How interpret $\||x|^\gamma u\|_{L^r}\leq C\|\nabla u\|_{L^p}^{a}\||x|^\beta u\|_{L^q}^{1-a}$ for $u\in\mathcal C_0^1(\mathbb R^d)$? 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 ?
 A: This inequalities are the famous Caffarelli-Kohn-Nirenberg  (CKN for short) inequalities from
L. Caffarelli, R. Kohn, L. Nirenberg, First order interpolation inequalities with weights. Compositio Math. 53 (1984), no. 3, 259–275
and you will find many papers in the literature about them, including this by myself (joint work by R. Durán and I. Drelichman) about the case of radial functions, which you may also find of interest.
https://arxiv.org/abs/1009.0484
Q1) No, there is no density argument behind. The condition on u means that is a function with continuous first order derivatives (in the classical sense)  and with compact support. So you don't need to interpret the derivatives in the inequality as weak derivatives! 
Once an inequality of this type is established, it can be usually extended by density to functions in the appropriate weighted Sobolev space (defined as the closure of the C1 functions with compact support under the weighted Sobolev norm).
Q2) When there are no weights, the CKN inequalities reduces to the 
(first order) Gagliardo-Nirenberg inequalities. The Sobolev embedding is a particular case of that.
https://en.wikipedia.org/wiki/Gagliardo%E2%80%93Nirenberg_interpolation_inequality
Q3) The notation $|x|$ stands for the Euclidean distance to the origin (i.e.: Euclidean norm) $|x|^\alpha$ is just a power of the distance to the origin.
Q4) The most famous application of the CKN inequalities is a theorem on partial regularity of solutions to the weak solutions to the Navier Stokes equations, by the same authors
http://onlinelibrary.wiley.com/doi/10.1002/cpa.3160350604/abstract
You can find more information in the Phd Thesis by Renato Luca
https://arxiv.org/abs/1308.4361
They are also a basic tool in the study of some ellitic or parabolic equations with singular or degenerate coefficients. 
A: 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. 

For given $p\ge 1$, If $\color{blue}{\frac{1}{p^*}= \frac{1}{p}-\frac{1}{d}>0}$ then there is a constant $C=C(d,p)$ such that  for all $u\in W^{1,p}(\Bbb R^d)$ with compact support , 
  $$\|u\|_{p^*} \le C\|\nabla u\|_{p}\tag{1}$$

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)
