Densely Defined Multiplication on the Sobolev Space

TL;DR

This paper classifies densely defined multiplication operators on Sobolev spaces, showing they are exactly the Sobolev space itself, and extends the classification to subspaces with boundary conditions, using constructive proofs.

Contribution

It provides a complete classification of densely defined multipliers on Sobolev spaces, extending Sarason's Hardy space results and sharpening Shields' bounded multiplier findings.

Findings

Densely defined multipliers on W^{1,2}[0,1] are exactly the Sobolev space itself.

For the subspace with zero boundary conditions, multipliers are ratios of functions in the subspace with non-vanishing denominator.

The classification is achieved through constructive proofs, providing explicit representations.

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.