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arXiv · 1306.2662

Densely Defined Multiplication on the Sobolev Space

Joel A. Rosenfeld
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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, $W^{1,2}[0,1]$. In this paper we find that the collection of such multipliers for the Sobolev space is exactly the Sobolev space itself. This sharpens a result of Shields concerning bounded multipliers. The densely defined multiplication operators over the subspace $W_0 = \{f \in W^{1,2}[0,1] : f(0)=f(1)=0 \}$ are also classified. In this case the densely defined multiplication operators can be written as a ratio of functions in $W_0$ where the denominator is non-vanishing. This is proved using a contructive argument.

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Joel A. Rosenfeld. 2014-04-03. Densely Defined Multiplication on the Sobolev Space. https://arxiv.org/abs/1306.2662

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