Fractional Laplacians on domains, a development of Hörmander's theory of mu-transmission pseudodifferential operators

Abstract

Let $P$ be a classical pseudodifferential operator of complex order $m$ on an $n$-dimensional smooth manifold ${\Omega }_1$. For the truncation $P_{\Omega }$ to a smooth subset $\Omega$ there is a well-known theory of boundary value problems when $P_{\Omega }$ has the transmission property (preserves $C^{\infty }(\bar{\Omega })$) and is of integer order; the calculus of Boutet de Monvel. Many interesting operators, such as for example complex powers of the Laplacian $(-\Delta {)}^{\mu }$ with noninteger mu, are not covered. They have instead the mu-transmission property defined in Hörmander's books, mapping $x_n^{\mu }{C}^{\infty }(\bar{\Omega })$ into $C^{\infty }(\bar{\Omega })$. In an unpublished lecture note from 1965, Hörmander described an $L_2$-solvability theory for mu-transmission operators, departing from Vishik and Eskin's results. We here develop the theory in $L_p$ Sobolev spaces ($1<p<\infty$) in a modern setting. It leads to not only Fredholm solvability statements but also regularity results in full scales of Sobolev spaces (for $s\to \infty$). The solution spaces have a singularity at the boundary that we describe in detail. We moreover obtain results in Hölder spaces, which radically improve recent regularity results for fractional Laplacians.

Accepted for publication in Advances in Mathematics. 44 pages

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