  In the paper, we introduce the concept of weight with reasonable growth on a
locally compact group $G$. We verify that these weights form a natural class to
work with, by examining the most common examples. We proceed with the
discussion of the $L^p-$conjecture. 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 $p'$. At last, we confirm the $L^p_{\omega}-$conjecture
for weights of $(p,q)-$reasonable growth.
