Is this simple-looking weighted poincare-sobolev inequality correct

Question

In Moser's famous paper harnack inequality for parabolic equations, 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.

Answer 1

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.)