Narrow Results

For a bounded convex domain Ω in R^N we prove refined Hardy inequalities that involve the Hardy potential corresponding to the distance to the boundary of Ω, the volume of Ω, as well as a finite number of sharp logarithmic corrections. We also discuss the best constant of these inequalities.

In this paper, we study the Hardy–Rellich inequalities for polyharmonic operators in the critical dimension and an analogue in the p-biharmonic case. We also develop some optimal weighted Hardy–Sobolev inequalities in the general case and discuss the related eigenvalue problem. We also prove W^2,q(Ω) estimates in the biharmonic case.

We obtain sharp Hardy inequalities on antisymmetric functions, where antisymmetry is understood for multi-dimensional particles. Partially it is an extension of the paper [Th. Hoffmann-Ostenhof and A. Laptev, Hardy inequalities with homogeneous weights, J. Funct. Anal. 268 (2015) 3278–3289], where Hardy’s inequalities were considered for the antisymmetric functions in the case of the 1D particles. As a byproduct we obtain some Sobolev and Gagliardo–Nirenberg type inequalities that are applied to the study of spectral properties of Schrödinger operators.