Coronizations and big pieces in metric spaces
Annales de l'Institut Fourier, Online first, 42 p.

We prove that coronizations with respect to arbitrary d-regular sets (not necessarily graphs) imply big pieces squared of these (approximating) sets. This is known (and due to David and Semmes in the case of sufficiently large co-dimension, and to Azzam and Schul in general) in the (classical) setting of Euclidean spaces with Hausdorff measure of integer dimension, where the approximating sets are Lipschitz graphs. Our result is a far reaching generalization of these results and we prove that coronizations imply big pieces squared is a generic property. In particular, our result applies, when suitably interpreted, in metric spaces having a fixed positive (perhaps non-integer) dimension, equipped with a Borel regular measure and with arbitrary approximating sets. As a novel application we highlight how to utilize this general setting in the context of parabolic uniform rectifiability.

Nous prouvons que la construction en couronnes relative à des ensembles d-réguliers arbitraires (non nécessairement des graphes) fournit de larges pièces carrées de ces ensembles (approximants). Cela est connu (et dû à David et Semmes dans le cas d’une co-dimension assez grande, et à Azzam et Schul en général) dans le cadre (classique) des espaces Euclidiens avec une mesure de Hausdorff de dimension entière, où les ensembles approximants sont des graphes Lipschitziens. Nos travaux sont une généralisation d’envergure de ces résultats, et nous montrons que le fait que la construction en couronnes fournisse de larges pièces carrées est une propriété générique. En particulier, nos résultats, convenablement interprétés, s’appliquent dans des espaces métriques de dimension strictement positive fixée (éventuellement non entière), munisavec une mesure de Borel régulière et avec des espaces approximants arbitraires. Comme application nouvelle, nous illustrons comment on peut utiliser ces outils généraux dans le cadre de la rectifiabilité parabolique uniforme.

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DOI: 10.5802/aif.3518
Classification: 28A75,  30L99,  43A85
Keywords: Carleson measures, corona decompositions, big pieces, geometric lemmas.
Bortz, Simon 1; Hoffman, John 2; Hofmann, Steve 2; Luna-Garcia, Jose Luis 3; Nyström, Kaj 4

1 Department of Mathematics University of Alabama Tuscaloosa, AL, 35487 (USA)
2 Department of Mathematics University of Missouri Columbia, MO 65211 (USA)
3 Department of Mathematics & Statistics McMaster University Hamilton, ON L8S 3L8 (Canada)
4 Department of Mathematics Uppsala University S-751 06 Uppsala (Sweden)
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Bortz, Simon; Hoffman, John; Hofmann, Steve; Luna-Garcia, Jose Luis; Nyström, Kaj. Coronizations and big pieces in metric spaces. Annales de l'Institut Fourier, Online first, 42 p.

[1] Azzam, Jonas; Schul, Raanan Hard Sard: quantitative implicit function and extension theorems for Lipschitz maps, Geom. Funct. Anal., Volume 22 (2012) no. 5, pp. 1062-1123 | DOI | MR | Zbl

[2] Bortz, Simon; Hoffman, John; Hofmann, Steve; Luna-Garcia, Jose Luis; Nyström, Kaj The corona decomposition for parabolic uniformly rectifiable sets (https://arxiv.org/abs/2103.12497)

[3] Bortz, Simon; Hoffman, John; Hofmann, Steve; Luna-Garcia, Jose Luis; Nyström, Kaj Parabolic Singular Integrals with Nonhomogeneous Kernels (not for publication, https://arxiv.org/abs/2103.12830)

[4] Bortz, Simon; Hofmann, Steve Harmonic measure and approximation of uniformly rectifiable sets, Rev. Mat. Iberoam., Volume 33 (2017) no. 1, pp. 351-373 | DOI | MR | Zbl

[5] Calderon, A. P.; Calderon, C. P.; Fabes, E.; Jodeit, M.; Rivière, N. M. Applications of the Cauchy integral on Lipschitz curves, Bull. Am. Math. Soc., Volume 84 (1978) no. 2, pp. 287-290 | DOI | MR | Zbl

[6] Christ, Michael A T(b) theorem with remarks on analytic capacity and the Cauchy integral, Colloq. Math., Volume 60/61 (1990) no. 2, pp. 601-628 | DOI | MR | Zbl

[7] Coifman, R.; David, G.; Meyer, Y. La solution des conjecture de Calderón, Adv. Math., Volume 48 (1983) no. 2, pp. 144-148 | DOI | MR | Zbl

[8] Coifman, R.; McIntosh, A.; Meyer, Y. L’intégrale de Cauchy définit un opérateur borné sur L 2 pour les courbes lipschitziennes, Ann. Math., Volume 116 (1982) no. 2, pp. 361-387 | DOI | MR | Zbl

[9] Coifman, R.; Semmes, S. L 2 estimates in nonlinear Fourier analysis, Harmonic analysis (Sendai, 1990) (ICM-90 Satell. Conf. Proc.), Springer, 1991, pp. 79-95 | MR

[10] David, G. Wavelets and singular integrals on curves and surfaces, Lecture Notes in Mathematics, 1465, Springer, 1991, x+107 pages | DOI | MR | Zbl

[11] David, G.; Schul, R. Quantitative decompositions of Lipschitz mappings into metric spaces (https://arxiv.org/abs/2002.10318)

[12] David, G.; Semmes, S. Singular integrals and rectifiable sets in R n : Beyond Lipschitz graphs, Astérisque, 193, Société Mathématique de France, 1991 | MR | Zbl

[13] David, G.; Semmes, S. Analysis of and on uniformly rectifiable sets, Mathematical Surveys and Monographs, 38, American Mathematical Society, 1993, xii+356 pages | DOI | MR

[14] Falconer, Kenneth Fractal geometry. Mathematical foundations and applications, John Wiley & Sons, 2014, xxx+368 pages | MR | Zbl

[15] Fässler, Katrin; Orponen, Tuomas Singular integrals on regular curves in the Heisenberg group (to appear in J. Math. Pures Appl., https://arxiv.org/abs/1911.03223)

[16] Hofmann, S. A characterization of commutators of parabolic singular integrals, Fourier analysis and partial differential equations (Miraflores de la Sierra, 1992) (Stud. Adv. Math.), CRC, Boca Raton, FL, 1995, pp. 195-210 | MR | Zbl

[17] Hofmann, S. Parabolic singular integrals of Calderón-type, rough operators, and caloric layer potentials, Duke Math. J., Volume 90 (1997) no. 2, pp. 209-259 | DOI | MR | Zbl

[18] Hofmann, S.; Lewis, J. L. L 2 solvability and representation by caloric layer potentials in time-varying domains, Ann. Math., Volume 144 (1996) no. 2, pp. 349-420 | DOI | MR | Zbl

[19] Hofmann, S.; Lewis, J. L.; Nyström, K. Existence of big pieces of graphs for parabolic problems, Ann. Acad. Sci. Fenn., Math., Volume 28 (2003) no. 2, pp. 355-384 | MR | Zbl

[20] Hofmann, S.; Lewis, J. L.; Nyström, K. Caloric measure in parabolic flat domains, Duke Math. J., Volume 122 (2004) no. 2, pp. 281-346 | DOI | MR | Zbl

[21] Hofmann, S.; Martell, J. M. Uniform rectifiability and harmonic measure I: Uniform rectifiability implies Poisson kernels in L p , Ann. Sci. Éc. Norm. Supér., Volume 47 (2014) no. 3, pp. 577-654 | DOI | MR | Zbl

[22] Hofmann, S.; Mitrea, D.; Mitrea, M.; Morris, A. J. L p -square function estimates on spaces of homogeneous type and on uniformly rectifiable sets, Mem. Am. Math. Soc., Volume 245 (2017) no. 1159, p. v+108 | DOI | MR | Zbl

[23] Hytönen, Tuomas; Kairema, Anna Systems of dyadic cubes in a doubling metric space, Colloq. Math., Volume 126 (2012) no. 1, pp. 1-33 | DOI | MR | Zbl

[24] Jones, Peter W. Rectifiable sets and the traveling salesman problem, Invent. Math., Volume 102 (1990) no. 1, pp. 1-15 | DOI | MR | Zbl

[25] Kaufman, Robert; Wu, Jang-Mei Parabolic measure on domains of class Lip1 2, Compos. Math., Volume 65 (1988) no. 2, pp. 201-207 | MR | Zbl

[26] Lewis, John L.; Murray, Margaret A. M. The method of layer potentials for the heat equation in time-varying domains, Mem. Am. Math. Soc., Volume 114 (1995) no. 545, p. viii+157 | DOI | MR | Zbl

[27] Lewis, John L.; Silver, Judy Parabolic measure and the Dirichlet problem for the heat equation in two dimensions, Indiana Univ. Math. J., Volume 37 (1988) no. 4, pp. 801-839 | DOI | MR | Zbl

[28] Mattila, Pertti Fourier analysis and Hausdorff dimension, Cambridge Studies in Advanced Mathematics, 150, Cambridge University Press, 2015, xiv+440 pages | DOI | MR | Zbl

[29] Murray, Margaret A. M. The Cauchy integral, Calderón commutators, and conjugations of singular integrals in R n , Trans. Am. Math. Soc., Volume 289 (1985) no. 2, pp. 497-518 | DOI | MR | Zbl

[30] Rigot, Séverine Quantitative notions of rectifiability in the Heisenberg groups (https://arxiv.org/abs/1904.06904)

[31] Rivera-Noriega, Jorge A parabolic version of Corona decompositions, Ill. J. Math., Volume 53 (2009) no. 2, pp. 533-559 | DOI | MR | Zbl

[32] Rivera-Noriega, Jorge Two results over sets with big pieces of parabolic Lipschitz graphs, Houston J. Math., Volume 36 (2010) no. 2, pp. 619-635 | MR | Zbl

[33] Rivera-Noriega, Jorge Parabolic singular integrals and uniformly rectifiable sets in the parabolic sense, J. Geom. Anal., Volume 23 (2013) no. 3, pp. 1140-1157 | DOI | MR | Zbl

[34] Tolsa, Xavier Uniform rectifiability, Calderón-Zygmund operators with odd kernel, and quasiorthogonality, Proc. Lond. Math. Soc., Volume 98 (2009) no. 2, pp. 393-426 | DOI | MR | Zbl

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