The sharp higher order Hardy--Rellich type inequalities on the homogeneous groups

Abstract

We prove several interesting equalities for the integrals of higher order derivatives on the homogeneous groups. As consequences, we obtain the sharp Hardy--Rellich type inequalities for higher order derivatives including both the subcritical and critical inequalities on the homogeneous groups. We also prove several uncertainty principles on the homogeneous groups. Our results seem to be new even in the case of Euclidean space ${\mathbb{R}}^n$ and give a simple proof of several classical Hardy--Rellich type inequalities in ${\mathbb{R}}^n$.

arXiv comment47 pages, comments are welcome

References in corpus (4)

• Hardy and Rellich inequalities, identities, and sharp remainders on homogeneous groups
• $L^p$-Caffarelli-Kohn-Nirenberg type inequalities on homogeneous groups
• Extended Caffarelli-Kohn-Nirenberg inequalities, and remainders, stability, and superweights for $L^p$-weighted Hardy inequalities
• Hardy-Littlewood, Bessel-Riesz, and fractional integral operators in anisotropic Morrey and Campanato spaces