The logarithm of the Poisson kernel of a domain has vanishing mean oscillation
Authors:
David S. Jerison and Carlos E. Kenig
Journal:
Trans. Amer. Math. Soc. 273 (1982), 781-794
MSC:
Primary 31B25; Secondary 42B99
DOI:
https://doi.org/10.1090/S0002-9947-1982-0667174-2
MathSciNet review:
667174
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: Let be a
domain in
, and
the harmonic measure of
, with respect to a fixed pole in
. Then,
, where
is the Poisson kernel of
. We show that log
has vanishing mean oscillation of
.
- [1] Lennart Carleson, On the existence of boundary values for harmonic functions in several variables, Ark. Mat. 4 (1962), 393–399 (1962). MR 159013, https://doi.org/10.1007/BF02591620
- [2] R. R. Coifman and C. Fefferman, Weighted norm inequalities for maximal functions and singular integrals, Studia Math. 51 (1974), 241–250. MR 358205, https://doi.org/10.4064/sm-51-3-241-250
- [3] Björn E. J. Dahlberg, Estimates of harmonic measure, Arch. Rational Mech. Anal. 65 (1977), no. 3, 275–288. MR 466593, https://doi.org/10.1007/BF00280445
- [4] Björn E. J. Dahlberg, On the Poisson integral for Lipschitz and 𝐶¹-domains, Studia Math. 66 (1979), no. 1, 13–24. MR 562447, https://doi.org/10.4064/sm-66-1-13-24
- [5] 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
- [6] Eugene B. Fabes, Carlos E. Kenig, and Umberto Neri, Carleson measures, 𝐻¹ duality and weighted BMO in nonsmooth domains, Indiana Univ. Math. J. 30 (1981), no. 4, 547–581. MR 620267, https://doi.org/10.1512/iumj.1981.30.30045
- [7] Richard A. Hunt and Richard L. Wheeden, On the boundary values of harmonic functions, Trans. Amer. Math. Soc. 132 (1968), 307–322. MR 226044, https://doi.org/10.1090/S0002-9947-1968-0226044-7
- [8] 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, https://doi.org/10.1090/S0273-0979-1980-14762-X
- [9] -, Boundary behavior of harmonic functions in non-tangentially accessible domains, preprint.
- [10] 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, https://doi.org/10.4064/sm-54-3-221-237
- [11] L. E. Payne and H. F. Weinberger, New bounds in harmonic and biharmonic problems, J. Math. and Phys. 33 (1955), 291–307. MR 68683, https://doi.org/10.1002/sapm1954331291
- [12] Ch. Pommerenke, On univalent functions, Bloch functions and VMOA, Math. Ann. 236 (1978), no. 3, 199–208. MR 492206, https://doi.org/10.1007/BF01351365
- [13] Donald Sarason, Functions of vanishing mean oscillation, Trans. Amer. Math. Soc. 207 (1975), 391–405. MR 377518, https://doi.org/10.1090/S0002-9947-1975-0377518-3
- [14] 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, https://doi.org/10.7146/math.scand.a-10841
Retrieve articles in Transactions of the American Mathematical Society with MSC: 31B25, 42B99
Retrieve articles in all journals with MSC: 31B25, 42B99
Additional Information
DOI:
https://doi.org/10.1090/S0002-9947-1982-0667174-2
Article copyright:
© Copyright 1982
American Mathematical Society