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With the use of Riesz-Thorin-Stein-Weiss interpolation, we establish that \\mbox{$L^p_{\\omega}(G)\\star L^p_{\\omega}(G)\\subset L^p_{\\omega}(G)$}, $p>1$ implies that $L^p_{\\omega}(G)\\star L^q_{\\omega}(G)\\subset L^q_{\\omega}(G)$ for $q$ which lies between $p$ and $p'$. 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