Abnormal boundary decay for stable operators

Assume α ∈ ( 0 , 2 ) and d ≥ 2 . Let L α be the generator of a symmetric, but not necessarily isotropic, α -stable process X in R d whose Lévy density is comparable with that of an isotropic α -stable process. In this paper, we show that the C 1 , Dini regularity assumption on an open set D ⊂ R d is optimal for the standard boundary decay property of nonnegative L α -harmonic functions in D , and for the standard boundary decay property of the heat kernel p D ( t , x , y ) of the part process X D of X on D by proving the following: (i) If D is a C 1 , Dini open set and h is a nonnegative function which is L α -harmonic in D and vanishes near a portion of ∂ D , then the rate at which h ( x ) decays to 0 near that portion of ∂ D is dist ( x , D c ) α / 2 . (ii) If D is a C 1 , Dini open set, then, as x → ∂ D , the rate at which p D ( t , x , y ) tends to 0 is dist ( x , D c ) α / 2 . (iii) For any non-Dini modulus of continuity ℓ , there exist non- C 1 , Dini open sets D , with ∂ D locally being the graph of a C 1 , ℓ function, such that the standard boundary decay properties above do not hold for D .