For radial multipliers, a necessary and sufficient condition for L^p\left(\mathbb{R}^n\right) boundedness is known for some partial range of p. Let n \geq 4 and 1 < p < 2\frac{n - 1}{n + 1}. Suppose that m is a radial multiplier compactly supported away from the origin. Then m is an L^p\left(\mathbb{R}^n\right) multiplier if and only if the Fourier transform of m belongs to L^p\left(\mathbb{R}^n\right).
