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• W2963485270 abstract "Abstract In this paper, we look for the weight functions (say g ) that admit the following generalized Hardy-Rellich type inequality: $$int_Omega g (x)u^2 dx les Cint_Omega vert Delta u vert ^2 dx,quad forall uin {rm {cal D}}_0^{2,2} (Omega ),$$ for some constant C > 0, where Ω is an open set in ℝ N with N ⩾ 1. We find various classes of such weight functions, depending on the dimension N and the geometry of Ω. Firstly, we use the Muckenhoupt condition for the one-dimensional weighted Hardy inequalities and a symmetrization inequality to obtain admissible weights in certain Lorentz-Zygmund spaces. Secondly, using the fundamental theorem of integration we obtain the weight functions in certain weighted Lebesgue spaces. As a consequence of our results, we obtain simple proofs for the embeddings of ${cal D}_0^{2,2} $ into certain Lorentz-Zygmund spaces proved by Hansson and later by Brezis and Wainger." @default.
• W2963485270 created "2019-07-30" @default.
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• W2963485270 date "2019-01-26" @default.
• W2963485270 modified "2023-09-25" @default.
• W2963485270 title "On the generalized Hardy-Rellich inequalities" @default.
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• W2963485270 doi "https://doi.org/10.1017/prm.2018.128" @default.
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