Uniform rectifiability and ε-approximability of harmonic functions in L p
Annales de l'Institut Fourier, Volume 70 (2020) no. 4, pp. 1595-1638.

Suppose that E n+1 is a uniformly rectifiable set of codimension 1. We show that every harmonic function is ε-approximable in L p (Ω) for every p(1,), where Ω:= n+1 E. Together with results of many authors this shows that pointwise, L and L p type ε-approximability properties of harmonic functions are all equivalent and they characterize uniform rectifiability for codimension 1 Ahlfors–David regular sets. Our results and techniques are generalizations of recent works of T. Hytönen and A. Rosén and the first author, J. M. Martell and S. Mayboroda.

Soit E un ensemble uniformément rectifiable de codimension 1 dans un espace euclidien de dimension n+1 et soit Ω son complémentaire. Nous montrons que toute fonction harmonique est ε-approchable dans L p (Ω) pour tout p fini strictement plus grand que 1. Cela montre, compte tenu de résultats précédents par différents auteurs, que ponctuellement, les propriétés d’ε-approximation de type L et L p de fonctions harmoniques sont équivalentes et elles caractérisent la rectifiabilité uniforme des ensembles réguliers au sens d’Ahlfors–David de codimension 1. Nos résultats et techniques sont des généralisations de travaux récents de T. Hytönen, A. Rosén et du premier auteur, J. M. Martell et S. Mayboroda.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/aif.3359
Classification: 42B37,  31B05,  42B20,  28A75
Keywords: ε-approximability, uniform rectifiability, Carleson measures, harmonic functions.
Hofmann, Steve 1; Tapiola, Olli 2

1 Department of Mathematics, University of Missouri, Columbia, MO 65211 (USA)
2 Department of Mathematics and Statistics, P.O. Box 35 (MaD), FI-40014 University of Jyväskylä (Finland)
@article{AIF_2020__70_4_1595_0,
     author = {Hofmann, Steve and Tapiola, Olli},
     title = {Uniform rectifiability and $\varepsilon $-approximability of harmonic functions in $L^p$},
     journal = {Annales de l'Institut Fourier},
     pages = {1595--1638},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {70},
     number = {4},
     year = {2020},
     doi = {10.5802/aif.3359},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3359/}
}
TY  - JOUR
TI  - Uniform rectifiability and $\varepsilon $-approximability of harmonic functions in $L^p$
JO  - Annales de l'Institut Fourier
PY  - 2020
DA  - 2020///
SP  - 1595
EP  - 1638
VL  - 70
IS  - 4
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.3359/
UR  - https://doi.org/10.5802/aif.3359
DO  - 10.5802/aif.3359
LA  - en
ID  - AIF_2020__70_4_1595_0
ER  - 
%0 Journal Article
%T Uniform rectifiability and $\varepsilon $-approximability of harmonic functions in $L^p$
%J Annales de l'Institut Fourier
%D 2020
%P 1595-1638
%V 70
%N 4
%I Association des Annales de l’institut Fourier
%U https://doi.org/10.5802/aif.3359
%R 10.5802/aif.3359
%G en
%F AIF_2020__70_4_1595_0
Hofmann, Steve; Tapiola, Olli. Uniform rectifiability and $\varepsilon $-approximability of harmonic functions in $L^p$. Annales de l'Institut Fourier, Volume 70 (2020) no. 4, pp. 1595-1638. doi : 10.5802/aif.3359. https://aif.centre-mersenne.org/articles/10.5802/aif.3359/

[1] Azzam, Jonas; Garnett, John B.; Mourgoglou, Mihalis; Tolsa, Xavier Uniform rectifiability, elliptic measure, square functions, and ε-approximability via an ACF monotonicity formula (2016) (https://arxiv.org/abs/1612.02650)

[2] Azzam, Jonas; Hofmann, Steve; Martell, José María; Nyström, Kaj; Toro, Tatiana A new characterization of chord-arc domains, J. Eur. Math. Soc., Volume 19 (2017) no. 4, pp. 967-981 | Article | MR: 3626548 | Zbl: 1366.28004

[3] Bishop, Christopher J.; Jones, Peter W. Harmonic measure and arclength, Ann. Math., Volume 132 (1990) no. 3, pp. 511-547 | Article | MR: 1078268 | Zbl: 0726.30019

[4] Bortz, Simon; Tapiola, Olli ε-approximability of harmonic functions in L p implies uniform rectifiability, Proc. Am. Math. Soc., Volume 147 (2019) no. 5, pp. 2107-2121 | Article | MR: 3937686 | Zbl: 07046532

[5] 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 | Article | MR: 1096400 | Zbl: 0758.42009

[6] Dahlberg, Björn E. J. Approximation of harmonic functions, Ann. Inst. Fourier, Volume 30 (1980) no. 2, pp. 97-107 | Article | Numdam | MR: 584274 | Zbl: 0417.31005

[7] David, Guy; Semmes, Stephen Singular integrals and rectifiable sets in R n : Beyond Lipschitz graphs, Astérisque, 193, Société Mathématique de France, 1991, 152 pages | Numdam | MR: 1113517 | Zbl: 0743.49018

[8] David, Guy; Semmes, Stephen Analysis of and on uniformly rectifiable sets, Mathematical Surveys and Monographs, 38, American Mathematical Society, 1993, xii+356 pages | Article | MR: 1251061 | Zbl: 0832.42008

[9] Duoandikoetxea, Javier Fourier analysis, Graduate Studies in Mathematics, 29, American Mathematical Society, 2001, xviii+222 pages (translated and revised from the 1995 Spanish original by David Cruz-Uribe) | MR: 1800316 | Zbl: 0969.42001

[10] Evans, Lawrence C.; Gariepy, Ronald F. Measure theory and fine properties of functions, Studies in Advanced Mathematics, CRC Press, 1992, viii+268 pages | MR: 1158660 | Zbl: 0804.28001

[11] Fefferman, Charles L.; Stein, Elias M. H p spaces of several variables, Acta Math., Volume 129 (1972) no. 3-4, pp. 137-193 | Article | MR: 0447953 | Zbl: 0257.46078

[12] Garnett, John B. Bounded analytic functions, Pure and Applied Mathematics, 96, Academic Press Inc., 1981, xvi+467 pages | MR: 628971 | Zbl: 0469.30024

[13] Garnett, John B.; Mourgoglou, Mihalis; Tolsa, Xavier Uniform rectifiability from Carleson measure estimates and ε-approximability of bounded harmonic functions, Duke Math. J., Volume 167 (2018) no. 8, pp. 1473-1524 | Article | MR: 3807315 | Zbl: 1396.28005

[14] Hänninen, Timo S. Equivalence of sparse and Carleson coefficients for general sets, Ark. Mat., Volume 56 (2018) no. 2, pp. 333-339 | Article | MR: 3893778 | Zbl: 1406.42028

[15] Hofmann, Steve; Kenig, Carlos E.; Mayboroda, Svitlana; Pipher, Jill C. Square function/non-tangential maximal function estimates and the Dirichlet problem for non-symmetric elliptic operators, J. Am. Math. Soc., Volume 28 (2015) no. 2, pp. 483-529 | Article | MR: 3300700 | Zbl: 1326.42028

[16] Hofmann, Steve; Martell, José María 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 | Article | MR: 3239100 | Zbl: 1302.31007

[17] Hofmann, Steve; Martell, José María; Mayboroda, Svitlana Uniform rectifiability, Carleson measure estimates, and approximation of harmonic functions (unpublished) | Zbl: 1359.28005

[18] Hofmann, Steve; Martell, José María; Mayboroda, Svitlana Uniform rectifiability, Carleson measure estimates, and approximation of harmonic functions, Duke Math. J., Volume 165 (2016) no. 12, pp. 2331-2389 | Article | MR: 3544283 | Zbl: 1359.28005

[19] Hytönen, Tuomas; Kairema, Anna Systems of dyadic cubes in a doubling metric space, Colloq. Math., Volume 126 (2012) no. 1, pp. 1-33 | Article | MR: 2901199 | Zbl: 1244.42010

[20] Hytönen, Tuomas; Rosén, Andreas Bounded variation approximation of L p dyadic martingales and solutions to elliptic equations, J. Eur. Math. Soc., Volume 20 (2018) no. 8, pp. 1819-1850 | Article | MR: 3854892 | Zbl: 1395.42065

[21] Jerison, David S.; Kenig, Carlos E. Boundary behavior of harmonic functions in nontangentially accessible domains, Adv. Math., Volume 46 (1982) no. 1, pp. 80-147 | Article | MR: 676987 | Zbl: 0514.31003

[22] Kenig, Carlos E.; Koch, Herbert; Pipher, Jill C.; Toro, Tatiana A new approach to absolute continuity of elliptic measure, with applications to non-symmetric equations, Adv. Math., Volume 153 (2000) no. 2, pp. 231-298 | Article | MR: 1770930 | Zbl: 0958.35025

[23] Lerner, Andrei K.; Nazarov, Fedor Intuitive dyadic calculus: the basics, Expo. Math., Volume 37 (2019) no. 3, pp. 225-265 | Article | MR: 4007575 | Zbl: 07127659

[24] Sawyer, Eric T.; Wheeden, Richard L. Weighted inequalities for fractional integrals on Euclidean and homogeneous spaces, Am. J. Math., Volume 114 (1992) no. 4, pp. 813-874 | Article | MR: 1175693 | Zbl: 0783.42011

[25] Varopoulos, Nicholas Th. A remark on functions of bounded mean oscillation and bounded harmonic functions, Pac. J. Math., Volume 74 (1978) no. 1, pp. 257-259 Addendum to: “BMO functions and the ¯-equation” (Pac. J. Math. 71 (1977), no. 1, p. 221–273) | Article | MR: 0508036 | Zbl: 0382.31004

[26] Verbitsky, Igor E. Imbedding and multiplier theorems for discrete Littlewood-Paley spaces, Pac. J. Math., Volume 176 (1996) no. 2, pp. 529-556 | Article | MR: 1435004 | Zbl: 0865.42009

Cited by Sources: