## Duke Mathematical Journal

### A dual characterization of the ${\mathcal C}^{1}$ harmonic capacity and applications

#### Abstract

The Lipschitz and ${\mathcal C}^{1}$ harmonic capacities $\kappa$ and $\kappa_{c}$ in ${\mathbb R}^{n}$ can be considered as high-dimensional versions of the so-called analytic and continuous analytic capacities $\gamma$ and $\alpha$ (resp.). In this article we provide a dual characterization of $\kappa_{c}$ in the spirit of the classical one for the capacity $\alpha$ by means of the Garabedian function. Using this new characterization, we show that $\kappa(E)=\kappa(\partial_{o}E)$ for any compact set $E\subset{\mathbb R}^{n}$, where $\partial_{o}E$ is the outer boundary of $E$, and we solve an open problem posed by A. Volberg, which consists in estimating from below the Lipschitz harmonic capacity of a graph of a continuous function.

#### Article information

Source
Duke Math. J., Volume 153, Number 1 (2010), 1-22.

Dates
First available in Project Euclid: 28 April 2010

https://projecteuclid.org/euclid.dmj/1272480930

Digital Object Identifier
doi:10.1215/00127094-2010-019

Mathematical Reviews number (MathSciNet)
MR2641938

Zentralblatt MATH identifier
1196.31002

Subjects
Primary: 31B05: Harmonic, subharmonic, superharmonic functions
Secondary: 31B15: Potentials and capacities, extremal length

#### Citation

Mas, Albert; Melnikov, Mark; Tolsa, Xavier. A dual characterization of the ${\mathcal C}^{1}$ harmonic capacity and applications. Duke Math. J. 153 (2010), no. 1, 1--22. doi:10.1215/00127094-2010-019. https://projecteuclid.org/euclid.dmj/1272480930

#### References

• L. V. Ahlfors, Bounded analytic functions, Duke Math. J. 14 (1947), 1–11.
• G. David, Unrectifiable $1$-sets have vanishing analytic capacity, Rev. Mat. Iberoamericana 14 (1998), 369–479.
• J. Diestel and J. J. Uhl Jr., Vector Measures, with a foreword by B. J. Pettis, Math. Surveys 15, Amer. Math. Soc., Providence, 1977.
• N. Dunford and J. T. Schwartz, Linear Operators, I: General Theory, Pure Appl. Math. 7, Wiley-Intersci., New York, 1958.
• P. L. Duren, Theory of $\HH^{p}$ Spaces, Pure Appl. Math. 28, Academic Press, New York, 1970.
• G. B. Folland, Introduction to Partial Differential Equations, 2nd ed., Princeton Univ. Press, Princeton, N.J., 1995.
• T. W. Gamelin, Uniform Algebras, Prentice-Hall, Englewood Cliffs, N.J., 1969.
• J. Garnett, Analytic Capacity and Measure, Lecture Notes in Math. 297, Springer, Berlin, 1972.
• B. Gustafsson and D. Khavinson, On annihilators of harmonic vector fields (in Russian), Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 232 (1996), Issled. po Lineĭn. Oper. i Teor. Funktsiĭ. 24, 90–108.; English translation in J. Math. Sci. (New York) 92 (1998), 3600–3612.
• S. Lang, Real and Functional Analysis, 3rd ed., Grad. Texts in Math. 142, Springer, New York, 1993.
• —, Fundamentals of Differential Geometry, Grad. Texts in Math. 191, Springer, New York, 1999.
• J. E. Marsden and M. Mccracken, The Hopf bifurcation and its applications, Applied Math. Sci. 19, Springer, New York, 1976.
• P. Mattila and P. V. Paramonov, On geometric properties of harmonic ${\rm Lip}_{1}$-capacity, Pacific J. Math. 171 (1995), 469–490.
• H. Pajot, Analytic Capacity, Rectifiability, Menger Curvature and the Cauchy Integral, Lecture Notes in Math. 1799, Springer, Berlin, 2002.
• P. V. Paramonov, Harmonic approximations in the $\CC^{1}$-norm (in Russian), Mat. Sb. 181 (1990), 1341–1365.; English translation in Math. USSR-Sb. 71 (1992), 183–207.
• A. Ruiz De Villa, Estimate of the normal derivative over the boundary of Lipschitz domains, Int. Math. Res. Not. IMRN 2008, art. ID rnn 109.
• A. Ruiz De Villa and X. Tolsa, Characterization and semiadditivity of the C$^1$-harmonic capacity, Trans. Amer. Math. Soc. 362 (2010), 3641–3675.
• X. Tolsa, On the analytic capacity $\gamma_{+}$, Indiana Univ. Math. J. 51 (2002), 317–343.
• —, Painlevé's problem and the semiadditivity of analytic capacity, Acta Math. 190 (2003), 105–149.
• A. G. Vituškin, The analytic capacity of sets in problems of approximation theory (in Russian), Uspehi Mat. Nauk. 22 (1967), 141–199.; English translation in Russian Math. Surveys 22 (1967), 139–200.
• A. Volberg, Calderón-Zygmund capacities and operators on nonhomogeneous spaces, CBMS Regional Conf. Ser. Math. 100, Amer. Math. Soc., Providence, 2003.
• F. W. Warner, Foundations of differentiable manifolds and Lie groups, Scott Foresman, Glenview, Ill., 1971.
• H. Whitney, Analytic extensions of differentiable functions defined in closed sets, Trans. Amer. Math. Soc. 36 (1934), 63–89.