Let $1\lt p\lt 2$. On several of the classical groups arising in Fourier analysis, including the $n$-torus and Euclidean $n$-space, we show that there exists a multiplier which is of weak type $(p,p)$, but which is not bounded on $L_p$.
It is well-known that many of the most important translation-invariant linear operators in Fourier analysis are not bounded on $L_1$, but rather satisfy a weaker condition, the so-called weak $(1,1)$ condition. Examples of such operators are the conjugate function operator on the circle group, the Hilbert transform on the real line, and their higher dimensional analogues—those singular integral operators of Calderón and Zygmund whose kernels are sufficiently smooth (see [11, Chapter 2]). On the other hand, every multiplier transformation (see the definition below) is bounded on $L_2$.
In this paper, we consider the case $1\lt p\lt 2$, and show that on several of the classical groups arising in Fourier analysis, including the $n$-torus and Euclidean $n$-space, there exist multipliers which are of weak type $(p,p)$ (see [16, Chapter 12, p. 111], and the definitions below), but which are not bounded in $L_p$. The proof involves the real interpolation spaces of Lions and Peetre, specifically a multilinear interpolation theorem for these spaces (see [5, Chapter 1, Theorem 4.1] and [15, Theorem 2.4]), certain results of de Leeuw concerning multipliers , and an analogue of a trigonometric series first considered by Hardy and Littlewood . Our principal result is Theorem 3.8. We begin our discussion with some notations and definitions.