In Moser's famous paper [harnack inequality for parabolic equations](http://onlinelibrary.wiley.com/doi/10.1002/cpa.3160170106/pdf), he used the following simple Poincare inequality(Lemma 3)


$\int (f(x)-k)^2 w(x) dx \leq c(w) \int |\nabla f|^2 w(x) dx$


where $k=\int f(x)w(x)dx / \int w(x)dx$. The weight function $w(x)$ is just a smooth axially decreasing cut-off function of compact support. 


Is this still true when LHS is replaced by a higher norm, i.e.


$\int (f(x)-k)^p w(x) dx \leq c(p,w) (\int |\nabla f|^2 w(x) dx)^\frac{p}{2} $ ?


(Of course $p$ should at least satisfy $p < 2^\*=\frac{2n}{n-2}$.)


References for weighted sobolev spaces often deal with complicated weight functions(for example satisfying the Muckenhoupt $A\_p$-condition). I can't find a clear answer to my question concerning just smooth cut-off weights while I need this inequality when trying to apply Moser iteration for drift-diffusion equations. 



Is this simple-looking weighted poincare-sobolev inequality correct
This inequality is not true. (For $2<p<2^\*$, that is. This is how I understood the question.) To see why, imagine that the function $f$ has a huge peak in the area where the weight $w$ is very small. 


To simplify the matters, assume that $w$ looks similar to (smoothed) "stairs" with infinite number of steps of width $\delta\_i\,(1\le i<\infty)$. Now, let $f$ be a usual (spherically symmetric) cap with support entirely on the $i$-th "step".
One can see that for the above inequality to hold in this case we must have the inequality 
$$\delta^{n(1-\frac{p}{2^\*})}\le Cw^{\frac{p}{2}-1},$$
where $w=w\_i$ is the "height" of this "step". Now, for $2<p<2^\*$ it is not a problem to design "stairs" where this inequality does not hold. (I presume that the inequality isn't true for any reasonable weight $w$, but to prove this in general may be a little more difficult.)


