On functions subharmonic in a Lipschitz domain

Author:
Jang-Mei Gloria Wu

Journal:
Proc. Amer. Math. Soc. **68** (1978), 309-316

MSC:
Primary 31B25

MathSciNet review:
0470234

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let *D* be a starlike Lipschitz domain in . If *w* is a subharmonic function in *D* with positive harmonic majorant, then at almost every point on the boundary of *D* (surface measure), *w* has radial limit. Results on limits along certain curves in general Lipschitz domains are also obtained.

**[1]**Björn E. J. Dahlberg,*Estimates of harmonic measure*, Arch. Rational Mech. Anal.**65**(1977), no. 3, 275–288. MR**0466593****[2]**J. R. Diederich,*Natural limits for harmonic and superharmonic functions*, Trans. Amer. Math. Soc.**224**(1976), no. 2, 381–397 (1977). MR**0419796**, 10.1090/S0002-9947-1976-0419796-9**[3]**L. L. Helms,*Introduction to potential theory*, Pure and Applied Mathematics, Vol. XXII, Wiley-Interscience A Division of John Wiley & Sons, New York-London-Sydney, 1969. MR**0261018****[4]**Richard A. Hunt and Richard L. Wheeden,*On the boundary values of harmonic functions*, Trans. Amer. Math. Soc.**132**(1968), 307–322. MR**0226044**, 10.1090/S0002-9947-1968-0226044-7**[5]**Oliver D. Kellogg,*On the derivatives of harmonic functions on the boundary*, Trans. Amer. Math. Soc.**33**(1931), no. 2, 486–510. MR**1501602**, 10.1090/S0002-9947-1931-1501602-2**[6]**J. E. Littlewood,*On functions subharmonic in a circle*. II, Proc. London Math. Soc. (2)**28**(1928), 383-394.**[7]**Martin L. Silverstein and Richard L. Wheeden,*Superharmonic functions on Lipschitz domains*, Studia Math.**39**(1971), 191–198. MR**0315150****[8]**J.-M. Wu,*Boundary limits of Green's potentials along curves*, Studia Math.**60**(1976), 137-144.**[9]**-,*Boundary limits of Green's potentials along curves*. II.*Lipschitz domains*, Studia Math. (to appear).

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
31B25

Retrieve articles in all journals with MSC: 31B25

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1978-0470234-7

Keywords:
Subharmonic,
Green's potential,
harmonic measure,
Lipschitz domain,
limits along curves,
Harnack's inequality,
maximum principle

Article copyright:
© Copyright 1978
American Mathematical Society