# Lewko's blog

## Fefferman’s ball multiplier counterexample

Posted in expository, Fourier Analysis, math.CA by Mark Lewko on August 4, 2009

In the previous post we saw the connection between the ball multiplier ${S_{1}}$ and spherical ${L^{p}}$ convergence of Fourier transforms. Recall that the operator ${S_{1}}$ is defined in ${d}$ dimensions by the relation

$\displaystyle \widehat{S_{1}f}(\xi) = \chi_{B}(\xi)\hat{f}(\xi)$

where ${B}$ denotes the ${d}$-dimensional unit ball.  The focus of this post will be to prove the following result

Theorem 1 (Fefferman, 1971) The operator ${S_{1}}$ is not bounded on ${L^{p}(\mathbb{R}^d)}$ if ${d>1}$ and ${p\neq2}$.