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Zbl 1201.42015
Tang, Canqin; Zhai, Zhichun
Generalized Poincaré embeddings and weighted Hardy operator on spaces.
(English)
[J] J. Math. Anal. Appl. 371, No. 2, 665-676 (2010). ISSN 0022-247X

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 $Q^{\alpha, q}_p(\mathbb{R}^n)$.
[Yang Dachun (Beijing)]
MSC 2000:
*42B30 Hp-spaces (Fourier analysis)
46E35 Sobolev spaces and generalizations
42B35 Function spaces arising in harmonic analysis

Keywords: Poincaré embedding; John-Nirenberg inequality; weighted Hardy space; $Q$-space

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