Singular Integral Operators and Nilpotent Groups

Abstract

The Hilbert transform and in fact the whole development of singular integrals begins with the theory of functions of one complex variable. The starting point is, in effect, the Cauchy integral:

$$ {\text{F}}\left( {\text{z}} \right) = \frac{1} {{2\pi {\text{i}}}}\int_{ - \infty }^\infty {\frac{{{\text{f}}\left( {\text{t}} \right)}} {{{\text{t - z}}}}{\text{dt}}} ,\,\,{\text{z = x + iy,}}\,\,{\text{y}} > 0. $$

((1.1))

Suppose for simplicity that we are dealing with f ∈ L² (R̰¹ ). What is the meaning of the mapping f → F? The elements F of the form (1.1) all belong to the Hardy space H². The Hardy space is the Hilbert space of functions F holomorphic to the upper half-plane y >o, vanishing at ∞ (i.e., as y →∞), and so that the boundary values $\mathop {\lim }\limits_{{\text{y}} \to 0} \,\,\,{\text{F}}\left( {{\text{x + iy}}} \right) = {\text{F}}_{\text{b}} \left( {\text{x}} \right)$ exists in the L² (R̰¹ ) norm. The norm of an element P in H² is the L² (R̰¹ ) norm of F_b. Thus H² maybe identified with a (closed) subspace of L². As a result of these considerations it may be shown that the mapping f → F_b, arising from ( 1.1), is the orthogonal projection of L² to H². For further reference we shall use the notation: f →F_b < C_b(f).