Let K(x) = Ω(x∕|x|)|x|−k
where Ω(ξ),|ξ| = 1, is a real valued function which is in Lipα,0 < α < 1,
on the unit (k − 1)-sphere S in k-dimensional Euclidean space, Ek,k ≧ 2
with the additional property that ∫sΩ(ξ)dσ(ξ) = 0 where σ is the natural
surface measure for S. (K(x) is usually called a Calderón-Zygmund kernel in
Lipα.) Let μ be a Borel measure of finite total variation on Ek and set
μ(y) = (2π)−k∫Eke−i(y,w)dfℓ(w). Also designate the principal-valued Fourier
transform of K by K(y) and the principal-valued convolution of K with μ by μ(x).
Define IR(x) = (2π)k∫Eke−|y|∕RK(y)μ(y)ei(y,x)dy. Then if k is an even integer or
if k = 3, the following result is established: limR→∞IR(x) =μ(x) almost
everywhere.
