离散型p次Dirichlet型第一特征值

设μ=(μ_i)_i≥0为Z_+上的测度且p 1,考虑下述离散型p次Dirichlet型D_p(f)=Σ_(i=0)~∞μ_ib_i(f_i-f_(i+1))(f_i~(p-1)-f_(i+1)~(p-1)),f≥0,其中(b_i)_(i≥0)为Z_+上的正序列.本文旨在给出空间L~p(μ)上p次Dirichlet型D_p(f)所对应的第一特征值λ_(0,p)=inf{D_p(f):‖f‖_p=1,f非负且具有紧支撑}的上下界精细估计.

First Eigenvalue of p-Dirichlet Forms in Discrete Settings

Let μ=(μ_i)_{i ≥ 0} be a measure on Z₊:={0, 1, 2,...}, and p > 1. Consider the following p-Dirichlet form D_p(f)=∑_i=0^∞ μ_ib_i(f_i-f_i+1)(f_i^p-1-f_i+1^p-1), f ≥ 0, where (b_i)_{i ≥ 0} is a positive sequence. The purpose of this paper is to obtain upper and lower bounds for the first eigenvalue of p-Dirichlet form D_p(f) λ_0,p=inf{D_p(f):‖f‖_p=1, f ≥ 0 and has compact support}, where ‖f‖_p=(∑_i=0^∞μ_if_i^p)^1/p.