球面上的Riesz位势

Riesz potential on sphere

设 Pρ(f) (x)表示 n维球面Ωn上的 Poisson积分 ,定义Ωn上的 Riesz位势为Iα(f) (x) =Cn,α∫Ωnf (y)|x - y|n-αdy, x∈Ωn.证明了若 0 <α<α+β <1 ,f (x)∈ Lipα,那么 Iβ(f) (x)∈ Lip (α+β) .若 q>1 ,nq <α<nq+1 ,f(x)∈ Lq(Ωn) ,那么 Iα(f ) (x)∈ Lip (α- nq)

Let P ρ(f)(x) denote poisson integral of f(x) defined on n sphere Ω n , Riesz potential on Ω n is defined by I α(f)(x)-C n,x ∫ Ω nf(y)｜x-y｜ n-α d y, x∈Ω n . The author shows that I β(f)(x)∈ Lip (α+β) when 0<α<α+β<1,f(x)∈ Lip α, and I α(f)(x)∈ Lip (α-nq) when q>1,nq<α<nq+1,f(x)∈L q(Ω n) .