Heat flow and quantitative differentiation

  • Tuomas Hytönen

    University of Helsinki, Finland
  • Assaf Naor

    Princeton University, USA
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For every Banach space (Y,Y)(Y,\|\cdot\|_Y) that admits an equivalent uniformly convex norm we prove that there exists c=c(Y)(0,)c=c(Y)\in (0,\infty) with the following property. Suppose that nNn\in \mathbb N and that XX is an nn-dimensional normed space with unit ball BXB_X. Then for every 11-Lipschitz function f:BXYf:B_X\to Y and for every \e(0,1/2]\e\in (0,1/2] there exists a radius rexp(1/ϵcn)r\ge \exp(-1/\epsilon^{cn}), a point xBXx\in B_X with x+rBXBXx+rB_X\subset B_X, and an affine mapping Λ:XY\Lambda:X\to Y such that f(y)Λ(y)Yϵr\|f(y)-\Lambda(y)\|_Y\le \epsilon r for every yx+rBXy\in x+rB_X. 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\Lambda to be the first order Taylor polynomial of a time-tt Poisson evolute of an extension of ff to all of XX and argues that, under appropriate assumptions on ff, there must exist a time t(0,)t\in (0,\infty) at which Λ\Lambda is (quantitatively) invertible. However, in the present context we desire a more stringent conclusion, namely that Λ\Lambda well-approximates ff 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\mathcal{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.

Cite this article

Tuomas Hytönen, Assaf Naor, Heat flow and quantitative differentiation. J. Eur. Math. Soc. 21 (2019), no. 11, pp. 3415–3466

DOI 10.4171/JEMS/906