Iterating the Big-Pieces operator and larger sets

Jared Krandel; Raanan Schul

doi:10.1112/blms.12683

Iterating the Big-Pieces operator and larger sets

Jared Krandel
, Raanan Schul

Mathematics

Stony Brook University

Stony Brook University

Research output: Contribution to journal › Article › peer-review

Abstract

We show that if an Ahlfors–David regular set (Formula presented.) of dimension (Formula presented.) has Big Pieces of Big Pieces of Lipschitz Graphs (denoted usually by (Formula presented.)), then (Formula presented.) where (Formula presented.) is Ahlfors–David regular of dimension (Formula presented.) and has Big Pieces of Lipschitz Graphs (denoted usually by (Formula presented.). Our results are quantitative and, in fact, are proven in the setting of a metric space for any family of Ahlfors–David regular sets (Formula presented.) replacing (Formula presented.). A simple corollary is the stability of the BP operator after two iterations. This was previously only known in the Euclidean setting for the case (Formula presented.) with substantially more complicated proofs.

Original language	English
Pages (from-to)	2151-2161
Number of pages	11
Journal	Bulletin of the London Mathematical Society
Volume	54
Issue number	6
DOIs	https://doi.org/10.1112/blms.12683
State	Published - Dec 2022

Access to Document

10.1112/blms.12683

Cite this

@article{105c5a228dba40d9a6a8f189da3cf00d,

title = "Iterating the Big-Pieces operator and larger sets",

abstract = "We show that if an Ahlfors–David regular set (Formula presented.) of dimension (Formula presented.) has Big Pieces of Big Pieces of Lipschitz Graphs (denoted usually by (Formula presented.)), then (Formula presented.) where (Formula presented.) is Ahlfors–David regular of dimension (Formula presented.) and has Big Pieces of Lipschitz Graphs (denoted usually by (Formula presented.). Our results are quantitative and, in fact, are proven in the setting of a metric space for any family of Ahlfors–David regular sets (Formula presented.) replacing (Formula presented.). A simple corollary is the stability of the BP operator after two iterations. This was previously only known in the Euclidean setting for the case (Formula presented.) with substantially more complicated proofs.",

author = "Jared Krandel and Raanan Schul",

note = "Publisher Copyright: {\textcopyright} 2022 The Authors. The publishing rights in this article are licensed to the London Mathematical Society under an exclusive licence.",

year = "2022",

month = dec,

doi = "10.1112/blms.12683",

language = "English",

volume = "54",

pages = "2151--2161",

journal = "Bulletin of the London Mathematical Society",

issn = "0024-6093",

publisher = "John Wiley and Sons Ltd",

number = "6",

}

TY - JOUR

T1 - Iterating the Big-Pieces operator and larger sets

AU - Krandel, Jared

AU - Schul, Raanan

PY - 2022/12

Y1 - 2022/12

N2 - We show that if an Ahlfors–David regular set (Formula presented.) of dimension (Formula presented.) has Big Pieces of Big Pieces of Lipschitz Graphs (denoted usually by (Formula presented.)), then (Formula presented.) where (Formula presented.) is Ahlfors–David regular of dimension (Formula presented.) and has Big Pieces of Lipschitz Graphs (denoted usually by (Formula presented.). Our results are quantitative and, in fact, are proven in the setting of a metric space for any family of Ahlfors–David regular sets (Formula presented.) replacing (Formula presented.). A simple corollary is the stability of the BP operator after two iterations. This was previously only known in the Euclidean setting for the case (Formula presented.) with substantially more complicated proofs.

AB - We show that if an Ahlfors–David regular set (Formula presented.) of dimension (Formula presented.) has Big Pieces of Big Pieces of Lipschitz Graphs (denoted usually by (Formula presented.)), then (Formula presented.) where (Formula presented.) is Ahlfors–David regular of dimension (Formula presented.) and has Big Pieces of Lipschitz Graphs (denoted usually by (Formula presented.). Our results are quantitative and, in fact, are proven in the setting of a metric space for any family of Ahlfors–David regular sets (Formula presented.) replacing (Formula presented.). A simple corollary is the stability of the BP operator after two iterations. This was previously only known in the Euclidean setting for the case (Formula presented.) with substantially more complicated proofs.

UR - https://www.scopus.com/pages/publications/85130622242

U2 - 10.1112/blms.12683

DO - 10.1112/blms.12683

M3 - Article

SN - 0024-6093

VL - 54

SP - 2151

EP - 2161

JO - Bulletin of the London Mathematical Society

JF - Bulletin of the London Mathematical Society

IS - 6

ER -

Iterating the Big-Pieces operator and larger sets

Abstract

Access to Document

Other files and links

Fingerprint

Cite this