On Jodeit’s multiplier extension theorems

Abstract

Let Δ(x) = max {1 - ¦x¦, 0} for all x ∈ ℝ, and let ξ_[0,1) be the characteristic function of the interval 0 ≤x < 1. Two seminal theorems of M. Jodeit assert that A and ξ_[0,1) act as summability kernels convertingp-multipliers for Fourier series to multipliers forL ^P (ℝ). The summability process corresponding to Δ extendsL ^P (T)-multipliers from ℤ to ℝ by linearity over the intervals [n, n + 1],n ∈ ℤ, when 1 ≤p < ∞, while the summability process corresponding to ξ_[0,1) extends L^P(T)-multipliers by constancy on the intervals [n, n + 1),n ∈ ℤ, when 1 <p < ∞. We describe how both these results have the following complete generalization: for 1 ≤p < ∞, an arbitrary compactly supported multiplier forL ^P (ℝ) will act as a summability kernel forL ^P (T)-multipliers, transferring maximal estimates from L^P(T) to L^P(ℝ). In particular, specialization of this maximal theorem to Jodeit’s summability kernel ξ_{[0, 1)} provides a quick structural way to recover the fact that the maximal partial sum operator on L^P(ℝ), 1 <p < ∞, inherits strong type (p,p)-boundedness from the Carleson-Hunt Theorem for Fourier series. Another result of Jodeit treats summability kernels lacking compact support, and we show that this aspect of multiplier theory sets up a lively interplay with entire functions of exponential type and sampling methods for band limited distributions.