Carleson–Sjölin theorem

Remarks

Lennart Carleson and Per Sjölin proved the sufficiency of the condition in 1972. The necessity holds in every dimension $n$. The kernel of $S^\delta$ decays like $\abs{x}^{-(n + 1)/2 - \delta}$ with oscillation, and it lies in $L^p$ only when $1/p - 1/2 < (1 + 2\delta)/(2n)$. Boundedness of $S^\delta$ on $L^p$ forces its kernel into $L^p$. Since $S^\delta$ is self-adjoint, boundedness on $L^p$ is equivalent to boundedness on $L^{p'}$, and this gives the other half of the condition. The proof of sufficiency decomposes the multiplier into pieces near the circle $\abs{\xi} = 1$ and uses an $L^4$ estimate for oscillatory integrals whose phase has curved level sets.

Example: for $\delta = 1/4$ the condition reads $\abs{1/p - 1/2} < 3/8$, that is $8/7 < p < 8$. Without $\delta > 0$ the result fails: for $\delta = 0$, Fefferman's ball multiplier theorem shows that $S^0$ is bounded only on $L^2(\R^2)$.

The same condition in dimension $n$, namely $\abs{1/p - 1/2} < (1 + 2\delta)/(2n)$, is the Bochner–Riesz conjecture; the theorem settles it for $n = 2$, while it is open for $n \ge 3$.

Going deeper

For a student

The theorem is the two-dimensional case of the Bochner–Riesz problem, and its proof has the same structure as the two-dimensional case of the restriction conjecture. For $\delta > 0$ and $4/3 \le p \le 4$, it gives $L^p$ convergence of the Bochner–Riesz means $S^\delta_R f \to f$ for $f \in L^p(\R^2)$. In higher dimensions, the Stein–Tomas restriction theorem gives a partial range, and the full range is open.