Some new Besov and Triebel-Lizorkin spaces associated with para-accretive functions on spaces of homogeneous type

Abstract

AbstractLet (X, ρ, μ)d, θbe a space of homogeneous type withd< 0 and θ ∈ (0, 1],bbe a para-accretive function, ε ∈ (0, θ], ∣s∣ > ∈ and a0∈ (0, 1) be some constant depending ond, ∈ ands. The authors introduce the Besov spacebBspq(X) with a0> p ≧ ∞, and the Triebel-Lizorkin spacebFspq(X) with a0>p> ∞ and a0> q ≧∞ by first establishing a Plancherel-Pôlya-type inequality. Moreover, the authors establish the frame and the Littlewood-Paley function characterizations of these spaces. Furthermore, the authors introduce the new Besov spaceb−1Bs(X) and the Triebel-Lizorkin spaceb−1Fspq(X). The relations among these spaces and the known Hardy-type spaces are presented. As applications, the authors also establish some real interpolation theorems, embedding theorems,T btheorems, and the lifting property by introducing some new Riesz operators of these spaces.