Cauchy integrals, Calderón projectors, and Toeplitz operators on uniformly rectifiable domains
The paper deals with first-order elliptic systems of differential operators \(D\) on bounded domains \(\Omega\) in a Riemannian manifold \(M\). The boundaries \(\partial\Omega\) are permitted to be rough. Theorem 2.2.1 gives a reproducing formula of Cauchy-Pompeiu type: \[ u(x) = i\int_{\partial\Omega} E(x,y)\sigma_{\mathcal{D}}(y,\nu(y))u(y)\,d\sigma(y) + \int_{\Omega} E(x,y)\mathcal{D}u(y)\,dV(y), \] where \(\Omega\) is Ahlfors regular, \(\mathcal{D}u\) is integrable, and \(u\) has nontangential boundary limit and \(L^p\)-bounded maximal function for some \(p>1\). Here \(\mathcal{D}\) is a \(2\times 2\)-system with Fredholm index zero. The off-diagonal elements are \(D^*\) and \(D\), and the diagonal elements are suitable multiplication operators. It is assumed that the unique continuation property (UCP) holds for \(D\) and \(D^*\). Then \(\mathcal{D}\) is invertible, and \(E\) denotes the integral kernel of the inverse. Moreover \(\sigma_{\mathcal{D}}\) denotes the principal symbol of \(\mathcal{D}\), and \(\nu\) is the outward pointing unit normal. From the reproducing formula the Calderón projector \(\mathcal{P}_{\mathcal{D}}\) is obtained, which is a projection operator onto the Hardy space of boundary traces of solutions of \(\mathcal{D}u=0\). The Calderón projector is shown to be \(L^p(\partial\Omega)\)-bounded for \(1<p<\infty\). If \(\Omega\) is a uniformly rectifiable (UR) domain, and if some additional assumptions hold, then the orthogonal projector onto the Hardy space, the Szegő projector, is an \(L^p(\partial\Omega)\)-bounded operator \(S_{\mathcal{D}}=\mathcal{P}_{\mathcal{D}}(I+A)^{-1}\). A Toeplitz operator associated with \(D\) is the \(L^p\)-bounded operator \[ \mathfrak{T}_{\mathcal{D}}:=\mathcal{P}_{\mathcal{D}}\Phi\mathcal{P}_{\mathcal{D}}+(I-\mathcal{P}_{\mathcal{D}}), \] where the coefficient \(\Phi\) is multiplication by a map, also denoted \(\Phi\), from \(\partial\Omega\) into the endomorphisms of \(\mathbb{C}^\ell\). If \(\Phi\) is an \(L^\infty\cap\mathrm{vmo}\,\)-section of \(GL(\ell,\mathbb{C})\), and \(\Omega\) is UR, then \(\mathfrak{T}_{\mathcal{D}}\) is Fredholm on \(L^p\) with index independent of \(p\in(1,\infty)\). Variants of this result are given when the Calderón projector is replaced by the Szegő projector, and when the trivial bundle with fiber \(\mathbb{C}^\ell\) is replaced by a more general bundle (twisted Toeplitz operator). Furthermore, a cobordism formula is given which states that the index over a rough boundary equals the index over a cobordant boundary, which could in applications be chosen smooth. Some index computations are given. An appendix contains some auxiliary results. The paper builds on [\textit{S. Hofmann} et al., Int. Math. Res. Not. 2010, No. 14, 2567--2865 (2010; Zbl 1221.31010)], notably on variable coefficient versions of the results of \textit{G. David} [Ann. Sci. École Norm. Sup. (4) 21, 225--258 (1988; Zbl 0655.42013)].
scientific article; zbMATH DE number 6370015
scientific article