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A weighted extremal function and equilibrium measure

Len Bos, Norman Levenberg, Sione Ma'u, Federico Piazzon


We find an explicit formula for the weighted extremal function of $\mathbb{R}^n\subset \mathbb{C}^n$ with weight $(1+x_1^2+\cdots +x_n^2)^{-1/2}$ as well as its Monge-Ampère measure. As a corollary, we compute the Alexander capacity of $\mathbb{RP}^n$.

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Baran, M., Plurisubharmonic extremal functions and complex foliations for the complement of convex sets in $bf R^n$, Michigan Math. J. 39 (1992), no. 3, 395–404.

Bedford, E. and Taylor, B. A., The complex equilibrium measure of a symmetric convex set in $bf R^n$, Trans. Amer. Math. Soc. 294 (1986), no. 2, 705–717.

Bloom, T., Weighted polynomials and weighted pluripotential theory, Trans. Amer. Math. Soc. 361 (2009), no. 4, 2163–2179.

Bloom, T. and Levenberg, N., Weighted pluripotential theory in $bf C^N$, Amer. J. Math. 125 (2003), no. 1, 57–103.

Bos, L., Levenberg, N., and Waldron, S., Pseudometrics, distances and multivariate polynomial inequalities, J. Approx. Theory 153 (2008), no. 1, 80–96.

Burns, D., Levenberg, N., and Ma'u, S., Pluripotential theory for convex bodies in $bf R^N$, Math. Z. 250 (2005), no. 1, 91–111.

Burns, D., Levenberg, N., Ma'u, S., and Révész, S., Monge-Ampère measures for convex bodies and Bernstein-Markov type inequalities, Trans. Amer. Math. Soc. 362 (2010), no. 12, 6325–6340.

Dinh, T.-C. and Sibony, N., Distribution des valeurs de transformations méromorphes et applications, Comment. Math. Helv. 81 (2006), no. 1, 221–258.

Guedj, V. and Zeriahi, A., Intrinsic capacities on compact Kähler manifolds, J. Geom. Anal. 15 (2005), no. 4, 607–639.

Klimek, M., Pluripotential theory, London Mathematical Society Monographs. New Series, vol. 6, The Clarendon Press, Oxford University Press, Oxford, 1991.

Lundin, M., The extremal plurisubharmonic function for the complement of the disk in $mathbb R^2$, unpublished preprint, 1984.

Sadullaev, A., Estimates of polynomials on analytic sets, Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), no. 3, 524–534, 671, English translation: Math. USSR Izvestiya 20 (1983), no. 3, 493–502.

Zelditch, S., Pluri-potential theory on Grauert tubes of real analytic Riemannian manifolds, I, in “Spectral geometry”, Proc. Sympos. Pure Math., vol. 84, Amer. Math. Soc., Providence, RI, 2012, pp. 299--339.



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