Generalized Poincaré embeddings and weighted Hardy operator on spaces

The Poincaré embeddings W ˙ 1 , n ( R n ) ⊂ BMO ( R n ) and the John–Nirenberg inequality in BMO ( R n ) are classical results in modern analysis and partial differential equations. This paper studies the generalized Poincaré embeddings and the John–Nirenberg inequality in Q p α , q ( R n ) spaces. Furthermore, for the weighted Hardy operator U ψ and its adjoint V ψ , this paper gives sufficient and necessary conditions on weight function ψ to ensure the boundedness of them on Q p α , q ( R n ) spaces.