On the Function Spaces of General Weights

The aim of this paper is twofold. Firstly, we establish the φ -transform characterizations of the Besov spaces Ḃ_p,q(Rⁿ,\ tₖ\ ) and the Triebel–Lizorkin spaces Ḟ_p,q(Rⁿ,\ tₖ\ ) for q = ∞ , in the sense of Frazier and Jawerth. Secondly, under some suitable assumptions on the p -admissible weight sequence \ tₖ\ , we prove that Ȧ_p,q(Rⁿ,\ tₖ\ ) = Ȧ_p,q(Rⁿ,tⱼ), j ∈ Z, in the sense of equivalent quasi-norms, with Ȧ ∈ \ Ḃ,Ḟ\ . Moreover, we find a necessary and sufficient conditions for the coincidence of the spaces Ȧ_p,q(Rⁿ,tᵢ) , i ∈ \ 1,2\ .