Harmonic majorants for eigenfunctions of the Laplacian with finite Dirichlet integrals ✩

In the paper we prove the following theorem. Theorem. Let Ω be a bounded domain in RN (N 2) with C1,1 boundary, and let f be a real-valued C2 function on Ω satisfying f = λf for some λ 0. (a) Suppose f satisfies Dα(f ) = Ω δ(x)α ∇f (x) 2 dx <∞ for some α,−1 < α 1 (0 < α 1 when N = 2). Then for p = 2(N − 1)/(N + α − 2) the function |f |p has a harmonic majorant on Ω. (b) Conversely, if |f |p has a harmonic majorant on Ω for some p, 1 < p 2, then Dα(f ) <∞for α = [2(N − 1) − p(N − 2)]/p. Part (a) of the theorem was originally proved by S. Yamashita [Illinois J.Math. 25 (1981) 626–631] in the unit disc in C, and for harmonic functions (λ = 0) in the unit ball B of RN, N 3, with 0<α 1 when N = 2, and 0 α 1 when N 3.  2002 Elsevier Science (USA). All rights reserved.