Discrete Singular Integrals in a Half-Space

Abstract

We consider Calderon–Zygmund singular integral in the discrete half-space $h\mathbf{Z}^{m}_{+}$, where Z ^m is entire lattice (h>0) in R ^m, and prove, that the discrete singular integral operator is invertible in $L_{2}(h\mathbf{Z}^{m}_{+})$ iff such is its continual analogue. The key point for this consideration takes solvability theory of so-called periodic Riemann boundary problem, which is constructed by authors.

This work was completed when the second author was a DAAD stipendiat and hosted in Institute of Analysis and Algebra, Technical University of Braunschweig.