This note is organised as follows: in section 1 we list a few basic definitions and results useful when dealing with ncRCD metric measure spaces.…”

Quantitative stratification and the regularity of harmonic map flow

“…Otherwise, there would be at least δ −1 Λ 2 disjoint intervalls of the form (γ α−q , γ α+q ) with W γ α−q ,γ α+q (u, X) > δ. This is an instance of quantitative differentiation (see [Che12] for a general perspective). For each point X ∈ P 1 , to keep track of its behavior at different scales, we define a {0, 1}-valued sequence (T α (X)) α≥1 as follows.…”

Heat flow and quantitative differentiation

“…Given an appropriate (case-specific) replacement for the notation of "affine mapping," one can formulate notions of "differentiation" in many settings that do not necessarily involve linear spaces; examples of such "qualitative" metric differentiation results include [71,46,16,73,45,51,18,18,19,20]. Corresponding results about quantitative differentiation, which lead to refined (often quite subtle and important) rigidity results can be found in [7,43,67,59,21,52,74,75,22,53,17,29,62,30,31,24,23,2,54,32]. Due to the prominence of this topic and the fact that many of the quoted results are probably not sharp, it would be of interest to develop new methods to prove sharper quantitative differentiation results.…”