Let $U$ be a bounded open set in $\mathbb{R}^2$, $g\in L^1_{\mathrm{loc}}(\mathbb{R}^2)$. Let moreover $w$ be a weight (i.e. a non vanishing locally integrable function) on $U$ and $p\geq2$. Does there exist a constant $C=C(p,U,g)$ such that $$\int_{U}|f\ast g|^pwd\lambda\leq C \int_{U}|f|^pwd\lambda$$ for every $f\in \mathcal{C}^\infty_c(U)$ (compactly supported smooth functions), with $d\lambda$ the Lebesgue measure on the plane?

I think that this cannot hold for a general weight $w$. What conditions on $w$ can we require in order to obtain such an estimate?

*Edit*: I'm not asking for this to hold for every $g$ locally integrable. I am interested in some particular functions. A meaningful example could be $g=1/z$ (identifying $\mathbb{R}^2$ with $\mathbb{C}$).

every$g$, you are out of luck because then the shift must be continuous in $L^p(w)$, which happens only if $w$ is essentially constant (the minimum is comparable to the maximum). – fedja Mar 16 '12 at 12:20