Multipliers of Sobolev spaces

Poornima, S (1982) Multipliers of Sobolev spaces. In: Journal of Functional Analysis, 45 (1). pp. 1-28.

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Abstract

Let Wm,p denote the Sobolev space of functions on Image n whose distributional derivatives of order up to m lie in Lp(Image n) for 1 less-than-or-equals, slant p less-than-or-equals, slant ∞. 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 less-than-or-equals, slant1 whose first order derivatives are also integrable of order less-than-or-equals, slant1, 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 less-than-or-equals, slant1 or less-than-or-equals, slant2 accordingly as m is odd or even. We have obtained the multipliers from L1(Image n) into Wm,p, 1 less-than-or-equals, slant p less-than-or-equals, slant ∞, and the multiplier space of Wm,1 is realised as a dual space of certain continuous functions on Image n which vanish at infinity.

Item Type:	Journal Article
Publication:	Journal of Functional Analysis
Publisher:	Elsevier Science
Additional Information:	Copyright of this article belongs to Elsevier Publisher.
Department/Centre:	Division of Physical & Mathematical Sciences > Mathematics
Date Deposited:	15 Dec 2009 06:23
Last Modified:	19 Sep 2010 05:46
URI:	http://eprints.iisc.ac.in/id/eprint/23509

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