[Paper Review] Functional properties of Generalized Hörmander spaces of distributions I : Duality theory, completions and bornologifications

Yoann Dabrowski|arXiv (Cornell University)|Nov 11, 2014

Mathematical and Theoretical Analysis26 references3 citations

TL;DR

This paper introduces generalized Hörmander spaces of distributions, denoted $\mathcal{D}'_{\gamma,\Lambda}$ and $\mathcal{E}'_{\gamma,\Lambda}$, by controlling both the distributional wave front set ($DWF$) and the standard wave front set ($WF$) within nested cones $\gamma \subset \Lambda \subset \overline{\gamma}$. It establishes duality, completion, and bornologification properties for these spaces, showing they are nuclear and ultrabornological, with key results on Mackey topologies and pullback maps under smooth maps, crucial for quantum field theory in curved spacetime.

ABSTRACT

The space $D'_Λ$ of distributions having their $C^\infty$ wavefront set in a cone $Λ$ has become important in physics because of its role in the formulation of quantum field theory in curved spacetime. It is also a basic object in microlocal analysis, but not well studied from a functional analytic viewpoint. In order to compute its completion in the open cone case, we introduce generalized spaces $D'_{γ,Λ}$ where we also control the union of H^s-wave front sets in a second cone $γ$ contained in $Λ$. We can compute bornological and topological duals, completions and bornologifications of natural topologies for spaces in this class. All our topologies are nuclear, ultrabornological when bornological and we can describe when they are quasi-LB. We also give concrete microlocal representations of bounded and equicontinuous sets in those spaces and work with general support conditions including future compact or space compact support conditions on globally hyperbolic manifolds, as motivated by physics applications to be developed in a second paper.

Motivation & Objective

• To address the lack of completeness and functional analytic structure in $\mathcal{E}'_{\Lambda}$, a key space in quantum field theory on curved spacetimes.
• To define generalized distribution spaces $\mathcal{D}'_{\gamma,\Lambda}$ and $\mathcal{E}'_{\gamma,\Lambda}$ by controlling both $DWF(u) \subset \gamma$ and $WF(u) \subset \Lambda$ for nested cones $\gamma \subset \Lambda \subset \overline{\gamma}$.
• To compute the bornological and topological duals, completions, and bornologifications of natural topologies on these spaces.
• To provide a concrete functional analytic characterization of the completion of $\mathcal{E}'_{\Lambda}$ as $\widehat{\mathcal{E}'_{\Lambda}} = \{ u \in \mathcal{E}' : DWF(u) \subset \Lambda \}$, resolving a gap in the physics literature.
• To establish continuity and wave front set control under pullbacks via smooth maps, essential for field theory applications on globally hyperbolic manifolds.

Proposed method

• Introduces the dual wave front set $DWF(u) = \bigcup_{s>0} WF_s(u)$ as a refinement of the standard wave front set, enabling control of regularity beyond $L^2$-based definitions.
• Defines generalized spaces $\mathcal{E}'_{\gamma,\Lambda}(U) = \{ u \in \mathcal{E}'(U) : DWF(u) \subset \gamma, \, WF(u) \subset \Lambda \}$ and $\mathcal{D}'_{\gamma,\Lambda}(U)$ with support conditions on globally hyperbolic manifolds.
• Uses Mackey topology $\mathcal{I}$ and inductive limit topologies $\mathcal{I}_{iii}$, $\mathcal{I}_{ppp}$, $\mathcal{I}_{ibi}$ to define natural topologies on the spaces, ensuring nuclearity and completeness.
• Applies bornologification and completion functors to derive the strong dual and bornological dual, showing that $\widehat{\mathcal{E}'_{\Lambda}}$ is the bornological dual of $\mathcal{D}'_{\Gamma}$ with $\Gamma = -\Lambda^c$.
• Establishes continuity of pullbacks $f^*$ via the pushforward wave front set estimate $DWF(f^*u) \subset df^* DWF(u)$, using results from Duistermaat and [BDH].
• Uses polarized support conditions and enlargeable families $\mathcal{C}$ to generalize support control, ensuring compatibility with pullbacks and topological continuity.

Experimental results

Research questions

• RQ1 What is the functional analytic structure of $\mathcal{E}'_{\Lambda}$, particularly its completion, given that it is not sequentially complete?
• RQ2 How can one naturally characterize the completion of $\mathcal{E}'_{\Lambda}$ using wave front set conditions?
• RQ3 What are the duality, bornological, and topological properties of generalized Hörmander spaces $\mathcal{D}'_{\gamma,\Lambda}$ and $\mathcal{E}'_{\gamma,\Lambda}$ with $\gamma \subset \Lambda \subset \overline{\gamma}$?
• RQ4 How do pullbacks of distributions behave under smooth maps in terms of wave front set propagation, especially for generalized spaces?
• RQ5 Under what conditions is the Mackey topology $\mathcal{I}$ on $\mathcal{D}'_{\gamma,\Lambda}$ nuclear or quasi-LB?

Key findings

• The completion of $\mathcal{E}'_{\Lambda}$ is precisely $\{ u \in \mathcal{E}' : DWF(u) \subset \Lambda \}$, providing a concrete microlocal characterization of the completion.
• The Mackey topology $\mathcal{I}$ on $\mathcal{D}'_{\gamma,\gamma}$ is nuclear and coincides with $\mathcal{I}_{iii}$, ensuring strong topological and duality properties.
• The space $\mathcal{D}'_{\gamma,\Lambda}$ is complete and quasi-LB if $\Lambda$ is closed and $\gamma$ is open, with the topology independent of chart choices.
• The pullback map $f^*$ is continuous from $\mathcal{D}'_{\gamma,\Lambda}(U_2, \mathcal{C}; E)$ to $\mathcal{D}'_{df^*\gamma, \overline{df^*\gamma}}(U_1, f_e^{-1}(\mathcal{C}); f^*E)$, with $DWF(f^*u) \subset df^* DWF(u)$.
• The bornologification of $\mathcal{D}'_{\Gamma}$ yields the strong dual of $\mathcal{E}'_{\Lambda}$, and the completion of $\mathcal{E}'_{\Lambda}$ is the bornological dual of $\mathcal{D}'_{\Gamma}$ with $\Gamma = -\Lambda^c$.
• The topology on $\mathcal{D}'_{\gamma,\Lambda}$ is independent of the choice of charts, as shown via sequential continuity and Mackey continuity arguments from [BD] and [Du].