Heat flow and quantitative differentiation

Abstract

For every Banach space (Y, · Y ) that admits an equivalent uniformly convex norm we prove that there exists c = c(Y ) ∈ (0, ∞) with the following property. Suppose that n ∈ N and that X is an n-dimensional normed space with unit ball BX . Then for every 1-Lipschitz function f : BX → Y and for every ε ∈ (0, 1/2] there exists a radius r exp(−1/ε cn ), a point x ∈ BX with x + rBX ⊆ BX , and an affine mapping Λ :εr for every y ∈ x+rBX. This is an improved bound for a fundamental quantitative differentiation problem that was formulated by Bates, Johnson, Lindenstrauss, Preiss and Schechtman (1999), and consequently it yields a new proof of Bourgain's discretization theorem (1987) for uniformly convex targets. The strategy of our proof is inspired by Bourgain's original approach to the discretization problem, which takes the affine mapping Λ to be the first order Taylor polynomial of a time-t Poisson evolute of an extension of f to all of X and argues that, under appropriate assumptions on f , there must exist a time t ∈ (0, ∞) at which Λ is (quantitatively) invertible. However, in the present context we desire a more stringent conclusion, namely that Λ well-approximates f on a macroscopically large ball, in which case we show that for our argument to work one cannot use the Poisson semigroup. Nevertheless, our strategy does succeed with the Poisson semigroup replaced by the heat semigroup. As a crucial step of our proof, we establish a new uniformly convex-valued Littlewood-Paley-Stein G-function inequality for the heat semigroup; influential work of Martínez, Torrea and Xu (2006) obtained such an inequality for subordinated Poisson semigroups but left the important case of the heat semigroup open. As a byproduct, our proof also yields a new and simple approach to the classical Dorronsoro theorem (1985) even for real-valued functions.

Metric Dimension Reduction: A Snapshot of the Ribe Program

“…This makes H n be qualitatively different from all the previously known examples which exhibit the impossibility of metric dimension reduction in 1 , and as such its existence has further ramifications that answer longstanding questions; see [198] for a detailed discussion. The proof of Theorem 23 is markedly different from (and more involved than) previous proofs of impossibility of dimension reduction in 1 , as it relies on new geometric input (a subtle property of the 3-dimensional Heisenberg group which fails for the 5-dimensional Heisenberg group) that is obtained in [198], in combination with results from [26,144,143,119]; full details appear in [198].…”

Vector-valued Littlewood-Paley-Stein Theory for Semigroups II

“…In the rest of this section, we will assume that X is a Banach space of martingale cotype q with 2 ≤ q < ∞. The following lemma, due to Hytönen and Naor [8,Lemma 24], will play an important role in our argument.…”

“…Proof. We will use the idea of the proof of Theorem 17 of [8]. By virtue of the identity ∂T t+s = ∂T t T s , we write…”

“…The main problem left open in [15] asks whether the theorem holds for the semigroup {T t } t>0 itself instead of its subordinated Poisson semigroup {P t } t>0 (see Problem 2 on page 447 of [15]). Very recently, Hytönen and Naor [8] proved that the answer is affirmative for the heat semigroup of R n and for p = q; the resulting inequality plays a key role in their work on the approximation of Lipschitz functions by affine maps. Stimulated by their result and using a clever idea of them, we are able to resolve the problem in full generality.…”

Bourgain discretization using Lebesgue-Bochner spaces

“…The proof of Bourgain's discretization theorem was clarified and simplified in [Beg99] and [GNS12] (see also its presentation in [Ost13, Section 9.2]). Different approaches to proving Bourgain's discretization theorem in special cases were found in [LN13], [HLN16], and [HN16+]. However these approaches do not improve the order of estimates for the discretization modulus.…”