  Let $G$ a locally compact abelian group with Haar measure $\mu$ and let
$1<p<\infty. $ In the present paper we determine necessary and sufficient
conditions on $G$ for the grand Lebesgue space $ L^{p),\theta}(G)$ to be a
Banach algebra under convolution.Later we characterize the multipliers of the
grand Lebesgues, $L^{p,)\theta}(G)$ and the small Lebesgue spaces
$L^{(P'\theta}$, where $\frac{1}{p}+\frac{1}{p'}=1$

Functional Analysis