Multipliers of Sobolev spaces

Let W^m,p denote the Sobolev space of functions on $\text{R}$ⁿ whose distributional derivatives of order up to m lie in L^p($\text{R}$ⁿ) for 1 ? p ? ∞. When 1 < p < ∞, it is known that the multipliers on W^m,p are the same as those on L^p. This result is true for p = 1 only if n = 1. For, we prove that the integrable distributions of order ?1 whose first order derivatives are also integrable of order ?1, belong to the class of multipliers on W^m,1 and there are such distributions which are not bounded measures. These distributions are also multipliers on L^p, for 1 < p < ∞. Moreover, they form exactly the multiplier space of a certain Segal algebra. We have also proved that the multipliers on W^m,l are necessarily integrable distributions of order ?1 or ?2 accordingly as m is odd or even. We have obtained the multipliers from L¹($\text{R}$ⁿ) into W^m,p, 1 ? p ? ∞, and the multiplier space of W^m,1 is realised as a dual space of certain continuous functions on $\text{R}$ⁿ which vanish at infinity.