$p$-harmonic measure is not additive on null sets

Llorente, José G.; Manfredi, Juan J.; Wu, Jang-Mei

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 4 (2005) no. 2, pp. 357-373.

When $1<p<\infty$ and $p\ne 2$ the $p$-harmonic measure on the boundary of the half plane ${ℝ}_{+}^2$ is not additive on null sets. In fact, there are finitely many sets $E_1$, $E_2$,...,$E_{\kappa }$ in $ℝ$, of $p$-harmonic measure zero, such that $E_1\cup E_2\cup ...\cup E_{\kappa }=ℝ$.

[ARY] V. Alvarez, J. M. Rodríguez and D. V. Yakubovich, Estimates for nonlinear harmonic “measures” on trees, Michigan Math. J. 48 (2001), 47-64. | MR | Zbl

[AM] P. Avilés and J. J. Manfredi, On null sets of $p$-harmonic measure, In: “Partial Differential Equations with minimal smoothness and applications”, Chicago, IL 1990, B. Dahlberg et al. (eds.), Springer Verlag, New York, 1992, 33-36. | MR | Zbl

[B] A. Baernstein, Comparison of $p$-harmonic measures of subsets of the unit circle, St. Petersburg Math. J. 9 (1998), 543-551. | MR | Zbl

[BBS] A. Björn, J. Björn and N. Shanmugalingam, A problem of Baernstein on the equality of the $p$-harmonic measure of a set and its closure, Proc. AMS, to appear. | Zbl

[DB] E. Dibenedetto, $C^{1+\alpha }$-local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal., 7 (1983), 827-850. | MR | Zbl

[CFPR] A. Cantón, J. L. Fernández, D. Pestana and J. M. Rodríguez, On harmonic functions on trees, Potential Anal. 15 (2001), 1999-244. | MR | Zbl

[FGMS] E. Fabes, N. Garofalo, S. Marín-Malave and S. Salsa, Fatou theorems for some nonlinear elliptic equations, Rev. Mat. Iberoamericana 4 (1988), 227-251. | MR | Zbl

[GLM] S. Granlund, P. Lindquist and O. Martio, $F$-harmonic measure in space, Ann. Acad. Sci. Fenn. Math. Diss. 7 (1982), 233-247 | MR | Zbl

[HM] J. Heinonen and O. Martio, Estimates for $F$-harmonic measures and Øksendal’s theorem for quasiconformal mappings, Indiana Univ. Math. J. 36 (1987), 659-683. | MR | Zbl

[HKM] J. Heinonen, T. Kilpeläinen and O. Martio, “Nonlinear potential theory of degenerate elliptic equations”, Clarendon Press, New York, 1993. | MR | Zbl

[KW] R. Kaufman and J.-M. Wu, Fatou theorem of $p$-harmonic functions on trees, Ann. Probab. 28 (2000), 1138-1148. | MR | Zbl

[KLW] R. Kaufman, J. G. Llorente and J.-M. Wu, Nonlinear harmonic measures on trees, Ann. Acad. Sci. Fenn. Math. Diss. 28 (2003), 279-302. | EuDML | MR | Zbl

[K] J. Kurki, Invariant sets for $𝒜$-harmonic measure, Ann. Acad. Sci. Fenn. Math. Diss. 20 (1995), 433-436. | EuDML | MR | Zbl

[L1] J. L. Lewis, Regularity of the derivatives of solutions to certain elliptic equations, Indiana Univ. Math. J. 32 (1983), 849-856. | MR | Zbl

[L2] J. L. Lewis, “Note on a theorem of Wolff”, Holomorphic Functions and Moduli, Vol. 1, Berkeley, CA, 1986, D. Drasin et al. (eds.), Math. Sci. Res. Inst. Publ., Vol. 10, Springer-Verlag, 1988, 93-100. | MR | Zbl

[MW] J. J. Manfredi and A. Weitsman, On the Fatou theorem for $p$-harmonic functions, Comm. Partial Differential Equations 13 (1988), 651-658. | MR | Zbl

[M1] O. Martio, Potential theoretic aspects of nonlinear elliptic partial differential equations, Bericht Report 44, University of Jyväskylä, Jyväskylä, 1989. | MR | Zbl

[M2] O. Martio, Sets of zero elliptic harmonic measures, Ann. Acad. Sci. Fenn. Math. Diss. 14 (1989), 47-55. | MR | Zbl

[Ma] V. G. Maz'Ja, On the continuity at a boundary point of solutions of quasi-linear elliptic equations (English translation), Vestnik Leningrad Univ. Math. 3(1976), 225-242. Original in Vestnik Leningrad. Univ. 25 (1970), 42-45 (in Russian). | MR | Zbl

[Wo1] T. Wolff, Gap series constructions for the $p$-Laplacian, Preprint, 1984. | MR

[Wo2] T. Wolff, Generalizations of Fatou's theorem, Proceedings of the International Congres of Mathematics, Berkeley, CA, 1986, Vol. 2, Amer. Math. Soc., Providence, RI, 1987, 990-993. | MR | Zbl

[W] J.-M. Wu, Null sets for doubling and dyadic doubling measures, Ann. Acad. Sci. Fenn. Math. 18 (1993), 77-91. | EuDML | MR | Zbl