Essentially, the article deals with a suitable pseudifferentiable operators which are Fredholm. Accordingly, relative to an open bounded smooth subset \(\Omega\) of a paracompact set, the author provides the definition of a pseudo-differential operator satisfying the \(\mu\)-transmission property, i.e., in local coordinates the symbol has the asymptotic expansion \(\sum_{j\geq0}\mathfrak{p}_j(x,\xi)\) such that the derivatives of \(\mathfrak{p}_j(x,\xi)\) at \((x,\xi)\in\partial \Omega\times\{\pm N\}\) satisfy meaningful conditions, where \(N\) stands for the interior normal vector at \(x\in \partial \Omega\), the boundary of \(\Omega\).
The Introduction states the main results. By recalling the definition of Hörmander’s \(\mu(s)\)-space, \(H^{\mu(s)}_p(\overline{\Omega})\), an explicit subspace of \(\bigcup_{\varepsilon>0} \dot{H}^{\mathrm{Re}(\mu)+1/p-1+\varepsilon}(\overline{\Omega})\), where the elements of this union are the \(p\)-Bessel potential spaces, the author considers the factorization index \(\mu_0\) (a particular value of \(\mu\in \mathbb C\)), an elliptic operator \(P\) of order \(m\in \mathbb C\), and \(r_\Omega\) the smooth distance function restricted to \(\Omega\). He shows that \(r_\Omega P\) is a Fredholm operator from \(H^{\mu_0(s)}_p(\overline{\Omega})\) to \(\overline{H}_p^{s-\mathrm{Re}(m)}(\Omega)\), a space homeomorphic to a quotient space (Theorem 2). Then, by considering \(\mu=\mu_0-M\) such that \(M>0\), the author states that the vector operator \(\big(r_\Omega P, \gamma_{\mu,j}\big)_{0\leq j\leq M-1}\) is Fredholm from \(H_p^{\mu(s)}(\overline{\Omega})\) to the product of \(\overline{H}_p^{s-\mathrm{Re}(m)}(\Omega)\) times a finite number of Besov spaces on \(\partial{\Omega}\) (Theorem 3).
In the first section, the author recalls the properties of the Bessel potentials, Besov and anisotropic spaces, and also recalls a theorem on homomorphisms between \(\dot{H}_p^s(\overline{\Omega})\) and \(\dot{H}_p^{s-\mathrm{Re}(\mu)}(\overline{\Omega})\) spaces of Besov-potential functions with compact support in \(\overline{\Omega}\) (Theorem 1.3).
The second section focuses on giving a necessary and sufficient condition for a pseudo-differential operator to satisfy the property of \(\mu\)-transmission (Theorem 2.6). The third section carries over the known result that, if \(P\), an elliptic operator of order \(m\in \mathbb C\), checking the \(\mu_0\)-transmission condition just for the principal symbol of \(P\), then the operator \(r_\Omega P\) is Fredholm from \(\dot{H}^s_p(\overline{\Omega})\) to \(\overline{H}^{s-\mathrm{Re}(\mu)}_p(\overline{\Omega})\) (Theorem 3.1), and deduces an embedding anisotropic theorem (Theorem 3.3).
Now, we describe one by one the results of the fourth section. Proposition 4.1 deals with the relation between \(H^{\mu(s)}_p\) and \(\mathcal{E}_\mu\), the space of distributions (Definition 2.1). Theorem 4.2 states a continuous embedding theorem from \(H^{\mu(s)}_p\) into \(\overline{H}_p^{s-\mathrm{Re}(m) }\). Theorem 4.4 states that \(r_\Omega P\) is a Fredholm operator from \(H^{\mu_0(s)}_p(\overline{\Omega})\) to \(\overline{H}^{s-\mathrm{Re}(m)}_p({\Omega})\); furthermore, this operator has a an explicit parametrix and, by using a completion argument, this theorem remains true on the set of smooth functions on \(\overline{\Omega}\).
The fifth section focuses on the properties of Hörmander spaces and their boundary values; precisely, by unfolding an element of \(\mathcal{E}_\mu\) in terms of power functions comprised by the quotient of the distance function and the gamma function, the author shows the continuity of a map on the Hörmander space on \(\overline{\Omega}\) and with values the finite product of Besov-spaces over \(\partial\Omega\). Besides, this map has an explicit null space (Theorem 5.1). Then, by using Boutet de Monvel calculus, the author provides a version of the last theorem for the case of the upper half-space (Theorem 5.4).
The sixth and the seventh sections focus on inhomogeneous boundary value problems and on a few applications for the fractional Laplace-Beltrami operator, respectively.

Reviewer: Mohammed El Aïdi (Bogotá)

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