On remainders and stability of anisotropic $L^{p}$-weighted Hardy inequalities

Michael Ruzhansky, Durvudkhan Suragan, Nurgissa Yessirkegenov

Published 2017-01-05Version 1

In this paper, we obtain remainder estimates with sharp constants for $L^{p}$-weighted Hardy inequalities on homogeneous groups, which is also a new result in the Euclidean setting of $\mathbb R^{n}$. The critical Hardy inequalities of logarithmic type and uncertainty type principles on homogeneous groups are proved. Moreover, we investigate another improved version of $L^{p}$-weighted Hardy inequalities involving a distance and the critical case $p=Q=2$ of the improved form of the Hardy inequalities on the quasi-ball. The relation between the critical and the subcritical Hardy inequalities on homogeneous groups is investigated. We also introduce a new type of Caffarelli-Kohn-Nirenberg inequalities, which are novel also in the isotropic case of $\mathbb R^{n}$, implying for $0<\delta<1$ and for all functions $f\in C_{0}^{\infty}(\mathbb R^{n}\backslash\{0\})$, the inequalities $$ \left\|\frac{f}{|x|^{\frac{n-p}{p}+\delta}}\right\|_{L^{p}(\mathbb R^{n})}\leq p^{\delta} \left\|\frac{\log|x|}{|x|^{\frac{n-p}{p}}}\nabla f\right\|^{\delta}_{L^{p}(\mathbb R^{n})} \left\|\frac{f}{|x|^{\frac{n-p}{p}}}\right\|^{1-\delta}_{L^{p}(\mathbb R^{n})}, \quad 1<p<\infty, $$ and $$ \left\|\frac{f}{|x|^{\frac{n-p}{p}+\delta}}\right\|_{L^{p}(\mathbb R^{n})}\leq \left\|\frac{\nabla f}{|x|^{\frac{n-2p}{p}}}\right\|^{1-\delta}_{L^{p}(\mathbb R^{n})} \left\|\frac{f}{|x|^{\frac{n}{p}}}\right\|^{\delta}_{L^{p}(\mathbb R^{n})} , \quad 1<p<\infty. $$