TY - JOUR

T1 - Heat flow and quantitative differentiation

AU - Hytönen, Tuomas

AU - Naor, Assaf

N1 - Funding Information:
T.H. was supported by the ERC Starting Grant “AnProb” and the Academy of Finland, CoE in Analysis and Dynamics Research. A.N. was supported by BSF grant 2010021, the Packard Foundation and the Simons Foundation.
Funding Information:
We thank Apostolos Giannopoulos and Gilles Pisier for providing pointers to the literature. We also thank the three anonymous referees for their feedback and corrections. T.H. was supported by the ERC Starting Grant ?AnProb? and the Academy of Finland, CoE in Analysis and Dynamics Research. A.N. was supported by BSF grant 2010021, the Packard Foundation and the Simons Foundation.
Publisher Copyright:
© European Mathematical Society 2019

PY - 2019

Y1 - 2019

N2 - For every Banach space (Y, k · kY ) 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/εcn), a point x ∈ BX with x + rBX ⊆ BX, and an affine mapping 3 : X → Y such that kf (y) − 3(y)kY ≤ 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 3 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 3 is (quantitatively) invertible. However, in the present context we desire a more stringent conclusion, namely that 3 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.

AB - For every Banach space (Y, k · kY ) 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/εcn), a point x ∈ BX with x + rBX ⊆ BX, and an affine mapping 3 : X → Y such that kf (y) − 3(y)kY ≤ 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 3 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 3 is (quantitatively) invertible. However, in the present context we desire a more stringent conclusion, namely that 3 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.

KW - Heat semigroup

KW - Littlewood–Paley theory

KW - Metric embeddings

KW - Quantitative differentiation

KW - Uniform convexity

UR - http://www.scopus.com/inward/record.url?scp=85072657304&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85072657304&partnerID=8YFLogxK

U2 - 10.4171/JEMS/906

DO - 10.4171/JEMS/906

M3 - Article

AN - SCOPUS:85072657304

SN - 1435-9855

VL - 21

SP - 3415

EP - 3466

JO - Journal of the European Mathematical Society

JF - Journal of the European Mathematical Society

IS - 11

ER -