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Calderón-Zygmund capacities and Wolff potentials on Cantor sets. (English) Zbl 1215.42024
The author shows that, for some Cantor sets in $$\mathbb R^d$$, the capacity $$\gamma_s$$ associated with the $$s$$-dimensional Riesz kernel $$x/|x|^{s+1}$$ is comparable to the capacity $$\dot{C}_{\frac{2}{3}(d-s),\frac{3}{2}}$$ from non-linear potential theory. It is an open problem to show that, when $$s$$ is positive and non-integer, they are comparable for all compact sets in $$\mathbb R^d$$. There are also discussed other open problems in connection with Riesz transforms and Wolff potentials.

##### MSC:
 42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.) 31C45 Other generalizations (nonlinear potential theory, etc.)
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##### References:
 [1] Adams, D.R., Hedberg, L.I.: Function Spaces and Potential Theory. Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences), vol. 314. Springer, Berlin (1996) · Zbl 0834.46021 [2] David, G., Semmes, S.: Singular integrals and rectifiable sets in R n : Beyond Lipschitz graphs. Astérisque 193, 152 (1991) [3] David, G., Semmes, S.: Analysis of and on Uniformly Rectifiable Sets. Mathematical Surveys and Monographs, vol. 38. American Mathematical Society, Providence (1993) · Zbl 0832.42008 [4] Èĭderman, V.Ya.: Hausdorff measure and capacity associated with Cauchy potentials. Mat. Zametki 63(6), 923–934 (1998) [5] Eiderman, V., Nazarov, F., Volberg, A.: Vector-valued Riesz potentials: Cartan type estimates and related capacities (2008) · Zbl 1209.42008 [6] Garnett, J.: Analytic Capacity and Measure. Lecture Notes in Mathematics, vol. 297. Springer, Berlin (1972). · Zbl 0253.30014 [7] Garnett, J., Prat, L., Tolsa, X.: Lipschitz harmonic capacity and bilipschitz images of Cantor sets. Math. Res. Lett. 13(5–6), 865–884 (2006) · Zbl 1108.31005 [8] Léger, J.C.: Menger curvature and rectifiability. Ann. Math. (2) 149(3), 831–869 (1999) · Zbl 0966.28003 · doi:10.2307/121074 [9] Marstrand, J.M.: The ($$\phi$$,s) regular subsets of n-space. Trans. Am. Math. Soc. 113, 369–392 (1964) · Zbl 0144.04902 [10] Mattila, P.: On the analytic capacity and curvature of some Cantor sets with non $$\sigma$$-finite length. Publ. Mat. 40, 303–309 (1996) · Zbl 0888.30026 [11] Mattila, P., Preiss, D.: Rectifiable measures in R n and existence of principal values for singular integrals. J. Lond. Math. Soc. (2) 52(3), 482–496 (1995) · Zbl 0880.28002 [12] Mattila, P., Melnikov, M.S., Verdera, J.: The Cauchy integral, analytic capacity, and uniform rectifiability. Ann. Math. (2) 144(1), 127–136 (1996) · Zbl 0897.42007 · doi:10.2307/2118585 [13] Mateu, J., Tolsa, X.: Riesz transforms and harmonic Lip1-capacity in Cantor sets. Proc. Lond. Math. Soc. (3) 89(3), 676–696 (2004) · Zbl 1089.42009 · doi:10.1112/S0024611504014790 [14] Mateu, J., Tolsa, X., Verdera, J.: The planar Cantor sets of zero analytic capacity and the local T(b)-theorem. J. Am. Math. Soc. 16(1), 19–28 (2003). (electronic) · Zbl 1016.30020 · doi:10.1090/S0894-0347-02-00401-0 [15] Mateu, J., Prat, L., Verdera, J.: The capacity associated to signed Riesz kernels, and Wolff potentials. J. Reine Angew. Math. 578, 201–223 (2005) · Zbl 1086.31005 · doi:10.1515/crll.2005.2005.578.201 [16] Mateu, J., Prat, L., Verdera, J.: Capacities associated with scalar signed Riesz kernels, and analytic capacity. Preprint (2010) · Zbl 1253.31007 [17] Mayboroda, S., Volberg, A.: Boundedness of the square function and rectifiability. C. R. Math. Acad. Sci. Paris 347(17–18), 1051–1056 (2009) · Zbl 1185.28009 [18] Mayboroda, S., Volberg, A.: Square function and Riesz transform in non-integer dimensions. Preprint (2009) · Zbl 1184.42013 [19] Melnikov, M.S.: Analytic capacity: a discrete approach and the curvature of measure. Mat. Sb. 186(6), 57–76 (1995) [20] Melnikov, M.S., Verdera, J.: A geometric proof of the L 2 boundedness of the Cauchy integral on Lipschitz graphs. Int. Math. Res. Not. 7, 325–331 (1995) · Zbl 0923.42006 · doi:10.1155/S1073792895000249 [21] Paramonov, P.V.: C m -approximations by harmonic polynomials on compact sets in R n . Mat. Sb. 184(2), 105–128 (1993) · Zbl 0851.41029 [22] Prat, L.: Potential theory of signed Riesz kernels: capacity and Hausdorff measure. Int. Math. Res. Not. 19, 937–981 (2004) · Zbl 1082.31002 · doi:10.1155/S107379280413033X [23] Prat, L.: On the semiadditivity of the capacities associated with vector valued Riesz kernels. Preprint (2010) · Zbl 1278.42020 [24] Preiss, D.: Geometry of measures in R n : distribution, rectifiability, and densities. Ann. Math. (2) 125(3), 537–643 (1987) · Zbl 0627.28008 · doi:10.2307/1971410 [25] Ruiz de Villa, A., Tolsa, X.: Non existence of principal values of signed Riesz transforms of non integer dimension. Indiana Univ. Math. J. (to appear) · Zbl 1200.28004 [26] Tolsa, X.: Painlevé’s problem and the semiadditivity of analytic capacity. Acta Math. 190(1), 105–149 (2003) · Zbl 1060.30031 · doi:10.1007/BF02393237 [27] Tolsa, X.: Bilipschitz maps, analytic capacity, and the Cauchy integral. Ann. Math. (2) 162(3), 1243–1304 (2005) · Zbl 1097.30020 · doi:10.4007/annals.2005.162.1243 [28] Tolsa, X.: Principal values for Riesz transforms and rectifiability. J. Funct. Anal. 254(7), 1811–1863 (2008) · Zbl 1153.28003 · doi:10.1016/j.jfa.2007.07.020 [29] Tolsa, X.: Uniform rectifiability, Calderón-Zygmund operators with odd kernel, and quasiorthogonality. Proc. Lond. Math. Soc. (3) 98(2), 393–426 (2009) · Zbl 1194.28005 · doi:10.1112/plms/pdn035 [30] Vihtilä, M.: The boundedness of Riesz s-transforms of measures in R n . Proc. Am. Math. Soc. 124(12), 3797–3804 (1996) · Zbl 0876.28008 · doi:10.1090/S0002-9939-96-03522-8 [31] Volberg, A.: Calderón-Zygmund Capacities and Operators on Nonhomogeneous Spaces. CBMS Regional Conference Series in Mathematics, vol. 100. Am. Math. Soc., Providence (2003) · Zbl 1053.42022
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