Let $$\frac{1}{r}+\frac{\gamma }{d}=a\left(\frac{1}{p}+\frac{\alpha -1}{d}\right)+(1-a)\left(\frac{1}{q}+\frac{\beta }{d}\right),$$ where $d\geq 1$, $a\in [0,1]$, $\alpha ,\beta,\gamma \in \mathbb R$.
There exists $C$ independent of $u$ s.t. $$\||x|^\gamma u\|_{L^r(\mathbb R^d)}\leq C\||x|^\alpha \nabla u\|_{L^p(\mathbb R^d)}^{a}\||x|^\beta u\|_{L^q(\mathbb R^d)}^{1-a}$$ for all $u\in\mathcal C_0^1(\mathbb R^d)$. How can I interpret this ?
For example, if $\frac{1}{r}=\frac{a}{p}+\frac{1-a}{q}$, I know that for all $u\in L^p\cap L^q$ we have that $$\|u\|_{L^r}\leq \|u\|_{L^p}^a\|u\|_{L^q}^{1-a}$$ what can be interpreted as if $u\in L^p\cap L^q$, then $u\in L^r$ for all $r\in [p,q]$. Or in other word $L^r\supset L^p\cap L^q$ for all $r\in [p,q]$.
Q1) For my case, we have that $u\in \mathcal C_0^1 (\mathbb R^d)$ so it seems to be normal that $u\in L^p$ and $u\in W^{1,p}$ for all $p\geq 1$... but may be there is a density argument behind ?
Q2) So may be if $\alpha =\gamma =\beta =0$ I can say that $L^r\supset W^{1,q}\cap L^{p^*}$ for all $r\in [p^*,q]$ where $\frac{1}{p^*}=\frac{1}{p}-\frac{1}{d}$ ? But it doesn't look incredible...
Q3) And what are those $|x|^\gamma, |x|^\alpha $ and $|x|^\beta $ ? Which information do they give us ?
Q4) Could you also give me an application of such an inequality ?
As I studied non-local operators in my master thesis I know these inequalities are Caffarelli-Kohn-Nirenberg inequalities
Let us illuminate our answer by first of all properly recall the classical Gagliardo-Nirenberg inequality.
The Sobolev embedding $W^{1,p}(\Bbb R^d)\hookrightarrow L^{p^*}(\Bbb R^d)$ is a direct consequence case of that (by density argument).
Remark One easily check that $p^*$ is the only possible value for which such embedding(the inequality in (1) ) occur. Indeed it suffices to for smooth function $u$ function to plug in the scaled function $$u_\lambda(x) = u(\lambda x)$$ and $\lambda\to 0,\infty$ aftermath.
By interpolation We have that, $W^{1,p}(\Bbb R^d)\hookrightarrow L^{q}(\Bbb R^d)$ for every $q\in [p, p^*]$. Indeed, the is $a\in (0,1)$ suhc that $$\frac{1}{q}= \frac{a}{p^*}+\frac{1-a}{p}$$ By interpolation inequality we have , $$\|u\|_{q} \le \| u\|_{p}^{a}\|u\|_{p^*}^{1-a}\le \| u\|_{p} +\|u\|_{p^*} \le \| u\|_{p}+\|\nabla u\|_{p}\tag{2}$$
Problem and my answer to Q3) Caffarelli-Kohn-Nirenberg tried to answer similar question in a weighted $L^p-$spaces and then the possible weight should be chosen in the such that the corresponding $p^*$ in the weighted setting is still unique as the in the remark above. The uniqueness of $p^*$ is proved using the scaled function $u_{\lambda}(x) = u(\lambda x)$. It then turned out there need some weighted with good scaling properties such as homogeneity (i.e $f(xt) = t^\ell f(x)$ for some $\ell\in\Bbb R$ and all $x\in \Bbb R^d,t>0$ ). The Best and simplest candidate for this job are thereafter functions of the form $|x|^\ell.$ See this screen-short below from there paper, which briefly explain the scaling issue
I shall also mention that however the definition of weighted Sobolev is a bite demanding since the definition of weak derivative has some obstruction if the weight is not smooth enough. in such cases Analysis people assume the weak the derivative to be the usual weak derivative and they agree on the fact that the weighted Sobolev spaces $W^{1,p}(w)$ is the space is the of class functions $u$ such that $u,\nabla u\in L^p(w)$ where $\nabla u$ is understood in the classical distributional sense.
Hence observing that $u,$ and $\nabla u$ does not have the same scaling factor (just use $u_\lambda$) it turn out the weights $|x|^\ell.$ could be chosen with different exponents namely, $|x|^\alpha,|x|^\beta$ and $|x|^\gamma$
My answer to Q2) we have, $$\frac{1}{r}+\frac{\gamma }{d}=a\left(\frac{1}{p}+\frac{\alpha -1}{d}\right)+(1-a)\left(\frac{1}{q}+\frac{\beta }{d}\right),$$ and $$\||x|^\gamma u\|_{L^r(\mathbb R^d)}\leq C\||x|^\alpha \nabla u\|_{L^p(\mathbb R^d)}^{a}\||x|^\beta u\|_{L^q(\mathbb R^d)}^{1-a}$$
taking $\alpha =\beta=\gamma = 0$ leads exactly to the inequality $(2)$ indeed, we have $$\frac{1}{r}=a\left(\frac{1}{p}-\frac{1}{d}\right)+\frac{a}{q}= \frac{a}{p^*}+\frac{1-a}{q},$$ and $$\| u\|_{L^r(\mathbb R^d)}\leq C\| \nabla u\|_{L^p(\mathbb R^d)}^{a}\| u\|_{L^q(\mathbb R^d)}^{1-a}$$ Therefore the actually formulation of Caffarelli-Kohn-Nirenberg inequality does look incredible and it is more general than the former one.
My answer to Q1) The density of $C^\infty_0-$ functions in the $L^p(|x|^\ell dx)$ can be easily proven using that $C^\infty_0-$ functions in the $L^p( dx)$. since the singularity of $|x|^\ell$ can be Annihilated by convolution with any good exponential functions such as $e^{-|x|^2}$.
However, I do not know if similar argument can be apply to the Weighted Sobolev spaces since I mentioned above that $u$ and $\nabla u$ have different scaling. I doubt that it is possible to prove the density in such $W^{1,p}-$spaces. It may be possible to have density argument when $\alpha = \beta$. Since that situation we see that we could have from the given inequality that, $W^{1,p}(|x|^\alpha) \subset L^r(|x|^\gamma) $ I am not fully sure about the density.
My answer to Q4) Theory of weighted Sobolev spaces has not been that much very well developed In my opinion theses inequalities can be useful to prove in the nearest future (may be already I am not aware yet) the Sobolev Embeddings in weighted Sobolev spaces with weight of the form $|x|^\ell$. such as I mentioned in Q1) $$W^{1,p}(|x|^\alpha) \subset L^r(|x|^\gamma) $$
Out of that, the theory of classical Sobolev embeddings are well known and renowned and highly useful in the study of regularity of solution of certain class of PDE's.
Another famous consequence and application Gagliardo-Nieremberg inequality is that the study of the sharpness of the constant $C(p,d$ leads to the so called Isoperimetric inequality.
Hence May be in the future one could invent the notion of weighted Isoperimetric inequality. which could be a consequence of Caffarelli-Kohn-Nirenberg inequalities (this could be a good project to look at if still unstudied ).
I do know similar inequality but with convolution kernel, of such weight in the fractional Sobolev spaces. Indeed we prove that $$ \|u\|_{p^*}^p\le C(d,p,s)\iint_{\Bbb R^d\Bbb R^d} \frac{|u(x)-u(y)|^p}{|x-y|^{d+2s}}dxdy\overset{s\to 1}{\to} \|\nabla u\|_{p}^p$$ with $$\color{blue}{\frac{1}{p^*}= \frac{1}{p}-\frac{s}{d}>0, 0<s<1} $$
Another convolution inequality, is the boundedness of the Riesz potential ($I_\alpha f= f*|\cdot|^\alpha$). There are some on-line notes establishing relationships between Riesz-Potential and the Gagliardo-Nieremberg inequality.
The study of the boundedness of Riesz potential in the weighted $L^p-$ space has been study. It is pretty, possible that Caffarelli-Kohn-Nirenberg inequalities were also used therein( I personally never see the proof yet)