Stein's Spherical Maximal Theorem
Speaker: Patrick LiDate of Talk: April 8, 2026
Upstream link: UCLA Participating Analysis Seminar
1. Setup
Recall the Hardy-Littlewood maximal function defined by \[(Mf)(x) = \sup _{t > 0} \left\lvert \int _{B_1} f(x - ty)\ dy \right\rvert.\] The map \(f\mapsto Mf\) is bounded on \(L^p \left( \mathbb{R}^d \right)\) for all \(1 < p \leq \infty\).
One can view \((Mf)(x)\) as the \(\sup\) over convolutions of \(f\) against rescaled indicators of balls. One can therefore generalise this maximal operator to \[\left( M_ \sigma f \right)(x) = \sup _{t > 0} \left\lvert \int f(x - ty)\ d \sigma (y) \right\rvert,\] where \(\sigma \) is a sufficiently nice operator. The central question is then, does \(M_ \sigma \) have good boundedness?
2. Main Results
Theorem 1.
When \(\sigma = \left. \mathcal{H} ^{d-1} \right\rvert_{ \mathbb{S} ^{d-1}}\), \(M_ \sigma \) is bounded on \(L^p\) for all \(p > \frac{d}{d-1}\).
This was later generalised by Rubio de Francia in 1986 to
Theorem 2.
If \(\sigma \) is a compactly supported Borel measure such that \(\left\lvert \hat \sigma \right\rvert \lesssim \left\langle \xi \right\rangle ^{- \alpha }\) for some \(\alpha > \frac{1}{2}\), then \(M_ \sigma \) is bounded on \(L^p\) for every \(p > 1 + \frac{1}{2 \alpha }\).
\(\hat \sigma \) is the Fourier transform of the measure, and Stein’s theorem follows after a bit of effort.
3. An Interpolation Technique
One defines an auxiliary operator \(T_*\) and proves \(L^2\to L^2\) boundedness, then proves \(H _{\textrm{at}}^{1}\to L^1\) boundedness (in my head, this is a weak \((1, 1)\) type bound). Interpolate the bounds! This is classic stuff. Unwrapping the relationship between \(T_*\) and \(M_ \sigma \) gives boundedness of \(M_ \sigma \) on \(L^p\) for all \(1 + \frac{1}{2 \alpha } < p \leq \infty\).
The ability to interpolate with a Hardy space as an endpoint is nontrivial and is central to this proof. Likewise, one can still interpolate with \(BMO\) as an endpoint (the dual to \(H^1\)), and it’s broadly useful in harmonic analysis to interpolate with things that aren’t \(L^p\)!