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Small-Constant Uniform Rectifiability

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Abstract

We provide several equivalent characterizations of locally flat, d-Ahlfors regular, uniformly rectifiable sets E in \({\mathbb {R}}^n\) with density close to 1 for any dimension \(d \in {\mathbb {N}}\), \(1 \le d < n\). In particular, we show that when E is Reifenberg flat with small constant and has Ahlfors regularity constant close to 1, then the Tolsa \(\alpha \) coefficients associated to E satisfy a small-constant Carleson measure estimate. This estimate is new, even when \(d= n-1\), and gives a new characterization of chord-arc domains with small constant.

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References

  1. Azzam, J., Dąbrowski, D.: An \(\alpha \)-number characterization of \(L^p\) spaces on uniformly rectifiable sets. Publ. Mat. 67(2), 819–850 (2023)

    Article  MathSciNet  Google Scholar 

  2. Azzam, J., Mourgoglou, M., Tolsa, X.: The one-phase problem for harmonic measure in two-sided NTA domains. Anal. PDE 10(3), 559–588 (2017)

    Article  MathSciNet  Google Scholar 

  3. Azzam, J., Hofmann, S., Martell, J.M., Mourgoglou, M., Tolsa, X.: Harmonic measure and quantitative connectivity: geometric characterization of the \(L^p\)-solvability of the Dirichlet problem. Invent. Math. 222(3), 881–993 (2020)

    Article  MathSciNet  Google Scholar 

  4. Badger, M., Engelstein, M., Toro, T.: Regularity of the singular set in a two-phase problem for harmonic measure with Hölder data. Rev. Mat. Iberoam. 36(5), 1375–1408 (2020)

    Article  MathSciNet  Google Scholar 

  5. Blatt, S.: Chord-arc constants for submanifolds of arbitrary codimension. Adv. Calc. Var. 2(3), 271–309 (2009)

    Article  MathSciNet  Google Scholar 

  6. Bortz, S., Hofmann, S.: A singular integral approach to a two phase free boundary problem. Proc. Amer. Math. Soc. 144(9), 3959–3973 (2016)

    Article  MathSciNet  Google Scholar 

  7. Bortz, S., Engelstein, M., Goering, M., Toro, T., Zhao, Z.: Two phase free boundary problem for Poisson kernels. Indiana Univ. Math. J. 71(1), 251–306 (2022)

    Article  MathSciNet  Google Scholar 

  8. Bortz, S., Toro, T., Zhao, Z.: Optimal Poisson kernel regularity for elliptic operators with Hölder continuous coefficients in vanishing chord-arc domains. J. Funct. Anal. 285(5), 64 (2023)

    Article  Google Scholar 

  9. Byun, S.-S., Jehan, O.: Global gradient estimates for the borderline case of double phase problems with BMO coefficients in nonsmooth domains. J. Differ. Eqs. 263(2), 1643–1693 (2017)

    Article  MathSciNet  Google Scholar 

  10. David, G., Semmes, S.: Analysis of and on uniformly rectifiable sets. Mathematical Surveys and Monographs, vol. 38. American Mathematical Society, Providence, RI (1993)

  11. David, G., Semmes, S.: Singular integrals and rectifiable sets in\({\mathbb{R}}^n\). Au-delà des graphes lipschitziens. Number 193 in Astérisque. Société mathématique de France (1991)

  12. David, G.: Wavelets and singular integrals on curves and surfaces, Lecture Notes in Mathematics, vol. 1465. Springer-Verlag, Berlin (1991)

    Book  Google Scholar 

  13. David, G., Toro, T.: Reifenberg parameterizations for sets with holes. Mem. Amer. Math. Soc. 215(1012), vi+102 (2012)

    MathSciNet  Google Scholar 

  14. David, G., Kenig, C., Toro, T.: Asymptotically optimally doubling measures and Reifenberg flat sets with vanishing constant. Comm. Pure Appl. Math. 54(4), 385–449 (2001)

    Article  MathSciNet  Google Scholar 

  15. David, G., Feneuil, J., Mayboroda, S.: Dahlberg’s theorem in higher co-dimension. J. Funct. Anal. 276(9), 2731–2820 (2019)

    Article  MathSciNet  Google Scholar 

  16. David, G., Li, L., Mayboroda, S.: Small \(A_\infty \) results for Dahlberg-Kenig-Pipher operators in sets with uniformly rectifiable boundaries. Trans. Amer. Math. Soc. 376(11), 7857–7909 (2023)

    MathSciNet  Google Scholar 

  17. Engelstein, M., Jeznach, C., Li, L., Mayboroda, S.: Small \(a_\infty \) results for dahlberg-kenig-pipher operators outside of uniformly rectifiable sets with high co-dimension. In preparation

  18. Engelstein, M.: A free boundary problem for the parabolic Poisson kernel. Adv. Math. 314, 835–947 (2017)

    Article  MathSciNet  Google Scholar 

  19. Hofmann, S., Martell, J.M.: Uniform rectifiability and harmonic measure I: Uniform rectifiability implies Poisson kernels in \(L^p\). Ann. Sci. Éc. Norm. Supér. (4) 47(3), 577–654 (2014)

  20. Hofmann, S., Mitrea, M., Taylor, M.: Singular Integrals and Elliptic Boundary Problems on Regular Semmes-Kenig-Toro Domains. Int. Math. Res. Notices 2010(14), 2567–2865 (2009)

    MathSciNet  Google Scholar 

  21. Hofmann, S., Martell, J.M., Mayboroda, S.: Uniform Rectifiability and Harmonic Measure III: Riesz Transform Bounds Imply Uniform Rectifiability of Boundaries of 1-sided NTA Domains. Int. Math. Res. Notices 2014(10), 2702–2729 (2013)

    Article  MathSciNet  Google Scholar 

  22. Hofmann, S., Martell, J.M., Uriarte-Tuero, I.: Uniform rectifiability and harmonic measure, II: Poisson kernels in \(L^p\) imply uniform rectifiability. Duke Math. J. 163(8), 1601–1654 (2014)

    Article  MathSciNet  Google Scholar 

  23. Hong, Guanghao: Wang, Lihe: A new proof of Reifenberg’s topological disc theorem. Pacific J. Math. 246(2), 325–332 (2010)

    Article  MathSciNet  Google Scholar 

  24. Hytönen, T., Tapiola, O.: Almost lipschitz-continuous wavelets in metric spaces via a new randomization of dyadic cubes. J. Approx. Theo. 185, 12–30 (2014)

    Article  MathSciNet  Google Scholar 

  25. Jerison, D.S., Kenig, C.E.: The logarithm of the Poisson kernel of a \(C^{1}\) domain has vanishing mean oscillation. Trans. Amer. Math. Soc. 273(2), 781–794 (1982)

    MathSciNet  Google Scholar 

  26. Jones, P.W.: Rectifiable sets and the traveling salesman problem. Invent. Math. 102(1), 1–15 (1990)

    Article  MathSciNet  Google Scholar 

  27. Kenig, C.E., Toro, T.: Harmonic measure on locally flat domains. Duke Math. J. 87(3), 509–551 (1997)

    Article  MathSciNet  Google Scholar 

  28. Kenig, C.E., Toro, T.: Free boundary regularity for harmonic measures and Poisson kernels. Ann. Math. 150(2), 369–454 (1999)

    Article  MathSciNet  Google Scholar 

  29. Kenig, C.E., Toro, T.: Poisson kernel characterization of Reifenberg flat chord arc domains. Ann. Sci. École Norm. Sup. (4) 36(3), 323–401 (2003)

    Article  MathSciNet  Google Scholar 

  30. Lewis, J.L., Nyström, K.: Regularity and free boundary regularity for the \(p\)-Laplace operator in Reifenberg flat and Ahlfors regular domains. J. Amer. Math. Soc. 25(3), 827–862 (2012)

    Article  MathSciNet  Google Scholar 

  31. Maggi, F.: Sets of finite perimeter and geometric variational problems, volume 135 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, (2012). An introduction to geometric measure theory

  32. Mattila, P.: Geometry of sets and measures in Euclidean spaces. Cambridge Studies in Advanced Mathematics, vol. 44. Cambridge University Press, Cambridge (1995)

  33. Milakis, Emmanouil: Toro, Tatiana: Divergence form operators in Reifenberg flat domains. Math. Z. 264(1), 15–41 (2010)

    Article  MathSciNet  Google Scholar 

  34. Milakis, E., Pipher, J., Toro, T.: Harmonic analysis on chord arc domains. J. Geom. Anal. 23(4), 2091–2157 (2013)

    Article  MathSciNet  Google Scholar 

  35. Mourgoglou, M., Puliatti, C.: Blow-ups of caloric measure in time varying domains and applications to two-phase problems. J. Math. Pures Appl. 9(152), 1–68 (2021)

    Article  MathSciNet  Google Scholar 

  36. Prats, M., Tolsa, X.: The two-phase problem for harmonic measure in VMO. Calc. Var. Partial Differential Equations 59(3), 58 (2020)

    Article  MathSciNet  Google Scholar 

  37. Reifenberg, E.R.: Solution of the Plateau Problem for \(m\)-dimensional surfaces of varying topological type. Acta Math. 104, 1–92 (1960)

    Article  MathSciNet  Google Scholar 

  38. Semmes, S.: Chord-arc surfaces with small constant. I. Adv. Math. 85(2), 198–223 (1991)

    Article  MathSciNet  Google Scholar 

  39. Semmes, S.: Chord-arc surfaces with small constant II. Good parameterizations. Adv. Math. 88(2), 170–199 (1991)

    Article  MathSciNet  Google Scholar 

  40. Tolsa, X.: Uniform rectifiability, Calderón-Zygmund operators with odd kernel, and quasiorthogonality. Proc. Lond. Math. Soc. (3) 98(2), 393–426 (2009)

    Article  MathSciNet  Google Scholar 

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  1. School of Mathematics, University of Minnesota, Minneapolis, MN, 55455, USA

    Cole Jeznach

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C. Jeznach was partially supported by the Simons Collaborations in MPS grant 563916, and NSF DMS grant 2000288. The author would like to thank his advisors Max Engelstein and Svitlana Mayboroda, as well as Guy David for many helpful conversations regarding the main result. The author also thanks the referees for many helpful comments which improved the paper.

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Jeznach, C. Small-Constant Uniform Rectifiability. J Geom Anal 34, 125 (2024). https://doi.org/10.1007/s12220-024-01567-z

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