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 $p+q\geq r$, $\delta \in [0,1]\cap \left [\frac {r-q}{r},\frac {p}{r}\right ]$ 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 \begin{equation*} \||x|^{c}f\|_{L^{r}(\mathbb {R}^{n})} \leq \left |\frac {p}{n-p(1-a)}\right |^{\delta } \left \||x|^{a}\nabla f\right \|^{\delta }_{L^{p}(\mathbb {R}^{n})} \left \||x|^{b}f\right \|^{1-\delta }_{L^{q}(\mathbb {R}^{n})} \end{equation*} for $n\neq p(1-a)$, where the constant $\left |\frac {p}{n-p(1-a)}\right |^{\delta }$ is sharp for $p=q$ with $a-b=1$ or $p\neq q$ with $p(1-a)+bq\neq 0$. In the critical case $n=p(1-a)$ we have \begin{equation*} \left \||x|^{c}f\right \|_{L^{r}(\mathbb {R}^{n})} \leq p^{\delta } \left \||x|^{a}\log |x|\nabla f\right \|^{\delta }_{L^{p}(\mathbb {R}^{n})} \left \||x|^{b}f\right \|^{1-\delta }_{L^{q}(\mathbb {R}^{n})}. \end{equation*} 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 estimate. The relation between the critical and the subcritical Hardy inequalities on homogeneous groups is also investigated. 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 $\beta$. There are two reasons why we call the appearing weights the superweights: the arbitrariness of the choice of any homogeneous quasi-norm and a wide range of parameters.