The Abel summability of conjugate multiple Fourier-Stieltjes integrals

On the summation of conjugate Fourier integrals of functions of several variables

“…For O 0 these assertions follow directly from (4) for j = 2, the fact that K has mean value 0 on S"" 1 and dominated convergence. Part (a) of the Corollary, for instance, implies Abel and Bochner-Riesz summability (above the critical index (n -l)/2) of "conjugate" Fourier transforms fcj of integrable functions ƒ at any point where the singular integral p. v. K * ƒ exists and which is in the Lebesgue set of ƒ (see [3], [4], [5], [6]). Part (b) could be used to prove the same assertion for #(x) = (1 -|x| 2 ) ( r 1)/2 [log(e/(l -|x| 2 ))r a where a > 1, in which case the Corollary may no longer apply.…”

Abel summability of conjugate integrals

“…The case k = 2 and 1/2 < a < 1 was treated in [4]. The cases in which 0 < a < 1 and k ..= 3 or A; is even were handled in [2]. Further references and motivation for the theorem are given in [2] and [4].…”

“…The cases in which 0 < a < 1 and k ..= 3 or A; is even were handled in [2]. Further references and motivation for the theorem are given in [2] and [4]. The proof given in the present paper covers all cases with 0 < a < 1 and k ^ 3; modifications could be made in the proof to cover the cases k = 2, 0 < a < 1 and k = 1, but this seems pointless.…”