Extended Caffarelli-Kohn-Nirenberg inequalities, and remainders, stability, and superweights for $L^p$-weighted Hardy inequalities

Abstract

In this paper we give an extension of the classical Caffarelli-Kohn-Nirenberg inequalities: we show that for $1<p,q<\infty$, $0<r<\infty$ with $p+q\ge r$, $\delta \in [0,1]\cap \left[\frac{r-q}{r},\frac{p}{r}\right]$ with $\frac{\delta r}{p}+\frac{(1-\delta )r}{q}=1$ and $a$, $b$, $c\in \mathbb{R}$ with $c=\delta (a-1)+b(1-\delta )$, and for all functions $f\in C_0^{\infty }({\mathbb{R}}^n\backslash \{0\})$ we have $$\Vert |x{|}^{c}f{\Vert }_{L^r({\mathbb{R}}^n)}\le {\left|\frac{p}{n-p(1-a)}\right|}^{\delta }{\Vert |x{|}^a\nabla f\Vert }_{L^p({\mathbb{R}}^n)}^{\delta }{\Vert |x{|}^{b}f\Vert }_{L^q({\mathbb{R}}^n)}^{1-\delta }$$ for $n\ne p(1-a)$, where the constant ${\left|\frac{p}{n-p(1-a)}\right|}^{\delta }$ is sharp for $p=q$ with $a-b=1$ or $p\ne q$ with $p(1-a)+bq\ne 0$. In the critical case $n=p(1-a)$ we have $${\Vert |x{|}^{c}f\Vert }_{L^r({\mathbb{R}}^n)}\le p^{\delta }{\Vert |x{|}^a\log |x|\nabla f\Vert }_{L^p({\mathbb{R}}^n)}^{\delta }{\Vert |x{|}^{b}f\Vert }_{L^q({\mathbb{R}}^n)}^{1-\delta }.$$ Moreover, we also obtain anisotropic versions of these inequalities which can be conveniently formulated in the language of Folland and Stein's homogeneous groups. Consequently, we obtain remainder estimates for $L^p$-weighted Hardy inequalities on homogeneous groups, which are also new in the Euclidean setting of ${\mathbb{R}}^n$. The critical Hardy inequalities of logarithmic type and uncertainty type principles on homogeneous groups are obtained. Moreover, we investigate another improved version of $L^p$-weighted Hardy inequalities involving a distance and stability estimates. We also establish sharp Hardy type inequalities in $L^p$, $1<p<\infty$, with superweights, i.e. with the weights of the form $\frac{(a+b|x{|}^{\alpha }{)}^{\frac{\beta }{p}}}{|x{|}^m}$ allowing for different choices of $\alpha$ and $\beta$.

31 pages; new inequalities have been added yielding also new results on Rn, namely, extended Caffarelli-Kohn-Nirenberg inequalities and $L^p$-weighted Hardy inequalities with superweights. Therefore, the title has been also changed

References in corpus (2)

• On the best possible remaining term in the Hardy Inequality
• Stability of trace theorems on the sphere

Cited by in corpus (2)

• Caffarelli-Kohn-Nirenberg and Sobolev type inequalities on stratified Lie groups
• The sharp higher order Hardy--Rellich type inequalities on the homogeneous groups