Vol. 23, No. 2, 1967

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Hilbert transforms for the p-adic and p-series fields

Keith Lowell Phillips

Vol. 23 (1967), No. 2, 329–347

In this paper, a class of singular integral transforms of the Calderón-Zygmund type is constructed for the spaces Lrp);Ψp is the p-adic or p-series field, λ is additive Haar measure, 7 > 1. The transforms have the form

Lf(y) = lim            [m (x)]−1w(x)f(y− x)dλ(x),
k→ ∞  {m(x)≦p−k}

where m is the modular function for the field and

w (x)dλ (x) = 0.

The fundamental result is the existence of the Lr-limit and the NI. Riesz inequality Lfr Arfr. Several examples of functions w defining transforms L are given. In particular, subsets Φ of Ψp such that Φ ∩−Φ = and Φ ∪−Φ = Ψp∖{0} together with functions w satisfying w(x) = w(x) yield transforms which are analogues of the classical Hilbert transform. Multipliers for L are also discussed. A preliminary theorem of independent interest states that the L2-Fourier transform on certain 0-dimensional locally compact Abelian groups converges pointwise.

Mathematical Subject Classification
Primary: 47.70
Secondary: 42.00
Received: 23 June 1965
Revised: 8 August 1966
Published: 1 November 1967
Keith Lowell Phillips