Higher-Codimensional Boundary Value Problems and $F$-Mild Microfunctions —Local and Microlocal Uniqueness—

Abstract

For the study of local and microlocal boundary value problems with a boundary of codimension greater than one, sheaves of $F$-mild hyperfunctions and $F$-mild microfunctions are introduced. They are refinements of the notions of hyperfunctions and microfunctions with real analytic parameters and have natural boundary values. $F$-mild solutions of a general $\mathcal{D}$-Module $\mathcal{M}$ (that is, a system of linear partial differential equations with analytic coefficients) are considered. In particular, local and microlocal uniqueness in the boundary value problem (the Holmgren type theorem) is proved if the boundary is non-characteristic for $\mathcal{M}$. or else if $\mathcal{M}$ is Fuchsian along the boundary.