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.