Multipliers of Sobolev spaces |
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Authors: | S Poornima |
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Affiliation: | Department of Applied Mathematics, Indian Institute of Science, Bangalore 560012, India |
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Abstract: | Let Wm,p denote the Sobolev space of functions on n whose distributional derivatives of order up to m lie in Lp(n) 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(n) into Wm,p, 1 ? p ? ∞, and the multiplier space of Wm,1 is realised as a dual space of certain continuous functions on n which vanish at infinity. |
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