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In this paper we give an extension of the classical Caffarelli-Kohn-Nirenberginequalities: 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 $\frac{\deltar}{p}+\frac{(1-\delta)r}{q}=1$ and $a$, $b$, $c\in\mathbb{R}$ with$c=\delta(a-1)+b(1-\delta)$, and for all functions $f\inC_{0}^{\infty}(\mathbb{R}^{n}\backslash\{0\})$ we have  $$ \||x|^{c}f\|_{L^{r}(\mathbb{R}^{n})} \leq\left|\frac{p}{n-p(1-a)}\right|^{\delta} \left\||x|^{a}\nablaf\right\|^{\delta}_{L^{p}(\mathbb{R}^{n})}\left\||x|^{b}f\right\|^{1-\delta}_{L^{q}(\mathbb{R}^{n})} $$ for $n\neqp(1-a)$, where the constant $\left|\frac{p}{n-p(1-a)}\right|^{\delta}$ is sharpfor $p=q$ with $a-b=1$ or $p\neq q$ with $p(1-a)+bq\neq0$.  In the critical case $n=p(1-a)$ we have  $$ \left\||x|^{c}f\right\|_{L^{r}(\mathbb{R}^{n})} \leq p^{\delta}\left\||x|^{a}\log|x|\nabla f\right\|^{\delta}_{L^{p}(\mathbb{R}^{n})}\left\||x|^{b}f\right\|^{1-\delta}_{L^{q}(\mathbb{R}^{n})}. $$ Moreover, wealso obtain anisotropic versions of these inequalities which can beconveniently formulated in the language of Folland and Stein's homogeneousgroups. Consequently, we obtain remainder estimates for $L^{p}$-weighted Hardyinequalities on homogeneous groups, which are also new in the Euclidean settingof $\mathbb{R}^{n}$. The critical Hardy inequalities of logarithmic type anduncertainty type principles on homogeneous groups are obtained. Moreover, weinvestigate another improved version of $L^{p}$-weighted Hardy inequalitiesinvolving a distance and stability estimates. We also establish sharp Hardytype inequalities in $L^{p}$, $1<p<\infty$, with superweights, i.e. with theweights of the form $\frac{(a+b|x|^{\alpha})^{\frac{\beta}{p}}}{|x|^{m}}$allowing for different choices of $\alpha$ and $\beta$.

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