Multipliers of Sobolev spaces

AbstractLet Wm,p denote the Sobolev space of functions on Rn whose distributional derivatives of order up to m lie in Lp(Rn) for 1 ⩽ p ⩽ ∞. When 1 < p < ∞, it is known that the multipliers on Wm,p are the same as those on Lp. 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 Wm,1 and there are such distributions which are not bounded measures. These distributions are also multipliers on Lp, for 1 < p < ∞. Moreover, they form exactly the multiplier space of a certain Segal algebra. We have also proved that the multipliers on Wm,l are necessarily integrable distributions of order ⩽1 or ⩽2 accordingly as m is odd or even. We have obtained the multipliers from L1(Rn) into Wm,p, 1 ⩽ p ⩽ ∞, and the multiplier space of Wm,1 is realised as a dual space of certain continuous functions on Rn which vanish at infinity