Capacities Associated with Calderón-Zygmund Kernels

Vasilis Chousionis1, Jоan Mateu1, Laura Prat1, Xavier Tolsa2
1Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193, Barcelona, Catalonia, Spain
2Institució Catalana de Recerca i Estudis Avançats (ICREA), Passeig Lluís Companys, 23, 08010, Barcelona, Spain

Tóm tắt

Từ khóa


Tài liệu tham khảo

Ahlfors, L.: Bounded analytic functions. Duke Math. J. 14, 1–11 (1947)

Aronszajn, N., Creese, T., Lipkin, L.: Polyharmonic Functions. Oxford Mathematical Monographs. Oxford University Press, New York (1983)

Calderón, A.P.: Acceptance speech for the Bocher price. Not. Am. Math. Soc. 26, 97–99 (1979)

Chousionis, V., Mateu, J., Prat, L., Tolsa, X.: Calderón-Zygmund kernels and rectifiability in the plane. Adv. Math. 231(1), 535–568 (2012)

Christ, M.: Lectures on singular integral operators. In: CBMS Regional Conference Series in Mathematics, Published for the Conference Board of the Mathematical Sciences, vol. 77. Washington, DC (1990)

David, G.: Unrectifiable 1-sets have vanishing analytic capacity. Rev. Mat. Iberoam. 14(2), 369–479 (1998)

Davie, A.M., Øksendal, B.: Analytic capacity and differentiability properties of finely harmonic functions. Acta Math. 149, 127–152 (1982)

Duoandikoetxea, J.: Fourier Analysis. American Mathematical Society (2001)

Farag, H.M.: The Riesz kernels do not give rise to higher-dimensional analogues of the Menger-Melnikov curvature. Publ. Mat. 43(1), 251–260 (1999)

Garnett, J.: Analytic Capacity and Measure. Lecture Notes in Mathematics, vol. 297. Springer, Berlin (1972)

Graham, R.L., Knuth, D.E., Patashnik, O.: Concrete Mathematics. A Foundation for Computer Science, 2nd edn. Addison-Wesley Publishing Company, Reading, MA (1994)

Huovinen, P.: A nicely behaved singular integral on a purely unrectifiable set. Proc. Am. Math. Soc. 129(11), 3345–3351 (2001)

Léger, J.C.: Menger curvature and rectifiability. Ann. Math. 149, 831–869 (1999)

Lyons, R., Zumbrun, K.: Homogeneous partial derivatives of radial functions. Proc. Am. Math. Soc. 121(1), 315–316 (1994)

Mateu, J., Prat, L., Verdera, J.: Potential theory of scalar Riesz kernels. Indiana Univ. Math. J. (to appear)

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)

Mattila, P., Melnikov, M.S., Verdera, J.: The Cauchy integral, analytic capacity, and uniform rectifiability. Ann. Math. 144(2), 127–136 (1996)

Mattila, P., Paramonov, P.V.: On geometric properties of harmonic Lip1-capacity. Pac. J. Math. 171, 469–491 (1995)

Maz’ya, V.G., Shaposhnikova, T.O.: Theory of Multipliers in Spaces of Differentiable Functions. Monographs and Studies in Mathematics, vol. 23. Pitman (Advanced Publishing Program), Boston, MA (1985)

Melnikov, M.S.: Analytic capacity: a discrete approach and the curvature of measure. (Russian) Mat. Sb. 186(6), 57–76 (1995); translation in Sb. Math. 186(6), 827–846 (1995)

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)

Nazarov, F., Treil, S., Volberg, A.: Weak type estimates and Cotlar inequalities for Calderón-Zygmund operators on nonhomogeneous spaces. Int. Math. Res. Not. 9, 463–487 (1998)

Nazarov, F., Treil, S., Volberg, A.: The T(b)-theorem on non-homogeneous spaces that proves a conjecture of Vitushkin. CRM (2002, preprint)

Pajot, H.: Analytic Capacity, Rectifiability, Menger Curvature and the Cauchy Integral. Lecture Notes in Mathematics, vol. 1799. Springer, Berlin (2002)

Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton, NJ (1970)

Tolsa, X.: On the analytic capacity γ +. Indiana Univ. Math. J. 51(2), 317–344 (2002)

Tolsa, X.: Painlevé’s problem and the semiadditivity of analytic capacity. Acta Math. 190(1), 105–149 (2003)

Tolsa, X.: The semiadditivity of continuous analytic capacity and the inner boundary conjecture. Am. J. Math. 126, 523–567 (2004)

Tolsa, X.: L 2 boundedness of the Cauchy transform implies L 2 boundedness of all Calderón-Zygmund operators associated to odd kernels. Publ. Mat. 48(2), 445–479 (2004)

Tolsa, X.: Bilipschitz maps, analytic capacity and the Cauchy integral. Ann. Math. 162(3), 1243–1304 (2005)

Uy, N.X.: Removable sets of analytic functions satisfying a Lipschitz condition. Ark. Mat. 17, 19–27 (1979)

Verdera, J.: Removability, Capacity and Approximation. Complex Potential Theory, NATO ASI Series, pp. 419–473. Kluwer Academic Publ., Dordrecht (1994)

Vitushkin, A.G.: The analytic capacity of sets in problems of approximation theory. Math. Surv. Monogr. 22, 139–200 (1967)