Distributional zero divisors with full support

Abstract

We show that every smooth manifold $M$ of positive dimension admits fully supported distributions $u,v\in {\mathcal{D}}^{\prime }(M)$ with $\mathrm{WF}(u)\cap (-\mathrm{WF}(v))=\varnothing$ and $uv=0$. Neither factor can be continuous on any non-empty open set, but $u$ may be chosen of optimal Sobolev order $\dim M/2$, with wavefront set contained in any prescribed closed conical set $\Gamma \subset T^{\ast }M\setminus 0$ with non-empty and non-maximal fibers. Simultaneously, the wavefront set of $v$ may be confined to any compatible open conical set or, in geometric situations, to closed conical sets such as conormal bundles of foliations or ray bundles generated by locally conformally closed one-forms. We give positivity criteria and show that full-support zero divisor pairs accumulate at $(1,0)$ in Sobolev-microlocal topologies. The one-dimensional case uses a Fourier null series of Kozma-Olevski\uı.

49 pages