The Integrability of Superharmonic Functions on Lipschitz Domains

Integrability of superharmonic functions in a John domain

“…By S + (D) we denote the family of all positive superharmonic functions in D. Armitage [5], [6] proved that S + (D) ⊂ L p (D) for 0 < p < n/(n − 1), provided D is smooth. This result was extended by Maeda-Suzuki [11] to a Lipschitz domain. They gave an estimate of p in terms of Lipschitz constant.…”

Nonintegrability of Superharmonic Functions

“…We denote by SD(x) the distance between x £ D and dD, the boundary of D. In contrast to [3], we obtain the following result: It is obvious that in the condition (*) the exponent np -n -2p cannot be replaced by any larger number (e.g., see the remark below). Incidentally, we note that if D is a bounded C ''-domain, then fDs(x)pôD(x)np~"~pdx = oo for any nonzero subharmonic function s > 0 on D (cf.…”

Integrability of superharmonic functions on Hölder domains of the plane

“…This result was generalized by Maeda and Suzuki [7] to the case of Lipschitz domains. In our previous work [9] we considered domains satisfying an interior wedge condition To prove the theorem we employ a method similar to that used in [9].…”