A sharp existence and uniqueness theorem for linear Fuchsian partial differential equations (Q1598361)

This paper is devoted to linear Fuchsian partial differential equations. Here the author considers the equation \(Pu=f\), where \(P\) is the linear Fuchsian partial differential operator \[ P=(tD_t)^m+\sum^{m-1}_{j= 0} \sum_{|\alpha |\leq m-j}a_{j,\alpha} (t,z)\bigl(\mu(t) D_z \bigr)^\alpha (tD_t)^j, \] where \(\mu(t)\) is a positive-valued function on the interval \((0,T)\). The aim of the author is to present a sharp form of unique solvability in the following sense: one can find a domain \(\Omega\) such that if \(f\) is defined on \(\Omega\), then the author proves a unique solution \(u\) also defined on \(\Omega\).