Abstract
Let μ be a finite Radon measure in ℝd with polynomial growth of degree n, although not necessarily n-AD regular. We prove that under some geometric conditions on μ that are closely related to rectifiability and involve the so-called β-numbers of Jones, David and Semmes, all singular integral operators with an odd and sufficiently smooth Calderón-Zygmund kernel are bounded in L2(μ). As a corollary, we obtain a lower bound for the Lipschitz harmonic capacity of a compact set in ℝd only in terms of its metric and geometric properties.
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References
J. Azzam and X. Tolsa, Characterization of n-rectifiability in terms of Jones’ square function: Part II, Geom. Funct Anal. 25 (2015), 1371–1412.
G. David, Wavelets and Singular Integrals on Curves and Surfaces, Springer-Verlag, Berlin, 1991.
G. David and P. Mattila, Removable sets for Lipschitz harmonic functions in the plane, Rev. Mat. Iberoamer. 16 (2000), 137–215.
G. David and S. Semmes, Singular integrals and rectifiable sets in Rn: Beyond Lipschitz graphs, Astérisque No. 193 (1991).
G. David and S. Semmes, Analysis of and on Uniformly Rectifiable Sets, Amer. Math. Soc., Providence, RI, 1993.
K. Falconer, The Geometry of Fractal Sets, Cambridge University Press, Cambridge, 1985.
P. W. Jones, Rectifiable sets and the travelling salesman theorem, Invent. Math. 102 (1990), 1–15.
P. Mattila and P. Paramonov, On geometric properties of harmonic Lip1-capacity, Pacific J. Math. 171, (1995), 469–491.
F. Nazarov, S. Treil, and A. Volberg, The T (b)-theorem on non-homogeneous spaces that proves a conjecture of Vitushkin, CRM Preprint 519 (2002), 1–84.
P. V. Paramonov, Harmonic approximations in the C1-norm, Mat. Sb. 181 (1990), 1341–1365.
E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, 1970.
X. Tolsa. Littlewood-Paley theory and the T (1) theorem with non-doubling measures, Adv. Math. 164 (2001), 57–116.
X. Tolsa, Painlevé’s problem and the semiadditivity of analytic capacity, Acta Math. 190 (2003), 105–149.
X. Tolsa, L2 boundedness of the Cauchy transform implies L2 boundedness of all Calderón-Zygmund operators associated to odd kernels, Publ. Mat. 48 (2004), 445–479.
X. Tolsa, Bilipschitz maps, analytic capacity, and the Cauchy integral, Ann. of Math. 162 (2005), 1241–1302.
X. Tolsa, Uniform rectifiability, Calderón-Zygmund operators with odd kernel, and quasiorthogonality, Proc. Lond. Math. Soc. (3) 98 (2009), 393–426.
X. Tolsa, Analytic capacity, the Cauchy Transform, and Non-Homogeneous Calderón-Zygmund Theory, Birkhäuser Verlag, Basel, 2014.
X. Tolsa, Characterization of n-rectifiability in terms of Jones’ square function: Part I, Calc. Var. Partial Differential Equations 54 (2015), 3643–3665.
X. Tolsa, Rectifiable Measures, Square Functions Involving Densities, and the Cauchy Transform, Amer. Math. Soc., Providence, RI, 2016.
A. Volberg, Calderón-Zygmund Capacities and Operators on nonhomogeneous spaces, Amer. Math. Soc., Providence, RI, 2003.
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Supported by project MTM-2010-16232 (MICINN, Spain) and partially supported by the ERC grant 320501 of the European Research Council (FP7/2007-2013), and projects MTM-2013-44304-P (MICINN, Spain) and 2014-SGR-75 (Catalonia).
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Girela-Sarrión, D. Geometric conditions for the L2-boundedness of singular integral operators with odd kernels with respect to measures with polynomial growth in ℝd. JAMA 137, 339–372 (2019). https://doi.org/10.1007/s11854-018-0075-2
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DOI: https://doi.org/10.1007/s11854-018-0075-2