In this paper, a class of
singular integral transforms of the Calderón-Zygmund type is constructed for the
spaces Lr(Ψp,λ);Ψp is the p-adic or p-series field, λ is additive Haar measure, 7⋅> 1.
The transforms have the form
where m is the modular function for the field and
The fundamental result is the existence of the Lr-limit and the NI. Riesz inequality
∥Lf∥r≤ Ar∥f∥r. 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.
![∫
Lf(y) = lim [m (x)]−1w(x)f(y− x)dλ(x),
k→ ∞ {m(x)≦p−k}](a120x.png)
