Note about square function estimates and uniformly rectifiable measures

Martikainen, Henri; Mourgoglou, Mihalis

doi:10.1090/proc/13128

Note about square function estimates and uniformly rectifiable measures
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by Henri Martikainen and Mihalis Mourgoglou PDF

Proc. Amer. Math. Soc. 144 (2016), 3455-3463 Request permission

Abstract:

We generalise and offer a different proof of a recent $L^2$ square function estimate on uniformly rectifiable sets by Hofmann, Mitrea, Mitrea and Morris. The proof is a short argument using the $\alpha$-numbers of Tolsa.

References

Jonas Azzam and Raanan Schul, Hard Sard: quantitative implicit function and extension theorems for Lipschitz maps, Geom. Funct. Anal. 22 (2012), no. 5, 1062–1123. MR 2989430, DOI 10.1007/s00039-012-0189-0
Michael Christ, Lectures on singular integral operators, CBMS Regional Conference Series in Mathematics, vol. 77, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1990. MR 1104656
Michael Christ, A $T(b)$ theorem with remarks on analytic capacity and the Cauchy integral, Colloq. Math. 60/61 (1990), no. 2, 601–628. MR 1096400, DOI 10.4064/cm-60-61-2-601-628
Vasilis Chousionis, John Garnett, Triet Le, and Xavier Tolsa, Square functions and uniform rectifiability, Trans. Amer. Math. Soc., to appear, available at http://arxiv.org/abs/1401.3382.
Guy David and Stephen Semmes, Analysis of and on uniformly rectifiable sets, Mathematical Surveys and Monographs, vol. 38, American Mathematical Society, Providence, RI, 1993. MR 1251061, DOI 10.1090/surv/038
Steve Hofmann, Parabolic singular integrals of Calderón-type, rough operators, and caloric layer potentials, Duke Math. J. 90 (1997), no. 2, 209–259. MR 1484857, DOI 10.1215/S0012-7094-97-09008-6
Steve Hofmann and John L. Lewis, Square functions of Calderón type and applications, Rev. Mat. Iberoamericana 17 (2001), no. 1, 1–20. MR 1846089, DOI 10.4171/RMI/287
Steve Hofmann, Dorina Mitrea, Marius Mitrea, and Andrew J. Morris, $L^p$-square function estimates on spaces of homogeneous type and on uniformly rectifiable sets, Mem. Amer. Math. Soc., to appear, available at http://arxiv.org/abs/1301.4943.
Peter W. Jones, Square functions, Cauchy integrals, analytic capacity, and harmonic measure, Harmonic analysis and partial differential equations (El Escorial, 1987) Lecture Notes in Math., vol. 1384, Springer, Berlin, 1989, pp. 24–68. MR 1013815, DOI 10.1007/BFb0086793
Xavier Tolsa, Uniform rectifiability, Calderón-Zygmund operators with odd kernel, and quasiorthogonality, Proc. Lond. Math. Soc. (3) 98 (2009), no. 2, 393–426. MR 2481953, DOI 10.1112/plms/pdn035