In this paper, we reintroduce the weighted multi-parameter Triebel-Lizorkin spaces F_p^(α,q) (ω; R^(n_1)× R^(n_2)) based on the Frazier and Jawerth' method in [11]. This space was′firstly introduced in [18]. Then we establish its dual space and get that(F_p^(α,q))*= CMO_p^(-α,q') for 0 < p ≤ 1.
Let M be an n-dimensional complete noncompact Riemannian manifold with sectional curvature bounded from below, dμ = eh(x) dV (x) the weighted measure and △μ,p the weighted p-Laplacian. In this paper we consider the non-linear elliptic equation △μ,pu = -λμ,p|u|p-2u for p ∈ (1, 2). We derive a sharp gradient estimate for positive smooth solutions of this equation.As applications, we get a Harnack inequality and a Liouville type theorem.
