The Multipliers for the Pair (H _p (G), H _q (G)),1 ≦ p, q ≦ ∞

Abstract

In this final chapter we shall discuss a few of the more elementary results concerning the multipliers for the Hardy spaces H _p (G). Throughout we shall assume that G is a compact connected Abelian group and that its discrete dual group Ĝ has been given some fixed order (B.3). We shall denote the set of nonnegative elements in Ĝ with respect to the given order by Ĝ₊. We define the space H _p (G), 1 ≦ p ≦ ∞, to be the closed ideal in the semi-simple Banach algebra L _p (G), l ≦ p ≦ ∞, consisting of all those f∈L _p (G) such that $\hat f\left( \gamma \right) = 0]$, γ∈Ĝ ~ Ĝ₊ = Ĝ₋. Clearly each H _p (G), l ≦ p ≦ ∞, is a semi-simple Banach algebra with convolution multiplication.