The logarithm of the Poisson kernel of a 𝐶¹ domain has vanishing mean oscillation

Jerison, David S.; Kenig, Carlos E.

doi:10.1090/S0002-9947-1982-0667174-2

The logarithm of the Poisson kernel of a $C^{1}$ domain has vanishing mean oscillation
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by David S. Jerison and Carlos E. Kenig PDF

Trans. Amer. Math. Soc. 273 (1982), 781-794 Request permission

Abstract:

Let $D$ be a ${C^1}$ domain in ${{\mathbf {R}}^n}$, and $\omega$ the harmonic measure of $\partial D$, with respect to a fixed pole in $D$. Then, $d\omega = kd\sigma$, where $k$ is the Poisson kernel of $D$. We show that log $k$ has vanishing mean oscillation of $\partial D$.

References

Lennart Carleson, On the existence of boundary values for harmonic functions in several variables, Ark. Mat. 4 (1962), 393–399 (1962). MR 159013, DOI 10.1007/BF02591620
R. R. Coifman and C. Fefferman, Weighted norm inequalities for maximal functions and singular integrals, Studia Math. 51 (1974), 241–250. MR 358205, DOI 10.4064/sm-51-3-241-250
Björn E. J. Dahlberg, Estimates of harmonic measure, Arch. Rational Mech. Anal. 65 (1977), no. 3, 275–288. MR 466593, DOI 10.1007/BF00280445
Björn E. J. Dahlberg, On the Poisson integral for Lipschitz and $C^{1}$-domains, Studia Math. 66 (1979), no. 1, 13–24. MR 562447, DOI 10.4064/sm-66-1-13-24
Björn E. J. Dahlberg, Harmonic functions in Lipschitz domains, Harmonic analysis in Euclidean spaces (Proc. Sympos. Pure Math., Williams Coll., Williamstown, Mass., 1978) Proc. Sympos. Pure Math., XXXV, Part, Amer. Math. Soc., Providence, R.I., 1979, pp. 313–322. MR 545271
Eugene B. Fabes, Carlos E. Kenig, and Umberto Neri, Carleson measures, $H^{1}$ duality and weighted BMO in nonsmooth domains, Indiana Univ. Math. J. 30 (1981), no. 4, 547–581. MR 620267, DOI 10.1512/iumj.1981.30.30045
Richard A. Hunt and Richard L. Wheeden, On the boundary values of harmonic functions, Trans. Amer. Math. Soc. 132 (1968), 307–322. MR 226044, DOI 10.1090/S0002-9947-1968-0226044-7
David S. Jerison and Carlos E. Kenig, An identity with applications to harmonic measure, Bull. Amer. Math. Soc. (N.S.) 2 (1980), no. 3, 447–451. MR 561530, DOI 10.1090/S0273-0979-1980-14762-X

Boundary behavior of harmonic functions in non-tangentially accessible domains

Benjamin Muckenhoupt and Richard L. Wheeden, Weighted bounded mean oscillation and the Hilbert transform, Studia Math. 54 (1975/76), no. 3, 221–237. MR 399741, DOI 10.4064/sm-54-3-221-237
L. E. Payne and H. F. Weinberger, New bounds in harmonic and biharmonic problems, J. Math. and Phys. 33 (1955), 291–307. MR 68683, DOI 10.1002/sapm1954331291
Ch. Pommerenke, On univalent functions, Bloch functions and VMOA, Math. Ann. 236 (1978), no. 3, 199–208. MR 492206, DOI 10.1007/BF01351365
Donald Sarason, Functions of vanishing mean oscillation, Trans. Amer. Math. Soc. 207 (1975), 391–405. MR 377518, DOI 10.1090/S0002-9947-1975-0377518-3
Kjell-Ove Widman, Inequalities for the Green function and boundary continuity of the gradient of solutions of elliptic differential equations, Math. Scand. 21 (1967), 17–37 (1968). MR 239264, DOI 10.7146/math.scand.a-10841