The reverse Littlewood–Paley inequality for the sphere in dimensions $n\ge 3$

Carbery's note (arXiv:1507.02515, work from the early 1990s) conjectures the reverse Littlewood–Paley inequality $\|S^\delta f\|_{L^{2n/(n-1)}(\mathbb{R}^n)}\le C_n\|(\sum_\alpha|S_\alpha f|^2)^{1/2}\|_{L^{2n/(n-1)}(\mathbb{R}^n)}$ for the Bochner–Riesz multipliers $S^\delta$ associated to a $\delta$-neighbourhood of the unit sphere, decomposed into angular pieces $S_\alpha$. It is known only in two dimensions (Fefferman). The note proves that the conjecture, equivalently a local extension inequality for the sphere, would imply sharp logarithmic-loss bounds for the Kakeya maximal function, the Fourier restriction operator, and the Bochner–Riesz means in all dimensions. The problem is to prove or refute the conjectured inequality in any dimension $n\ge 3$; it remains fully open as of late 2025, and an affirmative answer would resolve three of the central open problems of Euclidean harmonic analysis.

Background

Let $f$ be a Schwartz function on $\mathbb{R}^n$ with Fourier transform $\widehat{f}$. The Bochner–Riesz means are the Fourier multiplier operators $S^\delta$ given by $\widehat{S^\delta f}(\xi)=\Phi\big((|\xi|-1)/\delta\big)\widehat{f}(\xi)$, where $\Phi$ is a fixed smooth bump supported in $[-1,1]$; they are smoothed spectral truncations of the inverse Fourier transform near frequency $|\xi|=1$, and the behaviour of their operator norms as $\delta\to 0$ is the classical Bochner–Riesz problem. Decompose the $\delta$-neighbourhood of the unit sphere $\mathbb{S}^{n-1}\subset\mathbb{R}^n$ into roughly $\delta^{-(n-1)/2}$ 'coin-shaped' pieces $E_\alpha$ of tangential size $\delta^{1/2}$ and radial size $\delta$, and let $S_\alpha$ denote the multiplier with smooth symbol adapted to $E_\alpha$, so $S^\delta=\sum_\alpha S_\alpha$ up to negligible error. Classical Littlewood–Paley theory bounds such square functions $\big(\sum_\alpha|S_\alpha f|^2\big)^{1/2}$ from above by $f$; Carbery's note studies the reverse direction at the critical restriction exponent $q_n=2n/(n-1)$: $\|S^{\delta} f\|_{L^{\frac{2n}{n-1}}(\mathbb{R}^{n})}\le C_{n}\Big\|\Big(\sum_{\alpha}|S_{\alpha}f|^{2}\Big)^{1/2}\Big\|_{L^{\frac{2n}{n-1}}(\mathbb{R}^{n})},$ with $C_n$ independent of $\delta$ and $f$. When $n=2$ (so $q_n=4$) Fefferman proved this in 1973: the $L^4$ norm can be multiplied out and Plancherel applied, the curvature of the circle making the algebraic sums $E_\alpha-E_{\alpha'}$, $\alpha\ne\alpha'$, essentially disjoint. For $n\ge3$ the exponent $q_n$ is no longer an even integer and no proof is known.

Since the kernels of $S^\delta$ and $S_\alpha$ are essentially supported in the ball $B(0,\delta^{-1})$, the inequality is equivalent (up to the harmless caveat, recorded in the note, that one averages over spheres of radii in $[1-\delta,1+\delta]$) to a local extension inequality for the sphere: writing $g\,d\sigma\check{\;}$ for the inverse Fourier transform of the measure with density $g$ on $\mathbb{S}^{n-1}$, and $g_\alpha=g\chi_{E_\alpha}$ for a decomposition into $\delta^{1/2}$-caps, $\|g\,d\sigma\check{\;}\|_{L^{\frac{2n}{n-1}}(B(0,\delta^{-1}))}\le C_{n}\Big\|\Big(\sum_{\alpha}|g_{\alpha}\,d\sigma\check{\;}|^{2}\Big)^{1/2}\Big\|_{L^{\frac{2n}{n-1}}(B(0,\delta^{-1}))}.$ This is a square-function (almost-orthogonality) statement for the Fourier extension operator at the critical exponent, in the direction opposite to the $\ell^r$-decoupling inequalities of Bonami–Garrigós and Bourgain–Demeter, whose mixed norms run the other way.

The note's Proposition 1 shows that the extension inequality, assumed only for functions $g$ that are constant on $\delta^{1/2}$-cells, implies the Kakeya maximal conjecture with logarithmic loss $\|M_{N}f\|_{L^{n}(\mathbb{S}^{n-1})}\le C_{n}(\log N)^{\frac{n-1}{n}}\|f\|_{L^{n}(\mathbb{R}^{n})},$ where the Kakeya (directional) maximal function $M_N f(\omega)$ is the supremum of averages of $|f|$ over tubes of dimensions $1\times\cdots\times1\times N$ whose axis points in direction $\omega\in\mathbb{S}^{n-1}$. A two-scale argument then yields the log-loss restriction estimate $\|g\,d\sigma\check{\;}\|_{L^{2n/(n-1)}(B(0,R))}\le C_n(\log R)^{(n-1)/2n}\|g\|_{L^{2n/(n-1)}(\mathbb{S}^{n-1})}$, and via the standard machine (bounds for the Kakeya maximal function imply bounds for the Nikodym maximal function — the analogous maximal operator built from thin rectangles of arbitrary orientation that contain the point in question — which in turn imply Bochner–Riesz bounds) one obtains the Bochner–Riesz estimate $\|S^\delta f\|_{L^{2n/(n-1)}}\le C_n(\log(1/\delta))^{(n-1)/2n}\|f\|_{L^{2n/(n-1)}}$ in all dimensions. All of these deductions are conditional on the reverse Littlewood–Paley inequality, which is the sole unestablished input.

Problem Statement

Prove or refute the reverse Littlewood–Paley inequality in a single dimension $n\ge3$ of your choosing: determine whether there exists a constant $C_n<\infty$ such that for all $0<\delta\le\tfrac12$ and all Schwartz functions $f$ on $\mathbb{R}^n$ the reverse Littlewood–Paley inequality $\|S^{\delta} f\|_{L^{\frac{2n}{n-1}}(\mathbb{R}^{n})}\le C_{n}\Big\|\Big(\sum_{\alpha}|S_{\alpha}f|^{2}\Big)^{1/2}\Big\|_{L^{\frac{2n}{n-1}}(\mathbb{R}^{n})}$ holds, where $S^\delta$ is the Bochner–Riesz multiplier adapted to the $\delta$-neighbourhood of the unit sphere and $S_\alpha$ are its angular pieces, the multipliers with smooth symbols adapted to the coin-shaped pieces $E_\alpha$ of tangential size $\delta^{1/2}$ and radial size $\delta$ from the decomposition of the background. Equivalently (in the averaged-over-radii sense recorded in the note's footnote): does the corresponding local extension inequality $\|g\,d\sigma\check{\;}\|_{L^{2n/(n-1)}(B(0,\delta^{-1}))}\le C_n\|(\sum_\alpha|g_\alpha\,d\sigma\check{\;}|^2)^{1/2}\|_{L^{2n/(n-1)}(B(0,\delta^{-1}))}$ hold for all smooth $g$ on $\mathbb{S}^{n-1}$? An affirmative answer for a single $n\ge3$ would, via Carbery's conditional Proposition 1, establish unconditionally in that dimension the Kakeya maximal bound with loss $(\log N)^{(n-1)/n}$ and the logarithmic-loss restriction estimate — for these two corollaries the variant of the extension inequality for $g$ constant on $\delta^{1/2}$-cells suffices — and, with the full inequality combined with that variant through the standard Kakeya-to-Bochner–Riesz machine invoked in the note, the Bochner–Riesz estimate with loss $(\log(1/\delta))^{(n-1)/2n}$. A negative answer must exhibit, for some fixed $n\ge3$, a sequence $\delta_k\to0$ and functions $f_k$ for which the ratio of the two sides of the displayed inequality tends to infinity.

References

1. Anthony Carbery, A Remark on reverse Littlewood-Paley, restriction and Kakeya, arXiv:1507.02515 (2015), https://arxiv.org/abs/1507.02515
2. Charles Fefferman, A note on spherical summation multipliers, Israel Journal of Mathematics 15 (1973), 44–52, doi:10.1007/BF02771772, https://doi.org/10.1007/BF02771772
3. Anthony Carbery, Littlewood, Paley and Almost-Orthogonality: a theory well ahead of its time, Journal of the London Mathematical Society 113 (2026), arXiv:2511.22605, doi:10.1112/jlms.70528, https://arxiv.org/abs/2511.22605
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