Generalized Poincaré embeddings and weighted Hardy operator on spaces

The Poincaré embeddings and the John–Nirenberg inequality in BMO(Rn) are classical results in modern analysis and partial differential equations. This paper studies the generalized Poincaré embeddings and the John–Nirenberg inequality in spaces. Furthermore, for the weighted Hardy operator Uψ and its adjoint Vψ, this paper gives sufficient and necessary conditions on weight function ψ to ensure the boundedness of them on spaces.