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 ${p\neq2}$.

Let ${\chi_{B}(x)}$ denote the characteristic function of the unit ball ${B}$ in ${d}$ dimensions. For a smooth function of rapid decay, say ${f}$, we can define the linear operator ${S_{1}}$ by the relation
$\displaystyle \widehat{S_{1}f}(\xi) = \chi_{B}(\xi)\hat{f}(\xi)$
where ${\hat{f}(\xi)}$ denotes the Fourier transform of ${f}$, as usual. This operator naturally arises in problems regarding the convergence of Fourier transforms (which we discuss below). A fundamental problem regarding this operator is to determine for which values of ${p}$ and ${d}$ we can extended ${S_{1}}$ to a bounded linear operator on ${L^{p}(\mathbb{R}^d)}$. The ${1}$-dimensional case of this problem was settled around 1928 by M. Riesz, however the higher dimensional cases proved to be much more subtle. In 1954 Herz showed that $2d/(d+1) was a necessary condition for the boundedness of $S_{1}$, and sufficient in the special case of radial functions. It was widely conjectured that these conditions were also sufficient in general (this was known as the disc conjecture). However,  in 1971 Charles Fefferman proved, for ${d\geq2}$, that ${S_{1}}$ does not extend to a bounded operator on any ${L^p}$ space apart from the trivial case when ${p=2}$ (which follows from Parseval’s identity). Recently, I needed to look at Fefferman’s proof and decided to spend some time trying to figure out what is really going on. I will attempt to give a motivated account of Fefferman’s result, in a two post presentation. In this (the first) post I will describe the motivation for the problem, as well as develop some tools needed in the proof. The problems discussed here were first considered in the context of Fourier series (i.e. functions on the ${d}$-dimensional torus ${\mathbb{T}^d}$). It turns out, however, that these problems are slightly easier to address on Euclidean space, and are equivalent thanks to a result of de Leeuw. In light of this, we will work exclusively on ${\mathbb{R}^d}$. (more…)