The existence of the well-known Jacquet–Langlands correspondence was established by Jacquet and Langlands via the trace formula method in 1970. An explicit construction of such a correspondence was obtained by Shimizu via theta series in 1972. In this paper, we extend the automorphic descent method of Ginzburg–Rallis–Soudry to a new setting. As a consequence, we recover the classical Jacquet–Langlands correspondence for PGL(2) via a new explicit construction.

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This paper is devoted to a functional analytic approach to the subelliptic oblique derivative problem for the usual Laplacian with a complex parameter λ. We solve the long-standing open problem of the asymptotic eigenvalue distribution for the homogeneous oblique derivative problem when |λ| tends to ∞. We prove the spectral properties of the closed realization of the Laplacian similar to the elliptic (non-degenerate) case. In the proof we make use of Boutet de Monvel calculus in order to study the resolvents and their adjoints in the framework of L2 Sobolev spaces.

We extend a recent result of Avelin, Hed, and Persson about approximation of functions f that are plurisubharmonic on a domain Ω and continuous on ˙Ω, with functions that are plurisubharmonic on (shrinking) neighborhoods of ˙Ω. We show that such approximation is possible if the boundary of Ω is C0 outside a countable exceptional set E⊂∂Ω. In particular, approximation is possible on the Hartogs triangle. For Hölder continuous u, approximation is possible under less restrictive conditions on E. We next give examples of domains where this kind of approximation is not possible, even when approximation in the Hölder continuous case is possible.

Let F :RN→RN be a locally Lipschitz continuous function. We prove that F is a global homeomorphism or only injective, under suitable assumptions on the subdifferential ∂F(x). We use variational methods, nonsmooth inverse function theorem and extensions of the Hadamard-Levy Theorem. We also address questions on the Markus-Yamabe conjecture.