# A generalization of the microlocal regularity theorem

#### 20.10.2022, 16:15  –  Raum 1.10 Forschungsseminar Differentialgeometrie

Rubens Longhi (UP)

The theorem of microlocal elliptic regularity is useful to determine the singularities of the solution of linear partial differential equations on manifolds. It states that if $$P$$ is a linear partial differential operator of order $$m$$, then $$\mathrm{WF}(u)\subset \mathrm{WF}(Pu)\cup \mathrm{char}(P)$$, where $$\mathrm{char}(P)$$ is the characteristic variety of $$P$$ and $$u$$ is a distribution. We propose an extension $$\mathrm{WF}_{\mathcal{F}}$$ of such classical notion of wavefront set in order to capture different degrees of lack of $$\mathcal{F}$$-regularity of a distribution around a certain codirection $$\xi$$, e.g. for $$\mathcal{F}=$$ Sobolev or Hölder spaces. This allows us to investigate under what assumptions on $$\mathcal{F}$$, $$\mathcal{G}$$ the statement of the microlocal elliptic regularity theorem can be generalized to $$\mathrm{WF}_{\mathcal{F}}(u)\subset \mathrm{WF}_{\mathcal{G}}(Pu)\cup \mathrm{char}(P)$$.

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