Summary: For \(2a\)-order strongly elliptic operators \(P\) generalizing \(( - \Delta )^a\), \(0 < a < 1\), the homogeneous Dirichlet problem on a bounded open set \(\Omega \subset \mathbb{R}^n\) has been widely studied. Pseudodifferential methods have been applied by the present author when \(\Omega\) is smooth; this is extended in a recent joint work with Helmut Abels showing exact regularity theorems in the scale of \(L_q\)-Sobolev spaces \(H_q^s\) for \(1 < q < \infty \), when \(\Omega\) is \(C^{\tau + 1}\) with a finite \(\tau > 2 a\). We now develop this into existence-and-uniqueness theorems (or Fredholm theorems), by a study of the \(L_p\)-Dirichlet realizations of \(P\) and \(P^\ast \), showing that there are finite-dimensional kernels and cokernels lying in \(d^a C^\alpha( \overline{\Omega})\) with suitable \(\alpha > 0\), \(d(x) = \operatorname{dist}(x, \partial \Omega)\). Similar results are established for \(P - \lambda I\), \(\lambda \in \mathbb{C} \). The solution spaces equal \(a\)-transmission spaces \(H_q^{a ( t )}( \overline{\Omega})\).
Moreover, the results are extended to nonhomogeneous Dirichlet problems prescribing the local Dirichlet trace \((u / d^{a - 1}) |_{\partial \Omega} \). They are solvable in the larger spaces \(H_q^{( a - 1 ) ( t )}( \overline{\Omega})\). Furthermore, the nonhomogeneous problem with a spectral parameter \(\lambda \in \mathbb{C}\), \[ P u - \lambda u = f \text{ in } \Omega, \quad u = 0 \text{ in } \mathbb{R}^n \setminus \Omega, \quad( u / d^{a - 1} ) |_{\partial \Omega} = \varphi \text{ on } \partial \Omega, \] is for \(q < ( 1 - a )^{- 1}\) shown to be uniquely resp. Fredholm solvable when \(\lambda\) is in the resolvent set resp. the spectrum of the \(L_2\)-Dirichlet realization. The results open up for applications of functional analysis methods. Here we establish solvability results for evolution problems with a time-parameter \(t\), both in the case of the homogeneous Dirichlet condition, and the case where a nonhomogeneous Dirichlet trace \((u(x, t) / d^{a - 1}(x)) |_{x \in \partial \Omega}\) is prescribed.