The optimal region for uniform $L^p$ resolvent estimates on compact Riemannian manifolds and the sharp spectral-projection scale on the flat torus

The uniform resolvent estimate $\|u\|_{L^{2n/(n-2)}}\le C\|(\Delta_g+\zeta)u\|_{L^{2n/(n+2)}}$ on a compact boundaryless Riemannian manifold of dimension $n\ge 3$ holds on regions of the complex spectral plane whose size is governed by how tightly the eigenvalues of $-\Delta_g$ cluster. Dos Santos Ferreira, Kenig and Salo proved it on a parabolic region and asked whether the optimal region $\mathcal R_{\mathrm{opt}}=\{\zeta:\ |\Im\zeta|\ge\delta\}\cup\mathcal R^-_\delta$ - which outside a disc about the origin excludes only a fixed-width strip about the spectral axis, the compact analogue of the full Kenig-Ruiz-Sogge Euclidean region, corresponding to spectral windows of width $1/\lambda$ - is admissible. Bourgain, Shao, Sogge and Yao answered negatively for Zoll manifolds (including the round sphere), reduced the question on any manifold to sharp small-scale spectral projection estimates, and proved power improvements $\varepsilon(\lambda)=\lambda^{-\varepsilon_n}$ on the flat torus, asking whether these can be pushed further. Later work sharpened both the results (Hickman: $\lambda^{-1/3+\epsilon}$ in all dimensions; Germain-Rydin Myerson: $\lambda^{-1+2d/(5d-4)}$ in every dimension and $\lambda^{-1/2}$ in dimension 3; Pezzi: sharp at the Stein-Tomas exponent and for large $p$) and the target (an explicit conjecture that $\lambda^{-1+\epsilon}$ is admissible, closely tied to the discrete restriction conjecture). The optimal exponent and the admissibility of the optimal region nevertheless remain open in every dimension $n\ge 3$, and no geometry-by-geometry classification of admissible decay rates is known. The problem asks to settle these: the optimal region and exponent for the flat torus, and the classification of compact manifolds admitting resolvent improvement.

Background

Let $(M,g)$ be a smooth compact Riemannian manifold without boundary of dimension $n\ge 3$, and let $\Delta_g$ be its Laplace-Beltrami operator, the natural second-order self-adjoint operator whose spectrum governs vibration and diffusion on $M$. A uniform resolvent estimate is a bound

$$\|u\|_{L^{2n/(n-2)}(M)} \le C\,\|(\Delta_g+\zeta)u\|_{L^{2n/(n+2)}(M)},\qquad \zeta\in\mathcal R\subset\mathbb C,$$

with $C$ independent of $u$ and of the spectral parameter $\zeta$ ranging over a region $\mathcal R$. The exponent pair $(2n/(n+2),2n/(n-2))$ is forced by scaling: in Euclidean $\mathbb R^n$ these are exactly the dual exponents for which Kenig, Ruiz and Sogge (1987) proved such an estimate on the largest possible region, the whole plane minus a disc about the origin. On a compact manifold the spectrum of $-\Delta_g$ is a discrete sequence $0=\lambda_0^2\le\lambda_1^2\le\cdots$, so $\mathcal R$ must avoid the eigenvalues; the question, posed explicitly by Dos Santos Ferreira, Kenig and Salo (2014) after earlier work of Shen (2001) on the torus, is how close to the spectrum $\mathcal R$ can reach near infinity. They proved the estimate on the parabolic region $\{(\Im\zeta)^2\ge\delta\,\Re\zeta,\ \Re\zeta\ge\delta\}\cup\mathcal R^-_\delta$ - where $\mathcal R^-_\delta=\{\zeta:\ |\zeta|\ge\delta,\ \Re\zeta\le\delta\}$ is the part of the exterior of the disc $\{|\zeta|<\delta\}$ lying to the left of the line $\Re\zeta=\delta$ - and asked whether it extends to the optimal region $\mathcal R_{\mathrm{opt}}=\{\zeta:\ |\Im\zeta|\ge\delta\}\cup\mathcal R^-_\delta$, which outside the disc $\{|\zeta|<\delta\}$ excludes only a strip of fixed width $2\delta$ about the positive real axis carrying the eigenvalues $\lambda_j^2$: the compact analogue of the full Kenig-Ruiz-Sogge region.

Bourgain, Shao, Sogge and Yao (2015) showed that the answer depends on how tightly the eigenvalues cluster. Writing $\zeta=(\lambda+i\mu)^2$ with $\lambda\ge 1$, admissible regions correspond to lower bounds $|\mu|\ge\varepsilon(\lambda)$ with $\varepsilon(\lambda)\to 0$: the Dos Santos Ferreira-Kenig-Salo region corresponds to $\varepsilon\approx 1$, and $\mathcal R_{\mathrm{opt}}$ to the physically natural scale $\varepsilon(\lambda)\approx 1/\lambda$. They proved three structural results. First, on a Zoll manifold - one all of whose geodesics are closed, such as the round sphere $S^n$ - the eigenvalues of $-\Delta_g$ cluster in bands of multiplicity $\approx\lambda^{n-1}$, no improvement over the known region is possible, and the resolvent norm is comparable to the inverse distance from $\zeta$ to the spectrum; hence $\mathcal R_{\mathrm{opt}}$ fails for these geometries. Second, a necessary condition (their Theorem 1.2): if $\varepsilon(\lambda)$ is admissible then the eigenvalue counting function $N(\lambda)=\#\{j:\lambda_j\le\lambda\}$ must satisfy $N(\lambda+\varepsilon(\lambda))-N(\lambda-\varepsilon(\lambda))=O(\varepsilon(\lambda)\lambda^{n-1})$; for $\varepsilon(\lambda)\approx 1/\lambda$ this forces $N(\lambda+1/\lambda)-N(\lambda-1/\lambda)=O(\lambda^{n-2})$, a condition the authors describe as almost impossibly strong. Third, their Theorem 1.3 shows that admissibility of $\varepsilon(\lambda)$ (any monotone $\varepsilon$ with $\varepsilon(2\lambda)\ge\varepsilon(\lambda)/2$) is equivalent to bounds for the spectral projection operators $\chi_{[\lambda-\delta,\lambda+\delta]}(\sqrt{-\Delta_g}\,)f=\sum_{|\lambda_j-\lambda|\le\delta}E_jf$ (where $E_j$ projects onto the $j$-th eigenspace) of the sharp form $\|\chi_{[\lambda-\delta,\lambda+\delta]}\|_{L^{2n/(n+2)}\to L^{2n/(n-2)}}\lesssim\lambda\delta$, a strengthening on shrinking windows of Sogge's 1988 universal cluster estimates for unit windows. On the positive side they proved that $\varepsilon(\lambda)=\lambda^{-\varepsilon_n}$ is admissible on the flat torus $\mathbb T^n=\mathbb R^n/\mathbb Z^n$ for explicit $\varepsilon_n>0$ (any $\varepsilon_3<85/252$ in dimension 3), that $\varepsilon(\lambda)=(\log\lambda)^{-1}$ works under nonpositive sectional curvature, and they asked whether the torus exponents can be improved, pointing to Zygmund's classical $L^4$ theorem on $\mathbb T^2$ and to multilinear restriction techniques.

On the torus the projection problem is equivalent, at the level of the kernel, to counting lattice points in thin spherical shells: the eigenvalues of $-\Delta_{\mathbb T^n}$ are $4\pi^2|m|^2$ with $m\in\mathbb Z^n$, and the diagonal of the projector at scale $\delta$ counts lattice points with $||m|-\lambda|\lesssim\delta$. The sharp thin-shell problem is thus a hybrid of harmonic analysis and analytic number theory; it contains as special cases Bourgain's 1993 conjectural $L^p$ bounds for toral eigenfunctions (the discrete restriction conjecture for the sphere, on which the Bourgain-Demeter $\ell^2$-decoupling theorem - a sharp decomposition inequality for Fourier-localized functions - established the conjectured bounds for $d\ge 4$ and sufficiently large $p$) and the classical Gauss-circle-type lattice point remainder problems. At the resolvent exponent $p=2n/(n-2)$, Hickman (2020) proved admissibility of $\varepsilon(\lambda)=\lambda^{-1/3+\epsilon}$ in every dimension (refined to $\lambda^{-\beta_n}$ with $\beta_n=1/3+n/(3(21n^2-n-24))$) and posed the endpoint conjecture: $\varepsilon(\lambda)=\lambda^{-1+\epsilon}$ should be admissible for every $\epsilon>0$, i.e. the optimal region should hold for the torus up to subpolynomial loss. Germain and Rydin Myerson (2022) formulated the sharp two-term conjecture for the norms of thin-shell spectral projectors on arbitrary flat tori - a statement closely tied to admissibility of the optimal region, since the projector and resolvent norms are essentially equivalent up to logarithmic factors - and proved it below the Stein-Tomas exponent $p_{ST}=2(d+1)/(d-1)$ (the exponent at which Sogge's cluster bounds change form, by the Stein-Tomas Euclidean restriction theorem) for all windows down to $1/\lambda$, and partial ranges above it, including windows $\delta\ge\lambda^{-2/5}$ for all $p>2$ in dimension $d=3$, and - directly for the resolvent problem - admissibility of $|\mu|\ge\lambda^{-1+2d/(5d-4)}$ in every dimension and of $|\mu|\ge\lambda^{-1/2}$ in dimension $3$, up to subpolynomial losses; they also showed that current methods (Poisson summation and decoupling) cannot penetrate below windows of width $\lambda^{-(d-1)/(d+1)}$ for $p>p_{ST}$, and that for generic non-arithmetic tori the conjectured range extends strictly below the optimal-region scale, to $\delta>\lambda^{1-d+\epsilon}$ for every $\epsilon>0$. Pezzi (2024) proved the sharp bounds at the Stein-Tomas exponent itself for windows down to $\lambda^{-1+\kappa}$ without subpolynomial loss, and Pezzi (2026) proved the discrete restriction conjecture without loss for $p>2d/(d-4)$, $d\ge 5$; neither range contains the resolvent exponent. The resolvent-projection dictionary was consolidated by Cuenin (2024), who proved the converse implication (spectral cluster estimates imply uniform resolvent estimates) in general settings. On the classification side, Sogge-Zelditch (2002) and Sogge-Toth-Zelditch (2011) characterized maximal eigenfunction growth and showed that generic metrics admit some improvement $\varepsilon(\lambda)\to 0$ (non-quantitatively), and Huang-Sogge-Taylor (2022) showed products of manifolds preserve improvements; but no quantitative geometry-by-geometry classification is known, and no manifold is known to satisfy the $\varepsilon(\lambda)\approx 1/\lambda$ spectral-density condition.

Problem Statement

With the notation above ($\zeta=(\lambda+i\mu)^2$, $\lambda\ge 1$; $\chi_{[\lambda-\delta,\lambda+\delta]}$ the spectral projection of $\sqrt{-\Delta_g}$ onto windows of half-width $\delta$), determine the optimal admissible scale $\varepsilon(\lambda)$ (call $\varepsilon$ admissible for a given manifold if the estimate $\|u\|_{L^{2n/(n-2)}(M)}\le C\|(\Delta_g+\zeta)u\|_{L^{2n/(n+2)}(M)}$ holds with $C$ independent of $\zeta$ for all $\zeta=(\lambda+i\mu)^2$ with $\lambda\ge 1$ and $|\mu|\ge\varepsilon(\lambda)$) - equivalently the maximal resolvent region, equivalently the sharp thin-shell spectral projection norms - first for the flat torus and then for general compact manifolds.

1. The flat torus (primary open case). For $\mathbb T^n=\mathbb R^n/\mathbb Z^n$ with the flat metric, $n\ge 3$: prove or disprove that for every $\epsilon>0$ there is $C_\epsilon$ such that

$$\|u\|_{L^{2n/(n-2)}(\mathbb T^n)}\le C_\epsilon\|(\Delta_{\mathbb T^n}+\zeta)u\|_{L^{2n/(n+2)}(\mathbb T^n)}$$

holds for all $\zeta=(\lambda+i\mu)^2$ with $\lambda\ge 1$ and $|\mu|\ge\lambda^{-1+\epsilon}$ (Hickman's Conjecture 2, the torus case of the Dos Santos Ferreira-Kenig-Salo optimal-region question up to a subpolynomial loss: the original region corresponds to $|\mu|\gtrsim 1/\lambda$, i.e. to taking $\epsilon=0$). By the established equivalence this is the same as proving

$$\|\chi_{[\lambda-\delta,\lambda+\delta]}(\sqrt{-\Delta_{\mathbb T^n}}\,)f\|_{L^{2n/(n-2)}(\mathbb T^n)}\le C_\epsilon\,\lambda\delta\,\|f\|_{L^{2n/(n+2)}(\mathbb T^n)}\quad\text{for all }\delta\ge\lambda^{-1+\epsilon}.$$

The sharp two-term Germain-Rydin Myerson conjecture for the $L^2\to L^p$ projector norms is a finer quantification of the same window regime, from which the resolvent bound follows (their own summary of implications does exactly this). Determine the optimal exponent $\sigma_n=\sup\{\sigma:\ |\mu|\ge\lambda^{-\sigma}\text{ admissible}\}$: whether $\sigma_n=1$ (the optimal region is admissible up to subpolynomial loss) or whether the true exponent is $\sigma_n\in(0,1)$, with matching upper and lower bounds. The required target is the square flat torus $\mathbb T^n=\mathbb R^n/\mathbb Z^n$ of part 1; the conjectured bounds are expected to extend to arbitrary flat tori (any positive definite quadratic form in place of $|m|^2$) and, for generic tori, even below the $\lambda^{-1}$ scale, and answers may distinguish the arithmetic type of the torus, but these wider classes are optional extensions and are not required to resolve the problem.

2. General manifolds (governing classification). Characterize the compact boundaryless Riemannian $(M,g)$, $n\ge 3$, for which some $\varepsilon(\lambda)=o(1)$ is admissible - equivalently, which admit improved small-scale spectral projection estimates - and which quantitative decay rates occur for which geometries. In particular: does any compact manifold satisfy the spectral-density condition $N(\lambda+1/\lambda)-N(\lambda-1/\lambda)=O(\lambda^{n-2})$ (where $N(t)=\#\{j:\ \lambda_j\le t\}$ counts eigenvalues of $\sqrt{-\Delta_g}$), and is $\varepsilon(\lambda)\approx 1/\lambda$ (the optimal region $\mathcal R_{\mathrm{opt}}$) admissible for any geometry that is not Zoll (a Zoll manifold being one all of whose geodesics are closed)? Known data points to be reconciled: Zoll manifolds admit no improvement; nonpositive sectional curvature gives $\varepsilon(\lambda)=(\log\lambda)^{-1}$; the flat torus gives powers $\lambda^{-\sigma_n}$; generic metrics give non-quantitative $o(1)$; products preserve improvements.

The common objective of all parts is the maximal region $\mathcal R_g$ - equivalently the optimal scale $\varepsilon(\lambda)$, equivalently the sharp thin-shell projector norms - together with the geometric and spectral mechanism (eigenvalue density versus clustering) that governs it. Resolving part 1 for even one dimension $n\ge 3$, in either direction, settles the central open case of the Dos Santos Ferreira-Kenig-Salo question; part 2 is the general form of the same question and is expected to be answered by the same mechanism. The two parts are logically independent: part 2 is not a prerequisite for part 1, and resolving part 1 does not depend on the part-2 classification.

References

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