On the generalized Hardy-Rellich inequalities. Issue 2 (April 2020) - Anoop, T.V.; Das, Ujjal; Sarkar, Abhishek
Abstract: In this paper, we look for the weight functions (say g ) that admit the following generalized Hardy-Rellich type inequality: _Ω g (x)u^2 dx C_Ω Δ u ^2 dx, uD_0^2, 2 (Ω ), 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 D_0^2, 2 into certain Lorentz-Zygmund spaces proved by Hansson and later by Brezis and Wainger.