Vol. 36, No. 1, 1971

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ISSN: 0030-8730
The Abel summability of conjugate multiple Fourier-Stieltjes integrals

S. P. Philipp, Victor Lenard Shapiro and William Hall Sills

Vol. 36 (1971), No. 1, 231–238
Abstract

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)(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 Ekei(y,w)df(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.

Mathematical Subject Classification
Primary: 42.20
Secondary: 47.00
Milestones
Received: 16 April 1969
Published: 1 January 1971
Authors
S. P. Philipp
Victor Lenard Shapiro
William Hall Sills