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 -