Applications to Singular Integrals

Abstract

In the preceding chapter, using intermediate space theory, we presented a general theory concerning the subspace X _{α, r; q} (0 < α < r 1 ≦ q ≦ ∞ and/or α = r, q = ∞; r = 1, 2, . . .) of a Banach space X, generated by a uniformly bounded semi-group }T(t); 0 ≦ t < ∞{ of class (C₀) in ℰ (X). Here we shall apply this theory to three characteristic examples, namely, to the singular integral of Abel-Poisson for periodic functions already familiar to us, to the integral of Cauchy-Poisson for functions in L^p(E₁) 1 ≦ p < ∞, and to the integral of Gauss-Weierstrass for functions defined on Euclidean n-space E _n . This chapter will, in fact, serve to show a constant interplay between functional analysis and “hard” analysis.