Transference and Extension of Fourier Multipliers for L p (T)

Let X be a closed subspace of Lp(μ), where 1 < p < ∞ and μ is an arbitrary measure, and suppose that z → Rz is a strongly continuous representation of the circle group T in X. It is shown that R will transfer any Lp(T)-Fourier multiplier to X along with the multiplier bound. As applications of transference methods, this theorem provides, for p in the range 1 < p < ∞, short proofs of two theorems due to M. Jodeit on multiplier extensions. Specifically, each Lp(T)-Fourier multiplier can be extended to an Lp(R)- Fourier multiplier by piecewise constancy as well as by piecewise linearity. The fact that the piecewise linear extension (with nodes at the integers) of any Lp(T)-Fourier multiplier is an Lp(R)-Fourier multiplier was originally shown by Jodeit to be valid for any p in the range 1 ⩽ p < ∞. By suitable adaptations of his classical methods, we generalize this latter result in the same range of p by establishing a class of processes, including piecewise linearity, which extend Lp(T)-Fourier multipliers to continuous functions which are Lp(R)-Fourier multipliers. Our approach also leads to an analogue for Lp(R)-Fourier multipliers of the Paley-Wiener theorem.