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Densely Defined Multiplication on Several Sobolev Spaces of a Single Variable

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Abstract

Following Sarason’s classification of the densely defined multiplication operators over the Hardy space, we classify the densely defined multipliers over the Sobolev space, \(W^{1,2}[a,b]\). In this paper we find that the collection of such multipliers for the Sobolev space is exactly the Sobolev space itself, and each is a densely defined multiplier is bounded. This sharpens a result of Shields concerning bounded multipliers. The densely defined multiplication operators over the spaces \(W_0(a,b) = \{ f \in W^{1,2}[a,b] : f(a)=f(b)=0 \}\) and \(W^{1,2}(\mathbb {R})\) are also classified. In the case of \(W_0(a,b)\), the densely defined multiplication operators can be written as a ratio of functions in \(W_0(a,b)\) where the denominator is non-vanishing. This is proved using a constructive argument.

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  1. University of Florida, 358 Little Hall, PO Box 118105, Gainesville, FL, 32611-8105, USA

    Joel A. Rosenfeld

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  1. Joel A. Rosenfeld
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Correspondence to Joel A. Rosenfeld.

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Communicated by Vladimir Bolotnikov.

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Rosenfeld, J.A. Densely Defined Multiplication on Several Sobolev Spaces of a Single Variable. Complex Anal. Oper. Theory 9, 1303–1309 (2015). https://doi.org/10.1007/s11785-014-0411-1

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