Bochner–Riesz operator L^p bounds in higher dimensions

Establish L^p boundedness (determine the optimal ranges of p and order δ) for the Bochner–Riesz operator S^δ on ℝ^n for n ≥ 3, where the Fourier multiplier is m_δ(ξ) = (1 − |ξ|^2)_+^δ.

Background

Bochner–Riesz summability is a classical problem in harmonic analysis; in higher dimensions, sharp Lp bounds for Sδ are not fully known. These bounds are closely linked to Kakeya maximal estimates and Fourier restriction phenomena.

The survey explains that while two-dimensional cases are resolved, higher-dimensional Lp bounds remain one of the core open problems, with progress often tied to reverse Littlewood–Paley and decoupling methods.

References

Three of the most important open questions in euclidean harmonic analysis are: (i) Lp bounds for the Kakeya maximal operator (the corresponding dimensional consequence of these when n=3 has recently been established in spectacular work of Wang and Zahl), (ii) Lp bounds for the Bochner--Riesz operators S\delta, and (iii) Lp -Lq bounds for the Fourier restriction and extension operators for the sphere. These are known to be intimately related, and all of them are resolved when n=2, but all are open in all higher dimensions.

— Littlewood, Paley and Almost-Orthogonality: a theory well ahead of its time  (2511.22605 - Carbery, 27 Nov 2025) in Section 7.3 (Angular decompositions — reverse inequalities and curvature)

In dimensions $d\ge 3$, the Bochner--Riesz conjecture remains open for $p$ close to the endpoint $\frac{2d}{d+1}$.

— Weak type $(1,1)$ boundedness of Bochner--Riesz means at the critical index on quantum tori  (2610.03107 - Lai, 2 Oct 2026) in Section 1, Introduction and main result

The conjecture can be stated as follows: \begin{conjecture}[Bochner--Riesz conjecture] Let $p'$ denote the Hölder conjugate of $p$. Then, for every $1<p<\infty$, $B_\delta$ is bounded on $L^p(\mathbb R^4)$ whenever $\delta> 4\left|\frac12-\frac1p\right|-\frac12$. \end{conjecture}

— Refinements on the Restriction and Kakeya problems range in $\mathbb{R}^4$  (2610.06181 - Moreira, 5 Oct 2026) in Section 1, introduction; Conjecture (Bochner--Riesz conjecture)