Definition of the Hilbert transform in En

In this chapter the elementary properties of the n-dimensional Hilbert transform are discussed. Basic aspects of the Calderón–Zygmund theory of singular integral operators in the n-dimensional Euclidean space, En, are also considered.

Applications of Hilbert transforms in En for n ≥ 2 are significantly less numerous than the one-dimensional case; however, they do arise in important areas. These include problems in nonlinear optics that focus on deriving dispersion relations and sum rules for the nonlinear susceptibility. The publications of Smet and Smet (1974), Nieto-Vesperinas (1980), Peiponen, (1987b), 1988), Bassani and Scandolo (1991, 1992), and Peiponen, Vartiainen, and Asakura (1999), will give the reader a sense of the advances in this field. There are applications in signal processing (see Bose and Prabhu (1979), Zhu, Peyrin, and Goutte (1990), and Reddy et al. (1991a, 1991b)), and in spectroscopy (see Peiponen, Vartiainen, and Tsuboi (1990)). In scattering theory, the double dispersion relations, frequently referred to as the Mandelstam representation, express the scattering amplitude as a double iterated dispersion relation in the energy and momentum transfer variables; see Roman (1965) and Nussenzveig (1972). Some of these applications are touched upon in later chapters.

To proceed, some preliminary definitions are required. Let x denote the n-tuple {x1, x2, x3, …, xn}, and let s denote the n-tuple {s1, s2, s3, …, sn}. It is quite common in the literature to represent multi-dimensional integration factors by ds (or some similar variable), where the context is meant to signify an n-dimensional integration factor ds1ds2ds3 … dsn.