Generalized Poincaré embeddings and weighted Hardy operator on spaces (Q990851)

The well-known Poincaré embedding \(\dot{W}^{1,n}(\mathbb{R}^n)\subset \text{BMO}(\mathbb{R}^n)\) and the John-Nirenberg inequality in \(\text{BMO}(\mathbb{R}^n)\) are useful tools in modern analysis and partial differential equations. The authors establish the generalized Poincaré embeddings and the John-Nirenberg inequality in the \(Q\)-type spaces \(Q^{\alpha, q}_p(\mathbb{R}^n)\) for all \(\alpha \in (0,1)\), \(p\in(0,\infty]\) and \(q\in[1,\infty]\), which generalizes the corresponding classical results on \(\text{BMO}(\mathbb{R}^n)\). Moreover, the authors also give sufficient and necessary conditions on the function \(\psi\) to ensure that the corresponding weighted Hardy operator \(U_\psi\) and its adjoint, the weighted Cesàro average operator \(V_\psi\), are bounded on the spaces \(Q^{\alpha, q}_p(\mathbb{R}^n)\).