Let $ G $ a locally compact groups and let $ S(G) $ be a segah algebra in $ L ^{1}(G) $.For a compact group $ G $ , we determine the space $ Z ^{ 1 } (S(G) , X) $ of continuous derivations from $ S(G) $ into $ X $ , where $ S(G) $ is one of the naturally arising Segal algebras $ (C(G) , \\ast ) $ , $ (L ^{p} (G), \\ast ), $ $ 1 \\leq p \\leq 1 $ , where $ \\ast $ stands for the convolution product and $ X $ is a naturally arising $ S(G)- $ module. Along the way we determine all the left (right) multipliers from $ S(G) $ into $ X $ we also necessary conditions for $ S(G) $ to be Arens regular, these conditions are sometimes suffi cient as well. We also fi nd necessary and suffi cient conditions for $ S(G) $ to be an ideal in its second dual space, when the latter is equipped with the fi rst or the second Arens product. \\\\ Finalluy, when $ G $ is a abelian group, $ 1 p 1 $ , and $ A $ be a commutative Banach algebra. W e study the space of multipliers on $ L ^{p}(G, A) $ and characterize it as the space of multipliers of certain Banach algebra. We also study the multipliers space on $L ^{p}(G, A) \\cap L ^{p}(G, A) $ .