Dual Compatibility for Measure-Driven Function Space Scales

Abstract

We address a basic but recurring functional-analytic issue: given a continuous linear operator acting between nested scales of spaces, its dual and the dual of its restriction typically act on non-comparable abstract dual spaces, so that a naive ``restriction'' statement is ill-posed.
While one can, if appropriate conditions are met, resolve this difficulty on a case-by-case basis by introducing concrete identifications, we propose a global and systematic approach based on embedding dual spaces into common distribution spaces via anchor spaces.
This method is classical for Lebesgue measure on $\mathbb{R}^n$, where the Schwartz space anchors the embedding of many function spaces into the tempered distributions.
Here we extend the same philosophy to a generic Borel measure $\mu$ on $\mathbb{R}^n$: with $\mathcal{S}|_\Gamma$ denoting the restricted Schwartz space on $\Gamma=\operatorname{supp}(\mu)$, we obtain canonical realizations of dual spaces in $(\mathcal{S}|_\Gamma)'$ under an appropriate density condition and a mild local finiteness assumption on $\mu$.
Our abstract compatibility theorem then applies, in particular, to operators mapping from the classical scale on $\mathbb{R}^n$ (anchored by $\mathcal{S}(\mathbb{R}^n)$) into such a measure-driven scale on $\Gamma$ (anchored by $\mathcal{S}|_\Gamma$), showing that realized duals agree on their common domain.
As a concrete application, we consider the trace operator $\operatorname{tr}^s_\Gamma$ for Bessel potential spaces $H^s_p(\mathbb{R}^n)$ on $d$-sets.