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}.