Foliated corona decompositions

Abstract

Hence, c X (M )=∞, as required. For future reference we record in passing that we obtained the following bound when X =L q and 1<q<p:c Lq (M n ) ≳ q (log n) 1/ max{q,2}−1/p .(1.17)Note that the bound in [4], which is asymptotically weaker than that of [58], suffices for the qualitative conclusion c X (M )=∞ of Theorem 1.9. The above estimates seem to 65The above reasoning also leads to Theorem 1.12 below, which answers another natural question arising in the Ribe program, on the factorization of Lipschitz functions.We first briefly make preparatory observations that will be also useful elsewhere. Recall that for K ∈N a metric space X is said to be K -doubling if, for every r>0, any ball B ⊆X of radius r can be covered by K balls of radius 1 2 r. A metric space X is said to be doubling if it is K -doubling for some K ∈N. The metric space M of Theorem 1.9 can be taken to be doubling. Indeed, fix p>2 and n∈N. As in the proof of Theorem 1.9, write ϑ=1/ min{p, 4}. It was shown in [57] that ψ n,p,ϑ (H Z ) is a O(1)-doubling subset of ℓ p . Let S ⊆ℓ p be the disjoint union of translates in ℓ p of the finite sets {ψ n,p,ϑ (B n )} ∞ n=1 that are sufficiently widely-spaced so as to ensure that S is a doubling subset of ℓ p , and sup n∈N c S (M n )<∞. As in the proof of Theorem 1.9, using Theorem 1.7 we get an embedding φ: S !ℓ 1 satisfying ∥φ(x)−φ(y)∥ ℓ1 ≍ ∥x−y∥ ℓp for all x, y∈S. Thus, φ(S)=M is a doubling subset of ℓ 1 .Since S is doubling, by [62] we can extend φ to a Lipschitz function f : ℓ p !ℓ 1 . If there were Lipschitz mappings g: ℓ p !ℓ 2 and h: g(ℓ p )!ℓ 1 such that f =h¤g, then it would follow that, for all x, y∈S, ∥x−y∥ ℓp ≍ ∥φ(x)−φ(y)∥ ℓ1 = ∥h(g(x))−h(g(y))∥ ℓ1 ≲ ∥g(x)−g(y)∥ ℓ2 ≲ ∥x−y∥ ℓp .Therefore, g¤φ −1 would be a bi-Lipschitz embedding of M into ℓ 2 , which we proved above was impossible. We thus arrive at the following statement.Theorem 1.12. For any 2<p<∞ there is a Lipschitz mapping f : ℓ p !ℓ 1 that cannot be factored through a subset of a Hilbert space using Lipschitz mappings. Namely, there do not exist Lipschitz mappings g: ℓ p !ℓ 2 and h: g(ℓ p )!ℓ 1 such that f =h¤g. More generally, f cannot be factored using Lipschitz mappings through a subset of a Banach space whose modulus of uniform convexity has power type q for 2⩽q<min{4, p}.By [66, Theorem 5.2], for p⩾2 any linear operator from ℓ p to ℓ 1 factors through ℓ 2 (the factorization is via linear operators, though by [46] this is equivalent to factorization using Lipschitz functions as above). Theorem 1.12 demonstrates that there is no analogue of this factorization phenomenon for Lipschitz mappings.Such investigations arose in the Ribe program in the seminal work [45], which had a major influence on the subsequent fruitful efforts by many mathematicians in search of metric analogues of the extension and factorization paradigm of [72]. This search is itself intimately intertwined with the search for metric theories of type and cotype.

The strong geometric lemma for intrinsic Lipschitz graphs in Heisenberg groups

“…(The condition that η and R satisfy Proposition 2.8 is not part of the statement of Theorem 7.5 of [NY22], but in the proof, η and R are chosen to satisfy Proposition 2.8 for a suitable ζ and λ. )…”

“…The following Vitali-type covering lemma is proved in [NY22]. (In [NY22], it is stated for 1 32r 2 -rectilinear foliated patchworks, but all PSFCs are 1 32r 2 -rectilinear.) Lemma 5.7 ([NY22, Lem.…”

“…We prove Theorem 1.3 by using the foliated corona decompositions constructed in [NY22]. These decompose an intrinsic Lipschitz graph into rectangular regions called pseudoquads, whose aspect ratio depends on the shape of the corresponding intrinsic Lipschitz graph.…”

“…One reason for this is that, like Lipschitz graphs in R n , intrinsic Lipschitz graphs in H n satisfy a version of Rademacher's theorem; they infinitesimally resemble planes almost everywhere [FSSC11]. As in R n , answering questions about singular integrals and uniform rectifiability in H n requires more quantitative bounds, which has led to the study of various notions of quantitative rectifiability in intrinsic Lipschitz graphs [CFO19b,NY18,NY22,FOR18,Rig19,CLY22b,CLY22a].…”

Monotone sets and local minimizers for the perimeter in Carnot groups

“…Monotone sets have been introduced by Cheeger and Kleiner in [7] where the proof of the non biLipschitz embeddability of the Heisenberg group into L 1 is reduced to the classification of its monotone subsets, see also [8]. Later on, this classification played an important role in several works related to the study of locally finite perimeter sets and to geometric measure theory issues in the Heisenberg setting [24,10,25,29].…”