Transversally Lipschitz Harmonic Functions are Lipschitz

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Ravisankar, Sivaguru (2012)

Let \Omega\subset\mathbb{R}^n be a bounded domain with C^\infty boundary. We show that a harmonic function in \Omega that is Lipschitz along a family of curves transversal to b\Omega is Lipschitz in \Omega. The space of Lipschitz functions we consider is defined using the notion of a majorant which is a certain generalization of the power functions t^\alpha, 0<\alpha<1.
  • References (10)

    [1] Jacqueline De´traz. Classes de Bergman de fonctions harmoniques. Bull. Soc. Math. France, 109(2):259-268, 1981.

    [2] Konstantin M. Dyakonov. Equivalent norms on Lipschitz-type spaces of holomorphic functions. Acta Math., 178(2):143-167, 1997.

    [3] David Gilbarg and Neil S. Trudinger. Elliptic partial differential equations of second order, volume 224 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, second edition, 1983.

    [4] V. P. Havin. A generalization of the Privalov-Zygmund theorem on the modulus of continuity of the conjugate function. Izv. Akad. Nauk Armjan. SSR Ser. Mat., 6(2-3):252-258; ibid. 6 (1971), no. 4, 265-287, 1971.

    [5] A.-K. Herbig and J. McNeal. Convex defining functions for convex domains. J. Geom. Anal., to appear, arXiv:0912.4653v2.

    [6] Miroslav Pavlovic´. On K. M. Dyakonov's paper: “Equivalent norms on Lipschitz-type spaces of holomorphic functions” [Acta Math. 178 (1997), no. 2, 143-167; MR1459259 (98g:46029)]. Acta Math., 183(1):141-143, 1999.

    [7] Miroslav Pavlovic´. Lipschitz conditions on the modulus of a harmonic function. Rev. Mat. Iberoam., 23(3):831-845, 2007.

    [8] Sivaguru Ravisankar. Lipschitz properties of harmonic and holomorphic functions. Ph.D. diss., The Ohio State University, 2011.

    [9] A. Zygmund. Trigonometric series. 2nd ed. Vols. I, II. Cambridge University Press, New York, 1959.

    DEPARTMENT OF MATHEMATICS, THE OHIO STATE UNIVERSITY, COLUMBUS, OHIO 43210 E-mail address: sivaguru@math.ohio-state.edu

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