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Generalized Poincaré embeddings and weighted Hardy operator on spaces. (English) Zbl 1201.42015

The well-known Poincaré embedding W ˙ 1,n ( n )BMO( n ) and the John-Nirenberg inequality in BMO( 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 p α,q ( n ) for all α(0,1), p(0,] and q[1,], which generalizes the corresponding classical results on BMO( n ). Moreover, the authors also give sufficient and necessary conditions on the function ψ to ensure that the corresponding weighted Hardy operator U ψ and its adjoint, the weighted Cesàro average operator V ψ , are bounded on the spaces Q p α,q ( n ).

MSC:
42B30H p -spaces (Fourier analysis)
46E35Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
42B35Function spaces arising in harmonic analysis
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