A Hilbert space approach to singularities of functions

Abstract

We introduce the notion of a pseudomultiplier of a Hilbert space $\mathcal{H}$ of functions on a set $\Omega$. Roughly, a pseudomultiplier of $\mathcal{H}$ is a function which multiplies a finite-codimensional subspace of $\mathcal{H}$ into $\mathcal{H}$, where we allow the possibility that a pseudomultiplier is not defined on all of $\Omega$. A pseudomultiplier of $\mathcal{H}$ has singularities, which comprise a subspace of $\mathcal{H}$, and generalize the concept of singularities of an analytic function, even though the elements of $\mathcal{H}$ need not enjoy any sort of analyticity. We analyse the natures of these singularities, and obtain a broad classification of them in function-theoretic terms.

52 pages. This version includes minor revisions. The paper has been accepted for publication by the Journal of Functional Analysis