Fefferman ball multiplier obstruction

Assume Countable Choice for the library Fourier and $L^p$ conventions, and let $n\ge 2$. Fix a Euclidean ball $B\subseteq {\mathbb{R}}^n$ and let ${\mathbf{1}}_B$ be its indicator, a bounded measurable symbol. Recorded boundary result, not proved here: in the usual range $1\le p<\mathrm{\infty }$ the ball indicator is an $L^p({\mathbb{R}}^n)$ Fourier multiplier in the sense of Lp Fourier multiplier and its norm only at $p=2$.

The $p=2$ half is elementary on this page. Since $\mathrm{\mid }{\mathbf{1}}_B\mathrm{\mid }\le 1$ everywhere, Exact L2 Fourier multiplier norm gives the unique bounded $L^2$ extension of the Schwartz-core multiplier with symbol ${\mathbf{1}}_B$ and its operator norm is exactly $1$. The case $p\mathrm{\ne }2$ is the deep part: Fefferman proved in Theorem 1 of The multiplier problem for the ball that for $n>1$ the ball multiplier is bounded on $L^p$ if and only if $p=2$. Williams records the same statement in Remark 3.12 and refers to Grafakos §10.1 for the argument.

This remark is a bibliographic leaf: it records the exact limitation of the $L^2$ criterion of Exact L2 Fourier multiplier norm and supplies no proof or dependency for any later item. No Kakeya or geometric measure-theoretic argument is reproduced here, and no $L^p$ multiplier claim other than the recorded quotation is asserted. Countable Choice is inherited from the cited $L^2$ multiplier interface (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)).