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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 $\mathbb{C}$).

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If you want it for 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
    
Whops, I should have said it more clearly... no, for me, $g$ is of a particular form. A usefull (for me) example could be $g=1/z$ (after identifying $\mathbb{R}^2$ with $\mathbb{C}$). –  Samuele Mar 16 '12 at 14:31
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An interesting class of weights $w>0$ for functions on $\mathbb R^n$ are the so called $v$-moderate weights, that satisfy $w(x+y) \leq C v(x) w(y)$ for all $x,y \in \mathbb R^n$, for some constant $C>0$. Here $v>0$ is another weight, often chosen as submultiplicative. Then it follows from the unweighted Young's inequality that $L_w^p * L_v^1 \subseteq L_w^p$. For more on this, have a look at univie.ac.at/nuhag-php/bibtex/open_files/grXX_weights.pdf –  Patrik Wahlberg Mar 20 '12 at 10:41

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