Densely Defined Multiplication on Several Sobolev Spaces of a Single Variable

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.