Fefferman's ball multiplier theorem

Remarks

For $p = 2$, the indicator is bounded, so the Plancherel theorem gives the bound. For $p \ne 2$, Charles Fefferman's 1971 proof uses Kakeya sets of small measure in the plane. By a lemma of Yves Meyer, a bound for the ball multiplier would give a vector-valued bound for the multipliers of many half-planes with different normal directions. A Kakeya construction gives many thin rectangles whose union has small area while their translates in the direction of their long sides are disjoint, and this contradicts that vector-valued bound. By duality, it suffices to disprove the case $p > 2$.

The theorem contrasts with dimension one, where the indicator of an interval is an $L^p$ multiplier for all $1 < p < \infty$, since it is a combination of modulated Hilbert transforms. It also contrasts with cubes: the indicator of a cube is a product of indicators of intervals, so it is an $L^p$ multiplier for $1 < p < \infty$.

A consequence is that the spherical partial sums $(\mathbf{1}_{RB} \hat f)^\vee$ of the Fourier integral of $f \in L^p(\R^n)$ need not converge to $f$ in $L^p$ when $p \ne 2$ and $n \ge 2$.

Going deeper

For a student

This result explains why one studies the smoothed Bochner–Riesz means instead of sharp spherical partial sums in higher dimensions. Its proof brought the Kakeya problem into Fourier analysis, a link that runs through the restriction conjecture as well.