June 17, 2026
We study random Schrödinger operators on closed Riemannian manifolds with Anderson-type potentials. We prove high-probability spectral inclusion bounds showing that eigenvalues remain close to those of the Laplacian, with deviations controlled by a norm of the potential coefficients. Compared with deterministic bounds, this yields a square-root cancellation gain. The proof is based on a general principle showing that randomisation improves operator norm bounds for multiplier-type operators, which we formulate in both discrete and continuous settings.
On a closed (i.e., compact, boundaryless) Riemannian manifold \((M,g)\) of dimension \(d \geq 2\), we consider the random Schrödinger operator \[\begin{align} H_{\omega,\lambda} = -\Delta_g + V_{\omega,\lambda} \quad \text{on } L^2(M), \end{align}\] where \(V_{\omega,\lambda}\) is an Anderson-type potential built from bumps at scale \(\lambda^{-1}\). More precisely, we assume that \[\begin{align} \label{eq:Anderson-type} V_{\omega,\lambda}(x) = \sum_{j=1}^{N(\lambda)} \omega_j v_j(\lambda) \psi_j(x;\lambda), \quad x \in M, \end{align}\tag{1}\] where the functions \(\psi_j(\cdot;\lambda)\) are uniformly bounded and have bounded overlap, with spatial support at scale \(\lambda^{-1}\), \(v_j(\lambda) \in \mathbb{C}\) are deterministic coefficients, the random variables \(\omega_j\) are i.i.d. symmetric Bernoulli (Rademacher) or centred normalised Gaussians, \(\lambda\geq 2\) is a large parameter (the inverse randomisation length), and \(N(\lambda)\leq C\lambda^d\).
Since \(M\) is compact, the spectrum of \(-\Delta_g\) is discrete \[\mathop{\mathrm{spec}}(-\Delta_g)=\{\lambda_k^2\}_{k=0}^\infty, \qquad 0=\lambda_0^2\leq \lambda_1^2\leq \cdots, \qquad \lambda_k^2\to \infty.\] Our first main result gives a high-probability bound on the location of the eigenvalues of \(H_{\omega,\lambda}\), up to energy \(\lambda^2\), in terms of the \(\ell^{2q}\)-norm of \(v(\lambda)=(v_j(\lambda))_{j=1}^{N(\lambda)}\).
Theorem 1. Fix \(N\geq 1\). Then there exists a constant \(C_N\) such that for every \(d/2\leq q\leq\infty\), every \(\lambda\geq 2\), and every \(K\geq 1\), with probability at least \(1-\exp(-K^2)\), we have \[\{z\in\mathop{\mathrm{spec}}(H_{\omega,\lambda}) : |z|\leq \lambda^2,\,\mathop{\mathrm{dist}}(z, \mathop{\mathrm{spec}}(-\Delta_g))\geq \lambda^{-N}\} \subset \bigcup_{\lambda_k\leq \lambda} D(\lambda_k^2,C_NKr_k)\cup \Omega,\] where \[\begin{align} \label{def46r95k} r_k := (1+\lambda_k)^{\frac{d(q+1)}{2q}-\mu(q)} \, \lambda^{-\frac{d(q+1)}{2q}} (\log\lambda)^{11/2}\|v(\lambda)\|_{\ell^{2q}}, \end{align}\tag{2}\] \[\begin{align} \Omega:=\{z\in\mathbb{C}: |z|^{1/2}(1+|z|)^{-\frac{d(q+1)}{4q}+\frac{\mu(q)}{2}} \leq C_NK\, \lambda^{-\frac{d(q+1)}{2q}} (\log \lambda)^{11/2} \|v(\lambda)\|_{\ell^{2q}}\}, \end{align}\] and \[\begin{align} \label{def46mu40q41} \mu(q):=\begin{cases} 1, & 1\leq q\leq \frac{d+1}{2},\\ \frac{d+1}{4q}+\frac{1}{2}, & \frac{d+1}{2}\leq q\leq\infty. \end{cases} \end{align}\tag{3}\]
We postpone further remarks concerning Theorem 1 to the end of the introduction, including a comparison with the deterministic results of [1].
The main new ingredient in the proof of Theorem 1 is an abstract operator-theoretic principle showing that randomisation improves norm bounds for multiplier-type operators of the form \(T_1^* V T_2\), where \(V\) is a multiplication operator and \(T_1,T_2\) are structured linear operators, for instance spectral projectors, resolvents, or Fourier extension operators.
Random multiplier estimates play a crucial role in the work of Schlag–Shubin–Wolff [2] and Bourgain [3], [4] on weakly disordered random Schrödinger operators on \(\mathbb{Z}^2\). In Schlag–Shubin–Wolff, the relevant bounds are obtained by exploiting geometric considerations involving curvature. Bourgain introduced a different approach, based on a general entropy bound, which works in arbitrary dimension and does not rely on curvature. On the other hand, Bourgain’s method makes essential use of the Rademacher or Gaussian distribution of the random variables \(\omega_j\).
Our contribution is a general abstract framework inspired by Bourgain’s approach. An instance of such a principle for operators on \(\mathbb{R}^d\) was established in [5]. In the present paper, we develop this idea in a substantially broader form.
We formulate the abstract results in two settings. The first is a discrete model, where a function on a finite set is randomised over the elements of a partition. The second is a continuous analogue on Ahlfors regular metric measure spaces, where the randomisation is performed on Voronoi cells associated with a finite set. These results are of independent interest and form the abstract core of the paper.
Theorem 2 (Discrete abstract estimate). For \(i=1,2\), let \(\mathcal{H}_i\) be complex separable Hilbert spaces, \(E\) a nonempty finite set, and consider bounded linear operators \[S_i:\mathcal{H}_i\to \ell^\infty(E),\] which we also view as operators into \(\ell^{p'}(E)\) since \(E\) is finite. Let \(\{E_j\}_{j\in J}\) be a partition of \(E\), that is, \[E=\bigcup_{j\in J}E_j, \qquad E_j\cap E_k=\emptyset \quad \text{for } j\neq k.\] Let \(v\in \ell^\infty(E)\), set \(v_j:=v\mathbf{1}_{E_j}\), and let \(\{\omega_j\}_{j\in J}\) be i.i.d. symmetric Bernoulli or centred normalised Gaussian random variables. Define \[\begin{align} \label{def:v95omega32discrte} v_{\omega}(x)=\omega_j v(x), \qquad x\in E_j. \end{align}\tag{4}\] Let \(2\leq q\leq\infty\) and \(1\leq p\leq 2\leq p'\leq \infty\) satisfy \(\frac{1}{q}=\frac{1}{p}-\frac{1}{p'}\). Then \[\begin{align} \label{eq:abstract1} \!\! \mathbf{E}\|S_1^*v_{\omega}S_2\|_{\mathcal{H}_2\to \mathcal{H}_1} \lesssim (\log \#E)^{\frac{5}{2}} \|S_1\|_{\mathcal{H}_1\to \ell^{p'}(E)} \|S_2\|_{\mathcal{H}_2\to \ell^\infty(E)} \Big(\sum_{j\in J}\|v_j\|_{\ell^p(E_j)}^{2q}\Big)^{\frac{1}{2q}}, \end{align}\tag{5}\] with the obvious modification for \(q=\infty\).
Here and below, we use the notation \(A \lesssim B\) for functions \(A,B\geq0\) to indicate that there is a constant \(c\) such that \(A\leq c B\). If \(c\) depends on a parameter \(\tau\), we may write \(A\lesssim_\tau B\), but the dependence on \(d,M\) is usually ignored in the notation.
Remark 3. By Hölder’s inequality, \[\Big(\sum_{j\in J}\|v_j\|_{\ell^p(E_j)}^{2q}\Big)^{1/2q} \leq \sup_{j\in J}(\# E_j)^{\frac{1}{p}-\frac{1}{2q}}\|v\|_{\ell^{2q}(E)}.\] In particular, if all the \(E_j\) are singletons, then the right-hand side is just \(\|v(\lambda)\|_{\ell^{2q}(E)}\).
We now state a continuous version of Theorem 2. All technical notions appearing in the statement are defined in Section 3.2. A precise and stronger statement is given in Theorem 12.
Theorem 4 (Continuous abstract estimate). Let \(X\) be an Ahlfors \(d\)-regular metric measure space, and let \[T_i:\mathcal{H}_i\to L^\infty(X)\cap L^{p'}(X), \qquad i=1,2,\] be bounded linear operators, where \(\mathcal{H}_i\) are complex separable Hilbert spaces. Let \(\Lambda\) be a finite, maximal \(r\)-separated subset of \(X\), for some \(r>0\), and let \(V_\omega\) be the randomisation, by i.i.d. symmetric Bernoulli or centred normalised Gaussian random variables, of a measurable complex-valued function \(V\in L^1_{\mathrm{loc}}(X)\) on the Voronoi cells associated with \(\Lambda\). Assume moreover that \(X\), \(\Lambda\), and the ranges of \(T_1\) and \(T_2\) satisfy the uniform reparametrisation and local constancy hypotheses stated in Section 3.2. Then, whenever \(2\leq q\leq\infty\) and \(1\leq p\leq 2\leq p'\leq \infty\) satisfy \(\frac{1}{q}=\frac{1}{p}-\frac{1}{p'}\), one has \[\begin{align} \mathbf{E}\|T_1^*V_{\omega}T_2\|_{\mathcal{H}_2\to \mathcal{H}_1} &\lesssim r^{d/p} (\log \#\Lambda)^{5/2} \|T_1\|_{\mathcal{H}_1\to L^{p'}(X)} \|T_2\|_{\mathcal{H}_2\to L^\infty(X)} \\ &\qquad \times \Big(\sum_{x\in \Lambda}\|V\|_{L^\infty(B(x,2r))}^{2q}\Big)^{1/2q}, \end{align}\] with the obvious modification for \(q=\infty\).
In the compact-manifold setting, there are two principal choices of \(T_1,T_2\) for which this theorem becomes effective.
(i) Spectral projectors. Let \[P=\sqrt{-\Delta_g}, \qquad \Pi_\lambda=\mathbf{1}(P\in[\lambda,\lambda+1]).\] The range of \(\Pi_\lambda\) is locally constant at wavelength scale \(r=\lambda^{-1}\), and Sogge’s spectral cluster bounds give \[\|\Pi_\lambda\|_{L^2(M)\to L^{\frac{2(d+1)}{d-1}}(M)} \lesssim \lambda^{\frac{d-1}{2(d+1)}}.\] Inserting these estimates into Theorem 4 yields the randomised spectral cluster estimate \[\begin{align} \mathbf{E}\|\Pi_{\lambda} ^*V_{\omega,\lambda}\Pi_{\lambda} \|_{L^2(M)\to L^2(M)} & \lesssim \lambda^{-1}(\log \lambda )^{5/2}\|v(\lambda)\|_{\ell^{d+1}}, \end{align}\] see Proposition 13. This is the Riemannian analogue of the random Fourier extension estimate [5]. By sub-Gaussian tail bounds (see Lemma 1), it follows that with probability at least \(1-\exp(-K^2)\), \[\left|\int_M |\Pi_{\lambda}u|^2V_{\omega,\lambda}\mathrm{d}v_g\right|\lesssim K \lambda^{-1}(\log \lambda )^{5/2}\|v(\lambda)\|_{\ell^{d+1}}\|u\|_{L^2(M)}^2,\] while the deterministic bound holds only with the \({\ell^{\frac{d+1}{2}}}\) norm, without the logarithm.
Whereas \(L^2\to L^{p'}\) bounds measure the size of level sets of \(\Pi_\lambda u\), weighted expressions of the form \(\int_M |\Pi_\lambda u|^2 V\,\mathrm{d}v_g\) provide geometric information about the shape of level sets. In the present random setting, however, \(V_{\omega,\lambda}\) is signed, so this geometric interpretation is less direct.
In the Euclidean setting, weighted Fourier restriction theory is concerned with inequalities of the form \[\int_{\mathbb{R}^d}|E_Sf|^2w\,\mathrm{d}x\leq C(w)\|f\|_{L^2(S)}^2\] for deterministic nonnegative weights \(w\). We refer, for example, to [6], [7] for further background and results.
(ii) Half-resolvents. We will prove deterministic estimates for the square root of the resolvent, \[\bigl\||-\Delta_g-z|^{-1/2}\bigr\|_{L^2(M)\to L^{\frac{2(d+1)}{d-1}}(M)} \lesssim \max(d(z)^{-1/2},\langle z\rangle^{-1/4}) \log\langle z\rangle^{1/2} \langle z\rangle^{\frac{d-1}{4(d+1)}},\] see Proposition 15. Combining these bounds with Theorem 4 yields a probabilistic estimate for the Birman-Schwinger operator \[\mathcal{K}_{\omega,\lambda}(z) = |-\Delta_g-z|^{-1/2} V_{\omega,\lambda} (-\Delta_g-z)^{-1/2},\] see Proposition 17. The spectral inclusion in Theorem 2 is then obtained by combining this estimate with the Birman–Schwinger principle.
The compact-manifold Schrödinger problem considered in this paper is only one instance of a more general mechanism that separates the probabilistic from the analytic input. As an outlook, we briefly indicate a few problems in which we expect this mechanism to apply.
Random Schrödinger operators in other geometric settings. The argument is not specific to compact boundaryless manifolds. It should extend to compact manifolds with boundary, as well as to noncompact spaces such as asymptotically conic or hyperbolic manifolds. The appropriate deterministic spectral cluster and resolvent bounds may be found, e.g., in [8]–[11]. Another direction would be to consider random operators on \(\mathbb{R}^d\) with trapping background potentials or magnetic fields, see, e.g., [12], [13].
In all of these settings, the length of the spectral cluster window and the proximity of \(z\) to the spectrum of the unperturbed operator in the half-resolvent estimates, both of which play an important role in the proof of Theorem 2, are expected to shrink compared to the compact manifold setting. One then also expects the \(r_k\) in Theorem 2 to shrink. This improvement could also be explored for the torus, where shrinking spectral cluster bounds are well-studied, see, e.g., [14], [15].
Once the location of individual eigenvalues is well understood, one could then investigate their distribution via Lieb–Thirring type inequalities; this was done in the Euclidean setting in [16].
Existence and completeness of a.s.wave operators in \(\mathbb{R}^d\). Ionescu and Schlag [17] proved the existence and completeness of wave operators for Schrödinger operators on \(\mathbb{R}^d\) for a large class of deterministic potentials. In particular, their results apply when \(V\in L^{\frac{d+1}{2}}(\mathbb{R}^d)\). When the potential is random, we expect an improvement to \(L^{d+1-\varepsilon}(\mathbb{R}^d)\). This would be an analogue of Bourgain’s result [4] for \(\mathbb{Z}^2\); the first author and Schippa [18] also sketched the argument for \(\mathbb{Z}^3\). A related question is whether the results of Ionescu–Jerison [19] and Koch–Tataru [20] on the absence of embedded eigenvalues for Schrödinger operators on \(\mathbb{R}^d\) can be strengthened under randomisation. Finally, it would be interesting to investigate the \(L^p\) boundedness of wave operators for random potentials.
Discrete models. The discrete abstract estimate, Theorem 2, is designed for applications to lattice and graph operators. In particular, it is compatible with spectral projectors and resolvents for discrete Schrödinger operators, and may therefore be useful in weak-disorder problems on \(\mathbb{Z}^d\) and on more general finite-range graph models. This direction is close in spirit to the work of Schlag–Shubin–Wolff and Bourgain discussed above.
In [1], the first author proved \[\begin{align} \label{spectral32inclusion} \mathop{\mathrm{spec}}(-\Delta_{g}+V)\subset \bigcup_{k=0}^{\infty}D(\lambda_k^2,C\tilde{r}_k^{\rm det})\cup \{z\in\mathbb{C}: |z|^{\frac{1}{2}}(1+|z|)^{-\sigma(q)}\leq C \|V\|_{L^q(M)}\}, \end{align}\tag{6}\] where \(d/2<q< \infty\), \(\tilde{r}_k^{\rm det}:=\|V\|_{L^q(M)}(1+\lambda_k)^{2\sigma(q)},\) and \(\sigma(q)\) is given by 23 . Specializing to Anderson-type potentials 1 with \(\omega_j=1\) and applying this with \(q\) replaced by \(2q\), it follows that \[\{z\in\mathop{\mathrm{spec}}(H_{\omega,\lambda}) : |z|\leq \lambda^2\} \subset \bigcup_{\lambda_k\leq \lambda} D(\lambda_k^2,CKR_k')\cup \Omega^{\rm det},\] where \(d/2<2q< \infty\) and \[r_k^{\rm det}:=(1+\lambda_k)^{2\sigma(2q)}\lambda^{-\frac{d}{2q}}\|v(\lambda)\|_{\ell^{2q}},\] \[\Omega^{\rm det}:=\{z\in\mathbb{C}: |z|^{\frac{1}{2}}(1+|z|)^{-\sigma(2q)}\leq C\lambda^{-\frac{d}{2q}}\|v(\lambda)\|_{\ell^{2q}}\}.\]
For \(\lambda_k\ll \lambda\), one has \(\frac{r_k}{r_k^{\rm det}}\ll 1\); more precisely, \[\begin{align} \label{eq:32r95k47r95kdet} \frac{r_k}{r_k^{\rm det}} \leq \begin{cases} (1+\lambda_k)^{d/2}\lambda^{-d/2}(\log\lambda)^{11/2}, & 1\le q\le \frac{d+1}{4},\\[2mm] (1+\lambda_k)^{\frac{d-2}{2}+\frac{d+1}{4q}} \lambda^{-d/2}(\log\lambda)^{11/2}, & \frac{d+1}{4}\le q\le \frac{d+1}{2},\\[2mm] (1+\lambda_k)^{(d-1)/2}\lambda^{-d/2}(\log\lambda)^{11/2}, & \frac{d+1}{2}\le q\le \infty. \end{cases} \end{align}\tag{7}\]
We now compare \(\Omega\) and \(\Omega^{\rm det}\). From Theorem 1, it follows that for \(z\in\Omega\) and \(|z|\leq \lambda^{2-\varepsilon}\).
\[|z|^{1/2}(1+|z|)^{-\sigma(2q)} \leq C_N K \lambda^{-\frac{d}{2q}-c\varepsilon}(\ln\lambda)^{11/2}\|v(\lambda)\|_{\ell^{2q}}.\]
We also make the following remarks on Theorem 1.
For comparison of the compact manifold bounds with the Euclidean bounds of Frank [21], we refer to [1].
The trivial bound \[\mathop{\mathrm{spec}}(-\Delta_g+V) \subset \{z\in\mathbb{C}:\mathop{\mathrm{dist}}(z,\mathop{\mathrm{spec}}(-\Delta_g))\leq \|V\|_{L^\infty}\}\] implies that \(\mathop{\mathrm{spec}}(H_{\omega,\lambda})\) lies in the \(\|v(\lambda)\|_{\ell^\infty}\)-neighbourhood of \(\{\lambda_k^2\}_{k=0}^\infty\). Since \(\ell^q \subset \ell^\infty\), this gives the asymptotically (as \(k\to\infty\)) weaker bound \(r_k \leq \|v(\lambda)\|_{\ell^q}\).
The deterministic bounds of [1] are sharp in both the low-energy and high-energy regimes. The former follows from a scaling argument when \(M\) is a torus, while the latter follows from the optimality of Sogge’s spectral cluster bounds together with an operator-valued Rouché theorem (Gohberg–Sigal theory) when \(M\) is the sphere, or more generally a Zoll manifold. It would be interesting to know whether Theorem 1 is sharp up to the logarithmic loss. Due to the probabilistic nature of the bound, this seems to be a difficult question.
For additional background on deterministic and random eigenvalue bounds for Schrödinger operators with complex potentials, we refer to the survey [22], which also discusses fractional Laplacians.
Theorem 1 remains nontrivial even when \(v_j(\lambda) \in \mathbb{R}\), so that \(H_{\omega,\lambda}\) is self-adjoint; in this case, it gives a high-probability improvement over the trivial \(L^\infty\) spectral inclusion discussed in (3).
We collect some tools from probability theory that were used in [5], and that will also be important here to prove Theorems 2 and 4.
We recall that a (complex) scalar random variable \(X\) is called sub-Gaussian if it has finite sub-Gaussian norm, \[\begin{align} \|X\|_{\psi_2}=\inf\{t>0:\,\mathbf{E}\exp(|X|^2/t^2)\leq 2\}<\infty. \end{align}\] We will use the following properties of sub-Gaussian (e.g., centred normalised Gaussian or symmetric Bernoulli) random variables; see, e.g., [23].
Proposition 5 ([5]). Assume that \((X_j)_{j=1}^{N}\), \(N\geq 2\), is a finite collection of i.i.d. mean-zero sub-Gaussian random variables.
Then \(\sum_{j=1}^{N} X_j\) is also sub-Gaussian and \[\begin{align} \|\sum_{j=1}^{N} X_j\|_{\psi_2}^2 \lesssim \sum_{j=1}^{N}\|X_j\|_{\psi_2}^2. \end{align}\]
We have \[\begin{align} \label{eq:dudley} \mathbf{E}\max_{j\leq N}|X_j| \lesssim \sqrt{\log N}\max_{j\leq N}\|X_j\|_{\psi_2}. \end{align}\qquad{(1)}\]
We call ?? Dudley’s inequality, cf. [23].
We now consider tail bounds for (infinite-dimensional) vector-valued normalised Gaussian or Bernoulli random variables \(X\) which have finite \(\psi_2\)-norm. We have \((\mathbf{E}\|X\|^p)^{1/p}\asymp(\mathbf{E}\|X\|^q)^{1/q}\) for all \(p,q>0\) (cf. [24]), which, combined with [24] implies \[\begin{align} \mathbf{P}(\|X\| > t) \leq \exp\left(-\frac{ct^2}{(\mathbf{E}\|X\|)^2}\right) \end{align}\] for some \(c>0\). Thus, we have the following estimate.
Lemma 1 ([5]). Let \(X\) be a vector-valued Gaussian or Bernoulli random variable. If \(\mathbf{E}\|X\|\leq C\), then \[\begin{align} \mathbf{P}(\|X\| > KC) \leq \exp(-cK^2) \end{align}\] for any \(K>0\).
We are not aware of generalisations of this lemma to more general sub-Gaussian vector-valued random variables.
We state an abstract version of the chaining argument [5], whose proof was inspired by that of Bourgain [3]. The proof is the same.
Proposition 6. Let \(\mathcal{H}\) be a finite-dimensional Hilbert space, \(E\) a nonempty finite set and \(S:\mathcal{H}\to \ell^{\infty}(E)\) a linear operator. For every \(k\in \mathbb{N}\), there exist sets \(\mathcal{F}_k\subset \ell^{\infty}(E)\) with the following properties.
\(\log\#\mathcal{F}_k\lesssim 4^k \log \# E\).
For every \(\xi\in \mathcal{F}_k\), we have \[\begin{align} \label{eq:boundxiinfty} \|\xi\|_{\ell^{\infty}(E)}\lesssim 2^{-k}\|S\|_{\mathcal{H}\to \ell^{\infty}(E)}. \end{align}\qquad{(2)}\]
For each \(g\in \mathcal{H}\) with \(\|g\|_{\mathcal{H}}\leq 1\) there is a representation \[\begin{align} Sg = \sum_{k\in \mathbb{N}}\xi_k\quad \text{for some}\quad \xi_k\in\mathcal{F}_k. \end{align}\] Moreover, each \(\xi_k\) lies in the image of the unit ball in \(\mathcal{H}\) under \(S\).
In this section, we first prove the discrete abstract estimate (Theorem 2). We then introduce the appropriate setup for the continuous abstract estimate (Theorem 4) and prove a rigorous version of it (Theorem 12). The results of this section form the technical core of the paper and are of independent interest.
We now prove Theorem 2.
Proof. 1. We prove this for \(q<\infty\); the case \(q=\infty\) is analogous. We assume first that \(\mathcal{H}_1,\mathcal{H}_2\) are finite-dimensional. We apply Proposition 6 to both \(S_1\) and \(S_2\) and denote the resulting sets by \(\mathcal{F}_{1,k}\) and \(\mathcal{F}_{2,\ell}\), respectively. Then \[\begin{align} \|S_1^*v_{\omega}S_2\|_{\mathcal{H}_2\to \mathcal{H}_1} =\sup_{\|g_1\|_{\mathcal{H}_1}=\|g_2\|_{\mathcal{H}_2}=1}|\langle S_1g_1,v_{\omega}S_2g_2\rangle| \leq \sum_{k,\ell\in \mathbb{N}}\max_{(\xi_1,\xi_2)\in \mathcal{F}_{1,k}\times \mathcal{F}_{2,\ell}}|\langle\xi_1,v_{\omega}\xi_2\rangle|, \end{align}\] where \(\langle\cdot,\cdot\rangle\) is the inner product in \(\ell^2(E)\). By monotonicity of the expectation, \[\begin{align} \mathbf{E}\|S_1^*v_{\omega}S_2\|_{\mathcal{H}_2\to \mathcal{H}_1} \leq \sum_{k,\ell\in \mathbb{N}}\mathbf{E}\max_{(\xi_1,\xi_2)\in \mathcal{F}_{1,k}\times \mathcal{F}_{2,\ell}}|\langle\xi_1,v_{\omega}\xi_2\rangle|. \end{align}\] Note that, by definition of \(v_{\omega}\) in 4 and Proposition 5 (i), we have \[\begin{align} \label{eq:langle95xi195vomega95xi295rangle} \|\langle\xi_1,v_{\omega}\xi_2\rangle\|_{\psi_2} =\Big\|\sum_{j\in J}\omega_j\langle \xi_1,v\xi_2\rangle_{\ell^2(E_j)}\Big\|_{\psi_2}\lesssim \Big(\sum_{j\in J}|\langle \xi_1,v_j\xi_2\rangle_{\ell^2(E_j)}|^2\Big)^{1/2}, \end{align}\tag{8}\] where we have also used \(\sup_{j\in J}\|\omega_j\|_{\psi_2}<\infty\). By Proposition 6 (a), the cardinality \(N\) of the set \(\mathcal{F}_{1,k}\times \mathcal{F}_{2,\ell}\) satisfies \[\begin{align} \label{eq:logN} \log N= \log\# (\mathcal{F}_{1,k}\times \mathcal{F}_{2,\ell}) \lesssim (4^k+4^{\ell})\, \log \# E. \end{align}\tag{9}\] By Dudley’s inequality ?? , 8 , 9 , disjointness of the sets \(E_j\), and Hölder’s inequality for sums, ?? , \[\begin{align} & \mathbf{E}\max_{(\xi_1,\xi_2)\in \mathcal{F}_{1,k}\times \mathcal{F}_{2,\ell}}| \langle\xi_1,v_{\omega}\xi_2\rangle| \lesssim \sqrt{\log N}\Big(\sum_{j\in J}|\langle \xi_1,v_j\xi_2\rangle_{\ell^2(E_j)}|^2\Big)^{\frac{1}{2}} \\ & \leq \max_{(\xi_1,\xi_2)\in \mathcal{F}_{1,k}\times \mathcal{F}_{2,\ell}} \sqrt{\log N}\Big(\sum_{j\in J}\| \xi_1\|_{\ell^{p'}(E_j)}^2\|v\xi_2\|_{\ell^{p}(E_j)}^2\Big)^{\frac{1}{2}} \\ & \leq \max_{(\xi_1,\xi_2)\in \mathcal{F}_{1,k}\times \mathcal{F}_{2,\ell}} \sqrt{\log N}\Big(\sum_{j\in J}\| \xi_1\|_{\ell^{p'}(E_j)}^2\|\xi_2\|_{\ell^{\infty}(E_j)}^2\|v\|_{\ell^{p}(E_j)}^2\Big)^{\frac{1}{2}} \\ & \leq \max_{(\xi_1,\xi_2)\in \mathcal{F}_{1,k}\times \mathcal{F}_{2,\ell}} \sqrt{\log N}\|\xi_1\|_{\ell^{p'}(E)}\|\xi_2\|_{\ell^{\infty}(E)}\Big(\sum_{j\in J}\|v\|_{\ell^{p}(E_j)}^{2q}\Big)^{\frac{1}{2q}} \\ & \lesssim \sqrt{\log \# E}(2^k+2^{\ell})2^{-\ell}\|S_1\|_{\mathcal{H}_1\to \ell^{p'}(E)}\|S_2\|_{\mathcal{H}_2\to \ell^{\infty}(E)}\Big(\sum_{j\in J}\|v\|_{\ell^p(E_j)}^{2q}\Big)^{\frac{1}{2q}}. \end{align}\] Interchanging the roles of \(\xi_1,\xi_2\), we can replace \((2^k+2^{\ell})2^{-\ell}\) by \((2^k+2^{\ell})2^{-k}\), so that \[\begin{align} & \mathbf{E}\max_{(\xi_1,\xi_2)\in \mathcal{F}_{1,k}\times \mathcal{F}_{2,\ell}}| \langle\xi_1,v_{\omega}\xi_2\rangle| \\ & \quad \lesssim \sqrt{\log \# E}\|S_1\|_{\mathcal{H}_1\to \ell^{p'}(E)}\|S_2\|_{\mathcal{H}_2\to \ell^{\infty}(E)}\Big(\sum_{j\in J}\|v\|_{\ell^p(E_j)}^{2q}\Big)^{\frac{1}{2q}} =: A. \end{align}\] Combining this with the trivial bound \[\begin{align} \max_{(\xi_1,\xi_2)\in \mathcal{F}_{1,k}\times \mathcal{F}_{2,\ell}}|\langle\xi_1,v_{\omega}\xi_2\rangle|\lesssim 2^{-k-\ell}\|S_1\|_{\mathcal{H}_1\to \ell^{\infty}(E)}\|S_2\|_{\mathcal{H}_2\to \ell^{\infty}(E)}\|v(\lambda)\|_{\ell^1(E)}=:2^{-k-\ell}B, \end{align}\] we get \[\begin{align} \mathbf{E}\|S_1^*v_{\omega}S_2\|_{\mathcal{H}_2\to \mathcal{H}_1} &\lesssim \sum_{k,\ell\in \mathbb{N}}\min(A,2^{-k-\ell}B) \lesssim A(1+(\log B/A)^2). \end{align}\] By Hölder’s inequality and the inclusion \(\ell^{s}(E)\subset\ell^{r}(E)\), for \(s\leq r\), we have \[\begin{align} B \leq \|S_1\|_{\mathcal{H}_1\to \ell^{p'}(E)}\|S_2\|_{\mathcal{H}_2\to \ell^{\infty}(E)} (\# E)^{\frac{1}{p'}}\Big(\sum_{j\in J}\|v\|_{\ell^p(E_j)}^{2q}\Big)^{\frac{1}{2q}} \end{align}\] so that \(\log(B/A)\leq \frac{1}{p'}\log\#E\). This concludes the proof for the case where \(\mathcal{H}_1,\mathcal{H}_2\) are finite-dimensional.
2. In the case where \(\mathcal{H}_1,\mathcal{H}_2\) are infinite-dimensional and separable, let \[\begin{align} \mathcal{H}_i^{(1)}\subset \mathcal{H}_i^{(2)}\subset\ldots\subset \mathcal{H}_i,\quad i=1,2, \end{align}\] be an increasing sequence of finite-dimensional subspaces whose union is dense in \(\mathcal{H}_i\). Let \(P_i^{(k)}\) be the orthogonal projections onto \(\mathcal{H}_i^{(k)}\). For each \(\omega=\{\omega_j\}_{j\in J}\) and \(k\in\mathbb{N}\) we set \[\begin{align} N_k(\omega):=\|P_1^{(k)}S_1^*v_{\omega}S_2P_2^{(k)}\|_{\mathcal{H}_2^{(k)}\to \mathcal{H}_1^{(k)}}. \end{align}\] For fixed \(\omega\), the numbers \(N_k(\omega)\) are nondecreasing in \(k\) and \[\begin{align} \lim_{k\to\infty}N_k(\omega)=\sup_{k\in\mathbb{N}} N_k(\omega)=\|S_1^*v_{\omega}S_2\|_{\mathcal{H}_2\to \mathcal{H}_1}. \end{align}\] By the monotone convergence theorem, \[\begin{align} \mathbf{E}\|S_1^*v_{\omega}S_2\|_{\mathcal{H}_2\to \mathcal{H}_1}=\lim_{k\to\infty}\mathbf{E}N_k(\omega). \end{align}\] Since \(\mathbf{E}N_k(\omega)\) is uniformly bounded in \(k\) by the right-hand side of 5 , the general case is proved. ◻
We now introduce the setup for the continuous model. The main result of this subsection is Theorem 12.
Definition 1. Let \((X,\rho,\mu)\) be a metric measure space and \(d>0\). We say that \(X\) is Ahlfors \(d\)-regular if \(\mu\) is a Radon measure and there is \(C_A>0\) such that \[\begin{align} \label{def:32alpha32regular32measure} C_A^{-1}r^{d} \leq \mu(B(x,r)) \leq C_A r^{d} \end{align}\tag{10}\] for all balls \(B(x,r)\subset X\) with \(r\in (0,1)\). We say \(X\) is upper/lower Ahlfors \(d\)-regular if the upper/lower bound in 10 holds.
Clearly, if \(X\) is upper Ahlfors \(d\)-regular, then for all \(p\in [1,\infty]\), \[\begin{align} \label{eq:32abstract32Hoelder} \|f\|_{L^{p}(B(x,r))}\leq C_A^{1/p}r^{d/p}\|f\|_{L^{\infty}(B(x,r))}. \end{align}\tag{11}\]
Note furthermore that if \(X\) is Ahlfors \(d\)-regular, then \(X\) is \(\sigma\)-finite.
Definition 2. Let \((X,\rho,\mu)\) be a metric measure space and \(r>0\). We say that a collection of measurable functions \(\mathcal{F}\) on \(X\) satisfies the local constancy property at scale \(r\) if there exists a constant \(C_{\rm loc}\) such that for all \(f\in\mathcal{F}\) and all balls \(B\subset X\) with radii at most \(2r\), we have \[\begin{align} \label{eq4632local32constancy32property} \|f\|_{L^{\infty}(B)}\leq C_{\rm loc}\int_{X}|f|w_B\mathrm{d}\mu, \end{align}\tag{12}\] where the weights \(w_B\) are given by \[\begin{align} w_B(x):=\mu(B)^{-1}(1+r^{-1}\mathop{\mathrm{dist}}(x,B))^{-100d}. \end{align}\]
Remark 7. (1) By definition, if \(\mathcal{F}\) satisfies the local constancy property at scale \(r\), then the same is true for all scales \(\leq r\).
(2) As an example, for \(X=\mathbb{R}^d\), \(\rho\) the Euclidean metric, and \(\mu\) the \(d\)-dimensional Lebesgue measure, the family of functions \(\{f\in \mathcal{S}(\mathbb{R}^d) \,|\, \exists a\in\mathbb{R}^d,b>0:\,\mathop{\mathrm{supp}}\hat{f} \subseteq B_a(b/r)\}\) satisfies the local constancy property at scale \(r\).
Definition 3. Let \((X,\rho)\) be a metric space and \(r>0\). We call \(\Lambda\subset X\) an \(r\)-separated set if \[\begin{align} \rho(x,x')>r\quadfor all distinct x,x'\in \Lambda. \end{align}\] We call \(\Lambda\) maximal \(r\)-separated if \(\Lambda\) is \(r\)-separated and if for every \(x\in X\setminus\Lambda\) there exists \(x'\in\Lambda\) such that \(\rho(x,x')\leq r\).
Definition 4. Let \((X,\rho,\mu)\) be a metric measure space and \(r>0\). We say that a collection of measurable functions \(\mathcal{F}\) on \(X\) satisfies the weak local constancy property at scale \(r\) if for every \(s\in [1,\infty]\) there exists a constant \(C_s\) such that for all \(r\)-separated sets \(\Lambda\subset X\) and \(f\in \mathcal{F}\), we have \[\begin{align} \label{eq:abstract32comparison32ellp32Lp} \Big(\sum_{x\in\Lambda}|f(x)|^s\Big)^{1/s}\leq C_sr^{-d/s}\Big(\int_X|f|^s \mathrm{d}\mu\Big)^{1/s}. \end{align}\tag{13}\]
Proposition 8. Let \(X\) be Ahlfors \(d\)-regular, \(r>0\), and \(\Lambda\subset X\) be an \(r\)-separated set. Let \(T:\mathcal{H}\to L^{p}(X)\) be a bounded linear operator for some \(p\in [1,\infty]\). Assume \(\mathop{\rm Ran}T\) has the weak local constancy property at scale \(r\). Then for \(S_{\Lambda}:\mathcal{H}\to \ell^{p}(\Lambda)\) defined by \(S_{\Lambda}f(x):=Tf(x)\), \(x\in\Lambda\), we have \[\begin{align} \|S_{\Lambda}\|_{\mathcal{H}\to \ell^{p}(\Lambda)}\leq C_pr^{-d/p}\|T\|_{\mathcal{H}\to L^{p}(X)}. \end{align}\]
Proof. This follows immediately from the definition. ◻
Proposition 9. The local constancy property (at scale \(r\)) implies the weak local constancy property (at scale \(r\)).
Proof. Assume that \(\mathcal{F}\) has the local constancy property at scale \(r\). Let \(\Lambda\subset X\) be an \(r\)-separated set and \(f\in \mathcal{F}\). By 12 and Hölder’s inequality, \[\begin{align} \sum_{x\in\Lambda}|f(x)|^s & \leq \sum_{x\in\Lambda}\|f\|_{L^{\infty}(B(x,r))}^s \leq C_{\rm loc}^s \sum_{x\in\Lambda}\Big(\int_{X}|f|w_{B(x,r)}\mathrm{d}\mu\Big)^s \\ & \leq C_{\rm loc}^s \sum_{x\in\Lambda} \Big(\int_{X}|f|^sw_{B(x,r)}\mathrm{d}\mu\Big)\Big(\int_X w_{B(x,r)}\mathrm{d}\mu\Big)^{s-1} \\ &\leq C_{\rm loc}^s (C_A2^{2d+2})^{s-1}\sum_{x\in\Lambda}\int_{X}|f|^sw_{B(x,r)}\mathrm{d}\mu, \end{align}\] where we have also used both the upper and lower bounds in 10 to estimate \[\begin{align} \int_X w_B\mathrm{d}\mu & \leq r^{-d}\big(\mu(\{\mathop{\mathrm{dist}}(y,B)\leq r\})+\sum_{k=0}^{\infty}2^{-100dk}\mu(\{r2^k\leq\mathop{\mathrm{dist}}(y,B)\leq r2^{k+1}\})\big) \\ & \leq C_A\big(2^d+\sum_{k=0}^{\infty}(1+2^{k+1})^d2^{-100dk}\big)\leq 2^{2d+2}C_A. \end{align}\] Since Ahlfors regularity implies \(\sigma\)-finiteness of \(X\), Tonelli’s theorem and a similar calculation as above yield \[\begin{align} \sum_{x\in\Lambda}\int_{X}|f|^sw_{B(x,r)}\mathrm{d}\mu & = \int_{X}|f|^s\sum_{x\in\Lambda}w_{B(x,r)} \mathrm{d}\mu\\ & \leq r^{-d}\int_{X}|f|^s\sum_{x\in\Lambda}\big(\mathbf{1}_{B(x,2r)}+\sum_{k=0}^{\infty}2^{-100dk}\mathbf{1}_{B(x,2^{k+2}r)}\big)\mathrm{d}\mu \\ & \leq C_A^2\big(5^d+\sum_{k=0}^{\infty}2^{-100dk}2^{(k+4)d}\big)r^{-d}\int_X |f|^s\mathrm{d}\mu\\ & \leq 2^{4d+2}C_A^2r^{-d}\int_X |f|^s\mathrm{d}\mu, \end{align}\] from which 13 follows. In the penultimate inequality, we have used that for \(M\geq 1\), \[\begin{align} \label{eq:32abstract32overlap32bound} \sum_{x\in\Lambda}\mathbf{1}_{B(x,Mr)}\leq C_A^2(2M+1)^d. \end{align}\tag{14}\] Indeed, for fixed \(y\in X\), consider the set \[\begin{align} \Lambda_y := \{x\in \Lambda:y\in B(x,Mr)\}. \end{align}\] Since \(\Lambda\) is \(r\)-separated, the balls \(\{B(x,r/2)\}_{x\in\Lambda}\) are disjoint. By the triangle inequality, \[\begin{align} \bigcup_{x\in\Lambda_y}B(x,r/2)\subset B(y,(M+1/2)r), \end{align}\] so that, by 10 (both upper and lower bounds), \[\begin{align} C_A^{-1}2^{-d}r^d\#\Lambda_y \leq \sum_{x\in\Lambda_y}\mu(B(x,r/2)) \leq \mu(B(y,(M+1/2)r)) \leq C_A(M+1/2)^dr^d. \end{align}\] Since \(y\) was arbitrary, this proves 14 . ◻
The following development merely uses the weak local constancy property. In applications, the local constancy property can be verified rather directly; see Appendix 5 below in the case of the range of the spectral projector for \(-\Delta_g\).
Definition 5. Given a maximal \(r\)-separated subset \(\Lambda\) of a metric space \((X,\rho)\), we denote by \(\mathcal{V}_x\) the Voronoi cell associated to \(x\in\Lambda\), \[\begin{align} \mathcal{V}_x := \{y\in X:\rho(x,y)<\rho(x',y) for all x'\in\Lambda\setminus\{x\}\}. \end{align}\]
All Voronoi cells are mutually disjoint and, if \(X\) is Ahlfors \(d\)-regular, \(\{\mathcal{V}_x\}_{x\in\Lambda}\) covers \(X\) up to a set of measure zero. Thus, for any integrable function \(f:X\to \mathbb{C}\), \[\begin{align} \int_X f\mathrm{d}\mu=\sum_{x\in\Lambda}\int_{\mathcal{V}_x}f\mathrm{d}\mu. \end{align}\]
Lemma 2. Let \((X,\rho)\) be a metric space, \(r>0\), and \(\Lambda\subseteq X\) be a maximal \(r\)-separated subset. Then \(B(x,r/2)\subset \mathcal{V}_x\subset B(x,2r)\) for all \(x\in\Lambda\).
Proof. \(B(x,r/2)\subset \mathcal{V}_x\): Let \(y\in B(x,r/2)\) and \(x'\in\Lambda\setminus\{x\}\). Then \[\begin{align} \rho(x',y)\geq \rho(x',x)-\rho(x,y)\geq r-\frac{r}{2}=\frac{r}{2}, \end{align}\] which implies \(\rho(x,y)< r/2 \leq \rho(x',y)\) and hence \(y \in\mathcal{V}_{x}\).
\(\mathcal{V}_x\subset B(x,2r)\): Let \(y\in \mathcal{V}_x\). Without loss of generality, we may assume \(y\neq x\), otherwise there is nothing to prove. Assume for contradiction that \(\rho(x,y)\geq 2r\). Since \(\Lambda\) is an \(r\)-net (see e.g. [23]), there exists \(x'\in\Lambda\setminus\{x\}\) such that \(\rho(x',y)\leq r\). But this implies \[\begin{align} r\geq \rho(x',y)>\rho(x,y)\geq 2r, \end{align}\] a contradiction. ◻
Definition 6. Let \((X,\rho,\mu)\) be Ahlfors \(d\)-regular, \(r>0\), and let \(\Lambda\subset X\) be an \(r\)-separated set.
We say that the pair \((X,\Lambda)\) satisfies the uniform reparametrisation property at scale \(r\) if there exists a finite measure space \((Y,\nu)\) with \(\nu(Y)\leq C_Yr^d\) and, for each \(x\in\Lambda\), a measurable bijection \(\varphi_x : X\supset B(x,2r) \to Y\) with measurable inverse, such that the pushforward measures \(\mu_x:=(\varphi_x)_*(\mu|_{B(x,2r)})\) are absolutely continuous with respect to \(\nu\), and the Radon–Nikodym derivatives \(w_x = \frac{d\mu_x}{d\nu}\) satisfy the uniform bounds \[\begin{align} \label{eq:32uniform32bounds32on32Radon8211Nikodym32derivatives} 0\leq w_x(y) \leq C_{\rm RN}\qquad \text{for all x\in\Lambda and \nu-a.e.\;y\in Y}. \end{align}\tag{15}\]
We say that \((X,\Lambda)\) satisfies the strong uniform reparametrisation property at scale \(r\) if there exists a separable metric measure space \((Y,\delta,\nu)\) with finite Ahlfors \(d\)-regular measure \(\nu\) satisfying \(C_Y^{-1}r^d \leq \nu(Y) \leq C_Yr^d\) and, for each \(x\in\Lambda\), a bi-measurable bijection \(\varphi_x : X\supset B(x,2r) \to Y\) such that \(\varphi_x^{-1}\) is Lipschitz with uniform Lipschitz bounds in \(x\), i.e., there exists a constant \(C_{L}\) such that \[\begin{align} \rho(\varphi_x^{-1}(y),\varphi_x^{-1}(z))\leq C_{L} \delta(y,z) \quadfor all y,z\in Y. \end{align}\]
We say that \(X\) satisfies the (strong) uniform reparametrisation property at scales \(\geq r\) if for every \(r'\geq r\) and every \(r'\)-separated set \(\Lambda\subset X\), the pair \((X,\Lambda)\) satisfies the (strong) uniform reparametrisation property at scale \(r'\).
Remark 10. (1) In the uniform reparametrisation property, the upper bound \(\nu(Y)\leq C_Y r^d\) is in fact complemented by a lower bound of the same order. Indeed, by Ahlfors \(d\)-regularity of \(X\), for each \(x\in\Lambda\), \[\mu(B(x,2r)) \geq C_A^{-1} (2r)^d,\] while the Radon–Nikodym bound gives \[\mu(B(x,2r)) = \int_Y w_x\,d\nu \leq C_{\rm RN}\nu(Y).\] Hence \[\nu(Y)\geq 2^dC_A^{-1}C_{\rm RN}^{-1}r^d.\]
(2) The uniform reparametrisation property is a purely measure-theoretic condition: it provides a common measurable model for the balls \(B(x,2r)\), together with uniform control of the associated Radon–Nikodym densities. The strong uniform reparametrisation property is a strengthening, in which the common model is a metric measure space and the Radon–Nikodym condition is replaced by a uniform Lipschitz condition. The main abstract theorem in this subsection (Theorem 12) only uses the weaker assumption. However, in most applications, \(X\) will satisfy the stronger assumption (see Remark 11). Moreover, the strong assumption is usually easier to verify. This is the reason we included it in the above definition.
Remark 11 (Examples). The strong uniform reparametrisation properties are satisfied in a number of standard situations.
Let \(X=\mathbb{R}^d\), \(\rho\) be the Euclidean metric, and \(\mu\) be the \(d\)-dimensional Lebesgue measure \(\mathcal{L}^d\). For any \(r>0\) and any \(r\)-separated set \(\Lambda\subset\mathbb{R}^d\), one may take \[Y=B(0,2r),\qquad \nu=\mathcal{L}^d|_{B(0,2r)}, \qquad \varphi_x(y)=y-x .\] Then \(\varphi_x:B(x,2r)\to Y\) is an isometry and \((\varphi_x)_*(\mathcal{L}^d|_{B(x,2r)})=\nu\). Thus \(w_x\equiv 1\), and the strong uniform reparametrisation property holds at all scales. The same argument applies to finite-dimensional normed spaces with Lebesgue measure.
More generally, if \(X\) is a Carnot group equipped with a homogeneous distance and Haar measure, then left translation gives the required parametrisations. Namely, for \(Y=B(e,2r)\) and \[\varphi_x(y)=x^{-1}y ,\] the maps \(\varphi_x:B(x,2r)\to Y\) are isometries and preserve Haar measure. Hence again \(w_x\equiv 1\), and the strong uniform reparametrisation property holds at all scales.
Let \(M\) be a (not necessarily closed) \(d\)-dimensional Riemannian manifold of bounded geometry, equipped with its geodesic distance and Riemannian volume measure. At sufficiently small scales, one may use normal coordinates centred at \(x\) to show that \(M\) satisfies the strong uniform reparametrisation property locally. In particular, any closed manifold \(M\) with smooth Riemannian metric (as considered in this article) satisfies the strong uniform reparametrisation property (see the proof of Proposition 13 for details).
Lemma 3. The strong uniform reparametrisation property implies the uniform reparametrisation property.
Proof. 1. By the Lipschitz assumption, for any ball \(B(z,t)\) in \(Y\), we see \(\varphi_x^{-1}(B(z,t))\subset B(\varphi_x^{-1}(z),C_Lt)\).
Thus, by definition of the pushforward measure, the Ahlfors regularity of \(\mu\) and lower Ahlfors regularity of \(\nu\), \[\begin{align} \mu_x(B(z,t)) =
\mu(\varphi_x^{-1}(B(z,t))) \leq \mu(B(\varphi_x^{-1}(z),C_Lt)) \leq C_A(C_Lt)^d \leq C_A C_Y^2 C_L^d\nu(B(z,t)).
\end{align}\] 2. Let \(U\subset Y\) be an open set, and for each \(y\in U\) choose \(\delta_y\in (0,1)\) such that \(B(y,\delta_y)\subset U\). By the \(5r\)-covering theorem (see, e.g., [25]), there
exists a finite or countable sequence of disjoint balls \(B_i=B(y_i,\delta_{y_i})\) such that the collection \(\{5B_i\}\) covers \(U\). Thus, by the first
step, \[\begin{align} \mu_x(U)\leq \sum_i\mu_x(5B_i) \leq C_A C_Y^2 C_L^d \sum_i\nu(5B_i) \leq C_A^2 C_Y^2 (5C_L)^d \sum_i\nu(B_i) \leq C_A^2 C_Y^2 (5C_L)^d\nu(U).
\end{align}\] 3. Let \(E\subset Y\) be a measurable set with \(\nu(E)=0\), and let \(\varepsilon>0\). By outer regularity of \(\nu\), there exists an open set \(U\subset Y\) such that \(E\subset U\) and \(\nu(U)\leq \varepsilon\). By the second step, it
follows that \(\mu_x(U) \leq C_A^2 C_Y^2 (5C_L)^d\varepsilon\). Since \(\varepsilon\) was arbitrary, we have \(\mu_x(E)=0\), so that \(\mu_x\ll\nu\).
. By the Radon-Nikodym theorem and the estimate in the first step, for any ball \(B\subset Y\), we have \[\begin{align} \mu_x(B) = \int_Bw_x\mathrm{d}\nu \leq C_A C_Y^2 C_L^d\nu(B) \implies \int_B
(w_x- C_A C_Y^2 C_L^d)\mathrm{d}\nu\leq 0.
\end{align}\] Consider the sets \[\begin{align} E:=\{y\in Y:w_x- C_A C_Y^2 C_L^d>0\},\quad E_n:=\{y\in Y:w_x- C_A C_Y^2 C_L^d \geq 1/n\},
\end{align}\] and \(E_{n,k}:=E_n\cap B(y_0,k)\) for some fixed \(y_0\in Y\). Then \(E_{n,k}\nearrow_{k\to\infty}
E_n\) and \(E_n\nearrow_{n\to\infty} E\). Since \[\begin{align} 0 \leq \nu(E_{n,k}) \leq n\int_{B(y_0,k)} (w_x- C_A C_Y^2 C_L^d)\mathrm{d}\nu\leq 0,
\end{align}\] it follows that \(\nu(E_n)=\lim_{k\to\infty}\nu(E_{n,k})=0\) and \(\nu(E)=\lim_{n\to\infty}\nu(E_n)=0\). Hence, \(w_x \leq C_A C_Y^2
C_L^d\) almost everywhere. ◻
Lemma 4. Assume that \((X,\Lambda)\) satisfies the uniform reparametrisation property at scale \(r\). Then for every \(\mu\)-integrable function \(f:X\to\mathbb{C}\), \[\begin{align} \label{eq:32change32of32variables32formula} \sum_{x\in\Lambda} \int_{B(x,2r)} f \, d\mu = \int_{Y} \sum_{x\in\Lambda} f\circ \varphi_x^{-1} \, w_x\, \mathrm{d}\nu. \end{align}\tag{16}\]
Proof. This follows from the change-of-variables formula, the Radon–Nikodym theorem and Fubini’s theorem. More precisely, by change-of-variables (definition of pushforward) and the Radon–Nikodym theorem, we have the identity \[\int_{B(x,2r)} f\,\mathrm{d}\mu = \int_{Y} f\circ \varphi_x^{-1} \, w_x\, \mathrm{d}\nu.\] To justify Fubini’s theorem, we apply the same argument to \(|f|\), but in the reverse direction, \[\sum_{x\in\Lambda}\int_{Y} |f|\circ \varphi_x^{-1} \, w_x\, \mathrm{d}\nu = \sum_{x\in\Lambda}\int_{B(x,2r)} |f|\,\mathrm{d}\mu\leq C_A^2 5^d\|f\|_{L^1(X)},\] where we used the finite overlap bound 14 . ◻
Corollary 1. Assume that \((X,\Lambda)\) satisfies the uniform reparametrisation property at scale \(r\) and that \(\Lambda\) is maximal \(r\)-separated. Then for every \(\mu\)-integrable function \(f:X\to\mathbb{C}\), \[\begin{align} \label{eq:32change32of32variables32formula322} \int_X f\mathrm{d}\mu=\sum_{x\in \Lambda}\int_{\mathcal{V}_x}f\mathrm{d}\mu = \sum_{x\in \Lambda}\int_{B(x,2r)}f\mathbf{1}_{\mathcal{V}_x}\mathrm{d}\mu = \int_{Y} \sum_{x\in\Lambda} (f \mathbf{1}_{\mathcal{V}_x})\circ \varphi_x^{-1} \, w_x\, d\nu. \end{align}\tag{17}\]
Proof. The first equality follows from maximality, the second from \(\mathcal{V}_x\subset B(x,2r)\) (see Lemma 2), and the third from Lemma 4. ◻
Theorem 12. Let \(X\) be an Ahlfors \(d\)-regular metric measure space, let \(2\leq q\leq\infty\) and \(1\leq p\leq 2\leq p'\leq \infty\) satisfy \(\frac{1}{q}=\frac{1}{p}-\frac{1}{p'}\). Let \[T_i:\mathcal{H}_i\to L^\infty(X)\cap L^{p'}(X), \qquad i=1,2,\] be bounded linear operators, where \(\mathcal{H}_i\) are complex separable Hilbert spaces. Assume that \(\mathop{\rm Ran}(T_1)\cup\mathop{\rm Ran}(T_2)\) has the weak local constancy property at scale \(r\). Let \(\Lambda\) be a finite subset of \(X\), let \(V\in L^{1}_{\rm loc}(X)\) be a measurable complex-valued function, and let \[\begin{align} \label{eq:defrandomisation} V_{\omega}(y):=\omega_xV(y)\quadforx\in \Lambda,y\in \mathcal{V}_x, \end{align}\tag{18}\] where \(\omega_x\) are i.i.d. symmetric Bernoulli or centred normalised Gaussian random variables.
Assume that \(\Lambda\) is maximal \(r\)-separated and that \((X,\Lambda)\) has the uniform reparametrisation property at scale \(r\). Then we have \[\begin{align} \mathbf{E}\|T_1^*V_{\omega}T_2\|_{\mathcal{H}_2\to \mathcal{H}_1} &\lesssim r^{d/p} (\log\#\Lambda)^{5/2}\|T_1\|_{\mathcal{H}_1\to L^{p'}(X)}\|T_2\|_{\mathcal{H}_2\to L^{\infty}(X)}\\ &\qquad \times\Big(\sum_{x\in\Lambda}\|V\|_{L^{\infty}(B(x,2r))}^{2q}\Big)^{1/(2q)}. \end{align}\]
Assume that \(X\) has the uniform reparametrisation property at scales \(\geq r\) and that \(\Lambda\) is maximal \(r'\)-separated for some \(r'\geq r\). Then we have \[\begin{align} \mathbf{E}\|T_1^*V_{\omega}T_2\|_{\mathcal{H}_2\to \mathcal{H}_1} &\lesssim r^{d/p} (r'/r)^{\frac{d}{p}-\frac{d}{2q}}(\log\#\Lambda)^{5/2}\|T_1\|_{\mathcal{H}_1\to L^{p'}(X)}\|T_2\|_{\mathcal{H}_2\to L^{\infty}(X)}\\ &\qquad \times\Big(\sum_{x\in \Lambda}\|V\|_{L^{\infty}(B(x,2r'))}^{2q}\Big)^{1/2q}. \end{align}\]
The above inequalities are understood to hold with the obvious modification for \(q=\infty\).
Proof. (1) For fixed \(y\in Y\), we define \[\begin{align} \label{eq:32def32Siy} S_{i,y}:\mathcal{H}_i\to \ell^{\infty}(\Lambda),\quad (S_{i,y}g)(x):=(T_ig)(\varphi_x^{-1}(y)), \end{align}\tag{19}\] and \(v_{\omega,y}(x):=(V_{\omega}\mathbf{1}_{\mathcal{V}_x})(\varphi_x^{-1}(y))w_x(y)\) for \(x\in\Lambda\). Let \(g_1\in \mathcal{H}_1,g_2\in \mathcal{H}_2\). By 17 , we can write \[\begin{align} \int_X \overline{T_1g_1}T_2g_2V_{\omega}\mathrm{d}\mu = \int_Y\sum_{x\in\Lambda}\overline{(S_{1,y}g_1)(x)}(S_{2,y}g_2)(x)v_{\omega,y}(x)\mathrm{d}\nu(y). \end{align}\] Taking the modulus and then the supremum over all unit vectors \(g_1,g_2\), it follows that \[\begin{align} \|T_1^*V_{\omega}T_2\|_{\mathcal{H}_2\to \mathcal{H}_1}\leq \int_Y \|S_{1,y}^*v_{\omega,y}S_{2,y}\|_{\mathcal{H}_2\to \mathcal{H}_1} \mathrm{d}\nu(y), \end{align}\] and taking expectation, \[\begin{align} \mathbf{E}\|T_1^*V_{\omega}T_2\|_{\mathcal{H}_2\to \mathcal{H}_1}\leq \int_Y \mathbf{E}\|S_{1,y}^*v_{\omega,y}S_{2,y}\|_{\mathcal{H}_2\to \mathcal{H}_1} \mathrm{d}\nu(y). \end{align}\] The claimed bound now follows from Theorem 2, Proposition 8 and the assumption \(\nu(Y)\leq C_Yr^d\). More precisely, we have \[\begin{align} & \int_Y \mathbf{E}\|S_{1,y}^*v_{\omega,y}S_{2,y}\|_{\mathcal{H}_2\to \mathcal{H}_1} \mathrm{d}\nu(y) \lesssim (\log\#\Lambda)^{5/2}\sup_{y\in Y}\|S_{1,y}\|_{\mathcal{H}_1\to \ell^{p'}(\Lambda)} \|S_{2,y}\|_{\mathcal{H}_2\to \ell^{\infty}(\Lambda)} \\ & \times \int_Y \Big(\sum_{x\in\Lambda}|V\mathbf{1}_{\mathcal{V}_x}(\varphi_x^{-1}(y))w_x(y)|^{2q}\Big)^{1/2q}\mathrm{d}\nu(y) \\ & \lesssim r^{-d/p'} r^{d}(\log\#\Lambda)^{5/2}\|T_1\|_{\mathcal{H}_1\to L^{p'}(X)}\|T_2\|_{\mathcal{H}_2\to L^{\infty}(X)}\Big(\sum_{x\in\Lambda}\|V\|_{L^{\infty}(B(x,2r))}^{2q}\Big)^{1/2q}. \end{align}\]
(2) For each \(x\in\Lambda\), let \(\Lambda_r(x)\) be a maximal \(r\)-separated subset of \(\mathcal{V}_x\). Let \[\Lambda_r:=\bigcup_{x\in\Lambda}\Lambda_r(x).\] By assumption, \((X,\Lambda_r)\) satisfies the uniform reparametrisation property. For fixed \(y\in Y\), define \(S_{i,y}\) as in 19 and \(v_{\omega,y}(x):=(V_{\omega}\mathbf{1}_{\mathcal{V}_r(x)})(\varphi_x^{-1}(y))w_x(y)\) for \(x\in\Lambda_r\), where \[\mathcal{V}_r(x) := \{y\in X:\rho(x,y)<\rho(x',y) for all x'\in\Lambda_r\setminus\{x\}\}.\] The remainder of the proof is analogous to part (1); in addition, one uses Hölder and the bound \(\#\Lambda_r(x)\lesssim (r'/r)^d\) (see Remark 3). ◻
We start this section with an application of Theorem 12 to a dual form of Sogge’s spectral projection bounds. The result will not be used directly to prove Theorem 1 but is perhaps of independent interest. We then turn to resolvent estimates, which are similar to those in [1], but for the square root of the resolvent instead of the full resolvent. We then use these estimates, together with Theorem 12, to prove a Birman–Schwinger bound involving Anderson-type potentials. The proof of Theorem 1 is then a straightforward consequence of this bound.
Consider the spectral projection operators \[\begin{align} \Pi_{\lambda} := \mathbf{1}\left(P \in [\lambda,\lambda+1]\right):L^2(M)\to L^{p'}(M) \end{align}\] where \(P=\sqrt{-\Delta_g}\). By Sogge’s bounds [26], \[\begin{align} \label{eq:32Sogge32bound32large32p39} \|\Pi_{\lambda} \|_{L^2(M)\to L^{p'}(M)}\lesssim \lambda^{\nu(p')}. \end{align}\tag{20}\] where \[\begin{align} \label{def4632nu40p3941} \nu(p') := \begin{cases} \frac{d-1}{2}(\frac{1}{2}-\frac{1}{p'})\quad & 2\leq p'\leq \frac{2(d+1)}{d-1},\\ d(\frac{1}{2}-\frac{1}{p'})-\frac{1}{2}\quad & \frac{2(d+1)}{d-1}\leq p'\leq \infty. \end{cases} \end{align}\tag{21}\]
Here and in what follows, we always assume that \(1\leq p\leq 2\leq p'\leq \infty\), Given \(1\leq q\leq\infty\), we also adopt the convention that \(p\) is defined by \[\begin{align} \label{eq:147q61147p-147p39} \frac{1}{q}=\frac{1}{p}-\frac{1}{p'}. \end{align}\tag{22}\] More explicitly, this means that \[p=\frac{2q}{q+1},\quad p'=\frac{2q}{q-1}.\] We also set \[q_c:=\frac{d+1}{2}.\] Using this identification in 21 , we can write \[\begin{align} \label{sigma40q41} \nu(p')=\sigma(q):=\begin{cases} \frac{d}{2q}-\frac{1}{2}\quad & 1\leq q\leq q_c,\\ \frac{d-1}{4q}\quad & q_c\leq q\leq\infty. \end{cases} \end{align}\tag{23}\]
Then, by 20 and Hölder’s inequality, \[\begin{align} \label{eq:32dual32Sogge32bound32large32p39} \|\Pi_{\lambda} ^*V\Pi_{\lambda} \|_{L^2(M)\to L^2(M)} \lesssim \lambda^{2\sigma(q)}\|V\|_{L^q(M)},\quad V\in L^q(M). \end{align}\tag{24}\] Conversely, 24 implies 20 by duality.
The following proposition yields an improvement over 24 for random potentials.
Proposition 13. Let \(\lambda\geq 2\) and \[r\leq \min\Big(\frac{\mathrm{inj}(M)}{4},\lambda^{-1}\Big)\leq R\leq \mathop{\mathrm{diam}}(M).\] Assume that \(V\in L^1_{\rm {\rm loc}}(M)\) and that its support has diameter at most \(R\). Let \(\Lambda\subset M\) be a maximal \(r\)-separated set and let \(\{\mathcal{V}_x\}_{x\in\Lambda}\) be the associated Voronoi cells. Let \[\begin{align} V_{\omega}(y):=\omega_xV(y)\quadfor x\in \Lambda,y\in \mathcal{V}_x, \end{align}\] where \(\omega_j\) are i.i.d. symmetric Bernoulli or centred normalised Gaussian random variables. Then, for \(1\leq q\leq\infty\), we have \[\begin{align} \mathbf{E}\|\Pi_{\lambda} ^*V_{\omega}\Pi_{\lambda} \|_{L^2(M)\to L^2(M)} & \lesssim \lambda^{-\mu(q)}(\log R/r )^{5/2}\Big(\sum_{x\in\Lambda}\|V\|_{L^{\infty}(B(x,2r))}^{2q}\Big)^{1/(2q)}, \end{align}\] with the obvious modification for \(q=\infty\). Here, \(\mu(q)\) is given by 3 In particular, for any \(K\geq 1\), we have \[\begin{align} \|\Pi_{\lambda} ^*V_{\omega}\Pi_{\lambda} \|_{L^2(M)\to L^2(M)} & \lesssim K \lambda^{-\mu(q)}(\log R/r )^{5/2}\Big(\sum_{x\in\Lambda}\|V\|_{L^{\infty}(B(x,2r))}^{2q}\Big)^{1/(2q)} \end{align}\] with probability at least \(1-\exp(-K^2)\).
Proof. The first statement follows from Theorem 12 (1) together with 20 by taking
\(X=M\) with Riemannian volume measure and the metric space structure induced by the Riemannian metric,
\(Y=B(0,2r)\subset \mathbb{R}^d\) with \(d\)-dimensional Lebesgue measure and \(\varphi_x:=\exp_x^{-1}\),
\(\mathcal{H}_1=\mathcal{H}_2=L^2(M)\) and \(T_1=T_2=\Pi_{\lambda}\),
and observing that, due to 22 , 23 , \[-d/p+\nu(p')+\nu(\infty) = -\mu(q).\] By definition of the injectivity radius, \[\exp_x:B(0,\mathrm{inj}(M))\to B(x,\mathrm{inj}(M))\] is a diffeomorphism for every \(x\in M\). Since \(4r\leq \mathrm{inj}(M)\), it follows by compactness of \(M\) that \[\begin{align} \sup_{x\in M}\|\mathrm{d}\exp_x^{-1}|_{B(x,2r)}\|<\infty. \end{align}\] Hence, by Lemma 3, Definition 6 is satisfied. The assumption 13 is verified in Lemma 11. The second statement follows from Lemma 1. ◻
Remark 14. The class of potentials covered by Proposition 13 contains randomisations of any fixed \(V\) with finite \(\ell^{2q}L^{\infty}\) norm (as in 18 ), not just Anderson-type potentials of the form 1 . In the latter case, taking \(r=\lambda^{-1}\), \(R=\mathop{\mathrm{diam}}(M)\), we get \[\begin{align} \mathbf{E}\|\Pi_{\lambda} ^*V_{\omega}\Pi_{\lambda} \|_{L^2(M)\to L^2(M)} \lesssim \lambda^{-\mu(q)}(\log \lambda )^{5/2}\|v(\lambda)\|_{\ell^{2q}}. \end{align}\] In comparison, e.g. in the Rademacher case, the deterministic bound 24 yields \[\begin{align} \|\Pi_{\lambda} ^*V_{\omega}\Pi_{\lambda} \|_{L^2(M)\to L^2(M)} \lesssim \lambda^{2\sigma(2q)-\frac{d}{2q}}\|v(\lambda)\|_{\ell^{2q}}. \end{align}\] Since \[\begin{align} \label{eq:mu40q41432sigma402q41-d472q} \mu(q)+2\sigma(2q)-\frac{d}{2q} = \begin{cases} 0, & 1\le q\le \dfrac{q_c}{2},\\ 1-\frac{q_c}{2q}, & \dfrac{q_c}{2}\le q\le q_c,\\ \frac{1}{2}, & q_c\le q\leq\infty, \end{cases} \end{align}\tag{25}\] it follows that \(-\mu(q)<2\sigma(2q)-\frac{d}{2q}\) for \(q>q_c/2\). Hence, the probabilistic bound improves upon the deterministic one in this range.
We prove the following deterministic bounds for a larger range of exponents than needed for the proof of Theorem 1. Let \[\begin{align} \label{def:delta40z41} d(z):=\mathop{\mathrm{dist}}(z,\mathop{\mathrm{spec}}(-\Delta_g)),\quad \langle z\rangle:=2+|z|,\quad \delta(z):=\frac{\min\{d(z),|z|^{1/2}\}}{\log\langle z\rangle}. \end{align}\tag{26}\]
Proposition 15. Let \(\Delta_g\) be the Laplace–Beltrami operator on a \(d\)-dimensional closed Riemannian manifold \((M,g)\). Let \(2\leq p'\leq 2d/(d-2)\) if \(d\geq 3\) and \(2\leq p'<\infty\) if \(d=2\). Then for all \(z\in \mathbb{C}\setminus\mathop{\mathrm{spec}}(-\Delta_g)\), \[\begin{align} \||-\Delta_g-z|^{-1/2}\|_{L^{2}(M)\to L^{p'}(M)} \lesssim \delta(z)^{-1/2} \langle z\rangle^{\frac{\nu(p')}{2}}.\label{eq:32deterministic32half32resolvent32bound} \end{align}\qquad{(3)}\] The implicit constant depends only on \((M,g)\) and \(d,p\), but not on \(z\).
Proof. 1. The same argument as in [1] yields \[\||-\Delta_g-z|^{-1/2}\|_{L^{2}(M)\to L^{p'}(M)} \lesssim d(z)^{-1/2},\quad \mathop{\mathrm{Re}}z\leq 1,\] which is better than ?? .
2. It remains to consider the case \(\mathop{\mathrm{Re}}z>1\). Let \(\Sigma:=\{z\in\mathbb{C}:|\mathop{\mathrm{Im}}z|\geq \mathop{\mathrm{Re}}z\}\). Then for \(z\in \Sigma\), we have \[\Big\|\frac{(-\Delta_g+|z|)^{1/2}}{|-\Delta_g-z|^{1/2}}\Big\|\lesssim 1.\] Thus, by Sobolev embedding, \[\begin{align} \||-\Delta_g-z|^{-1/2}f\|_{L^{p'}} &=\|(-\Delta_g+|z|)^{-1/2}\frac{(-\Delta_g+|z|)^{1/2}}{|-\Delta_g-z|^{1/2}}f\|_{L^{p'}}\\ & \lesssim |z|^{\frac{d}{2}(\frac{1}{p}-\frac{1}{2})-\frac{1}{2}}\|\frac{(-\Delta_g+|z|)^{1/2}}{|-\Delta_g-z|^{1/2}}f\|_{L^2} \lesssim |z|^{\frac{d}{2}(\frac{1}{p}-\frac{1}{2})-\frac{1}{2}}\|f\|_{L^2} \end{align}\] for \(z\in \Sigma\), which is again better than ?? for \(\mathop{\mathrm{Re}}z>1\). A similar argument shows \[\begin{align} \||-\Delta_g-z|^{-1/2}\mathbf{1}(-\Delta_g\notin[\mathop{\mathrm{Re}}z/2,2\mathop{\mathrm{Re}}z])\|_{L^{2}(M)\to L^{p'}(M)} & \lesssim|z|^{\frac{d}{2}(\frac{1}{p}-\frac{1}{2})-\frac{1}{2}}. \end{align}\]
. It remains to estimate the half resolvent, spectrally localised to \([\mathop{\mathrm{Re}}z/2,2\mathop{\mathrm{Re}}z]\), for \(\mathop{\mathrm{Re}}z>\max(1,|\mathop{\mathrm{Im}}z|)\). Let \(f\in \mathop{\rm Ran}\mathbf{1}(-\Delta_g\in[\mathop{\mathrm{Re}}z/2,2\mathop{\mathrm{Re}}z])\). We first observe that \[\begin{align} |-\Delta_g-z|^{-\frac{1}{2}}f=\sum_{k\in \mathbb{Z}\cap [0,2\sqrt{\mathop{\mathrm{Re}}z}]}\Pi_k\Upsilon_{k,z}f, \end{align}\] where \[\Upsilon_{k,z}:=\sum_{\lambda_j^2\in [k^2,(k+1)^2)\cap [\mathop{\mathrm{Re}}z/2,2\mathop{\mathrm{Re}}z] }\Big(\frac{1}{|\lambda_j^2-z|}\Big)^{1/2}E_j,\] and \(E_j\) is the orthogonal projection onto the eigenspace of \(-\Delta_g\) corresponding to \(\lambda_j^2\). Let \(k_0:=\left \lceil{\sqrt{\mathop{\mathrm{Re}}z}}\right \rceil\). By the spectral theorem, we have the \(L^2(M)\)-operator norm estimates \[\begin{align} \|\Upsilon_{k,z}\|^2\leq \begin{cases} \frac{1}{d(z)},\quad &for |k-k_0|\leq 10,\\ \frac{5}{\sqrt{\mathop{\mathrm{Re}}z}|k-k_0|},\quad &for |k-k_0|> 10. \end{cases} \end{align}\] Indeed, for \(|k-k_0|> 10\), \(k\in \mathbb{Z}\cap [0,2\sqrt{\mathop{\mathrm{Re}}z}]\) and \(\mathop{\mathrm{Re}}z\geq 1\), we have \[\begin{align} |\lambda_j^2-z| & \geq |k^2-\mathop{\mathrm{Re}}z|-(2k+1) \geq \sqrt{\mathop{\mathrm{Re}}z}|k-k_0|-2(\sqrt{\mathop{\mathrm{Re}}z}+k+1) \\ & \geq \frac{1}{5}\sqrt{\mathop{\mathrm{Re}}z}|k-k_0|. \end{align}\] By orthogonality and Sogge’s spectral cluster bounds 20 , it follows that \[\begin{align} \||-\Delta_g-z|^{-\frac{1}{2}}f\|_{L^2}^2 &\leq \sum_{k\in \mathbb{Z}\cap [0,2\sqrt{\mathop{\mathrm{Re}}z}]} \|\Upsilon_{k,z}\|^2\|\Pi_kf\|_{L^2}^2\\ &\lesssim (\sqrt{\mathop{\mathrm{Re}}z})^{2\nu(p')}\|f\|_{L^p}^2\sum_{k\in \mathbb{Z}\cap [0,2\sqrt{\mathop{\mathrm{Re}}z}]} \|\Upsilon_{k,z}\|^2\\ &\lesssim (\sqrt{\mathop{\mathrm{Re}}z})^{2\nu(p')}\Big(\frac{1}{d(z)}+\frac{\log\sqrt{\mathop{\mathrm{Re}}z}}{\sqrt{\mathop{\mathrm{Re}}z}}\Big)\|f\|_{L^p}^2. \end{align}\] By duality (using that estimates are invariant under \(z\to \overline{z}\)), we conclude that \[\begin{align} \||-\Delta_g-z|^{-1/2}\mathbf{1}(-\Delta_g\in[\mathop{\mathrm{Re}}z/2,2\mathop{\mathrm{Re}}z])\|_{L^{2}(M)\to L^{p'}(M)} & \lesssim \Big(\frac{1}{d(z)}+\frac{\log |z|}{|z|^{1/2}}\Big)^{1/2} |z|^{\frac{\nu(p')}{2}} \end{align}\] for \(\mathop{\mathrm{Re}}z>\max(1,|\mathop{\mathrm{Im}}z|)\). ◻
Remark 16. We used \(p'\leq 2d/(d-2)\) (or \(p'<\infty\) if \(d=2\)) only in Step 2. The estimates in Steps 1 and 3 hold for \(2\leq p'\leq \infty\).
We can also draw the following consequence from the above proof.
Corollary 2. Let \(P=\sqrt{-\Delta_g}\) and \(\lambda\geq 2\). Let \(\chi_-\) and \(\chi\) be smooth functions on \(\mathbb{R}\) supported on the intervals \((-\infty,4)\) and \((1/4,4)\), respectively. Then for all \(\lambda\geq 2\), \(k\in\mathbb{N}_0\) and all \(2\leq p'\leq\infty\), \[\begin{align} \||-\Delta_g-z|^{-1/2}\chi_-(P/\lambda)\|_{L^{2}(M)\to L^{p'}(M)}&\lesssim \delta(z)^{-1/2} \langle z\rangle^{\frac{\nu(p')}{2}},\tag{27}\\ \||-\Delta_g-z|^{-1/2}\chi(P/2^k)\|_{L^{2}(M)\to L^{p'}(M)}&\lesssim 2^{k(\frac{d}{2}-\frac{d}{p'}-1)}\quad for|z|\leq 2^{2k-5}.\tag{28} \end{align}\]
Proof. By the spectral theorem, \[\Big\|\frac{\chi(P/2^k)(-\Delta_g+2^{2k})^{1/2}}{|-\Delta_g-z|^{1/2}}\Big\|\leq\sup_{2^{k-2}<\tau<2^{k+2}}\left|\frac{\tau^2+2^{2k}}{\tau^2-z}\right|^{1/2}\lesssim 1\quad for|z|\leq 2^{2k-5}.\] Thus, by Bernstein inequalities, \[\begin{align} \||-\Delta_g-z|^{-1/2}\chi(P/2^k)f\|_{L^{p'}} & \lesssim 2^{k(\frac{d}{2}-\frac{d}{p'})} \|(-\Delta_g+2^{2k})^{-1/2}f\|_{L^{2}} \\ & \lesssim 2^{k(\frac{d}{2}-\frac{d}{p'}-1)} \|f\|_{L^2}. \end{align}\] The bound 27 follows from ?? and Remark 16. ◻
Let us consider the Birman–Schwinger operator \[\mathcal{K}_{\omega,\lambda}(z) := |-\Delta_g-z|^{-\frac{1}{2}} V_{\omega,\lambda} (-\Delta_g-z)^{-\frac{1}{2}},\quad z\in\mathbb{C}\setminus\mathop{\mathrm{spec}}(-\Delta_g).\] where \(V_{\omega,\lambda}\) is of Anderson-type, as in 1 . More precisely, we assume that \[\begin{align} \label{eq:Anderson-type-repeated} V_{\omega,\lambda}(x) = \sum_{j=1}^{N(\lambda)} \omega_j v_j(\lambda) \psi_j(x;\lambda), \quad x \in M, \end{align}\tag{29}\] where \(\psi_j(\cdot;\lambda)\in C_c^{\infty}(M;[0,1])\), and \(E_j(\lambda)=\mathop{\mathrm{supp}}\psi_j(\cdot;\lambda)\) have diameter \(\approx \lambda^{-1}\), with bounded overlap \[\sup_{x\in M}\sum_{j=1}^{N(\lambda)}\mathbf{1}_{E_j(\lambda)}(x)\leq C.\] The coefficients \(v_j(\lambda)\) are complex numbers, \(\omega_j\) are i.i.d. symmetric Bernoulli or centred normalised Gaussian random variables, and \(\lambda\geq 2\) is a large parameter.
The following is the main result of this subsection.
Proposition 17. Let \(V_{\omega,\lambda}\) be as in 29 , \(\lambda\geq 2\). Then for \(z\in\mathbb{C}\setminus\mathop{\mathrm{spec}}(-\Delta_g)\), \(|z|\leq \lambda^2\), \(\frac{d}{2}\leq q\leq \infty\), \[\begin{align} \label{eq:EnormBSexp} \mathbf{E}\|\mathcal{K}_{\omega,\lambda}(z)\| \lesssim \lambda^{-\frac{d(q+1)}{2q}}\,\delta(z)^{-1} \,\langle z\rangle^{\frac{d(q+1)}{4q}-\frac{\mu(q)}{2}} (\log\lambda)^{5/2}\|v(\lambda)\|_{\ell^{2q}}, \end{align}\qquad{(4)}\] where \(\delta(z)\) and \(\mu(q)\) were defined in 26 and 3 , respectively. Moreover, for any \(K\geq 1\), and \(z,q,\lambda\) as above, there exists an event \(E_0(z,q,\lambda,K)\) with probability \[\begin{align} \label{eq:P40E95040z44q44lambda44K4141} \mathbf{P} (E_0(z,q,\lambda,K)) \geq 1-\exp(-K^2) \end{align}\qquad{(5)}\] and such that for all \(\omega\in E_0(z,q,\lambda,K)\), \[\begin{align} \label{eq:EnormBS} \|\mathcal{K}_{\omega,\lambda}(z)\| \lesssim K\lambda^{-\frac{d(q+1)}{2q}}\,\delta(z)^{-1} \,\langle z\rangle^{\frac{d(q+1)}{4q}-\frac{\mu(q)}{2}} (\log\lambda)^{5/2}\|v(\lambda)\|_{\ell^{2q}}. \end{align}\qquad{(6)}\]
Remark 18. Similarly to Proposition 13, our method could handle randomisations of a fixed potential, see Remark 14. For the sake of exposition and compactness of notation, we restrict our attention to Anderson-type potentials.
In the following, to simplify notation, we use the abbreviations \[\begin{align} A_{\lambda} & := \lambda^{-\frac{d(q+1)}{2q}} (\log\lambda)^{5/2}\|v(\lambda)\|_{\ell^{2q}}, \tag{30} \\ B(z) & := \delta(z) \,\langle z\rangle^{\frac{\mu(q)}{2}-\frac{d(q+1)}{4q}}.\tag{31} \end{align}\] The proof of Proposition 17 will be given at the end of this subsection.
Lemma 5. Under the assumptions of Corollary 2, we have, for \(1\leq q\leq \infty\), \[\begin{align} \label{eq:chi95-chi95-} \mathbf{E}\|\chi_-(P/\lambda)\mathcal{K}_{\omega,\lambda}(z)\chi_-(P/\lambda)\|&\lesssim A_{\lambda}B(z)^{-1}. \end{align}\tag{32}\] Moreover, for \(|z|\leq 2^{2k-5}\), \(\lambda\lesssim 2^k\), \[\begin{align} \mathbf{E}\|\chi(P/2^k)\mathcal{K}_{\omega,\lambda}(z)\chi_-(P/\lambda)\|&\lesssim 2^{-k}\lambda^{-\frac{d}{2}}(\log\lambda)^{5/2}\delta(z)^{-\frac{1}{2}} \langle z\rangle^{\frac{d-1}{4}}\|v(\lambda)\|_{\ell^{2q}}, \label{eq:chi95k95chi95-} \end{align}\tag{33}\] and for \(|z|\leq 2^{2\ell-5}\) and \(k\geq \ell\), \[\begin{align} \mathbf{E}\|\chi(P/2^\ell)\mathcal{K}_{\omega,\lambda}(z)\chi(P/2^k)\|&\lesssim 2^{-k}\lambda^{-\frac{d}{2}}(\log\lambda)^{5/2}2^{\ell(\frac{d}{2}-1)}\|v(\lambda)\|_{\ell^{2q}}. \label{eq:chi95l95chi95k} \end{align}\tag{34}\]
Proof. The proof uses Theorem 12, whose assumptions we verified in Proposition 13.
. Theorem 12 (1), with \(r=\lambda^{-1}\), together with 27 yields \[\begin{align} \mathbf{E}\|\chi_-(P/\lambda)\mathcal{K}_{\omega,\lambda}(z)\chi_-(P/\lambda)\| &\lesssim \lambda^{-d/p} (\log\lambda)^{5/2} \delta(z)^{-1} \langle z\rangle^{\frac{\nu(p')+\nu(\infty)}{2}}\|v(\lambda)\|_{\ell^{2q}}. \end{align}\] Observing that \(\nu(p')+\nu(\infty)=d/p-\mu(q)\) and recalling 22 , 23 , this proves 32 .
2. Theorem 12 (2), with \(r=2^{-k}\), \(r'=\lambda^{-1}\), together with 27 and 28 implies \[\begin{align} \mathbf{E}\|\chi_-(P/\lambda)\mathcal{K}_{\omega,\lambda}(z)\chi(P/2^k)\| &\lesssim 2^{-\frac{kd}{p}} (2^k/\lambda)^{\frac{d}{p}-\frac{d}{2q}}(\log\lambda)^{5/2}\\ &\quad\times 2^{k(\frac{d}{2}-\frac{d}{p'}-1)}\delta(z)^{-1/2} \langle z\rangle^{\frac{\nu(\infty)}{2}}\|v(\lambda)\|_{\ell^{2q}}. \end{align}\] Since \(\nu(\infty)=\frac{d-1}{2}\), \(2^{k(-\frac{d}{2q}+\frac{d}{2}-\frac{d}{p'}-1)}=2^{-k}\) and \(\lambda^{\frac{d}{2q}-\frac{d}{p}}=\lambda^{-\frac{d}{2}}\), this proves 33 .
3. Similarly, Theorem 12 (2) and 28 yield \[\begin{align} \mathbf{E}\|\chi(P/2^\ell)\mathcal{K}_{\omega,\lambda}(z)\chi(P/2^k)\| &\lesssim 2^{-\frac{kd}{p}} (2^k/\lambda)^{\frac{d}{p}-\frac{d}{2q}}(\log\lambda)^{5/2}\\ &\quad\times 2^{k(\frac{d}{2}-\frac{d}{p'}-1)}2^{\ell(\frac{d}{2}-1)}\|v(\lambda)\|_{\ell^{2q}}, \end{align}\] which proves 34 . ◻
Corollary 3. Let \(\lambda\geq 2\) and \(\chi_-,\chi_+\in C^{\infty}(\mathbb{R};[0,1])\) such that \(\mathop{\mathrm{supp}}\chi_-\subset (-\infty,8)\) and \(\mathop{\mathrm{supp}}\chi_+\subset (4,\infty)\). Then for \(z\in\mathbb{C}\setminus\mathop{\mathrm{spec}}(-\Delta_g)\), \(|z|\leq \lambda^2\), \(1\leq q\leq q_c\), \[\begin{align} \mathbf{E}\|\chi_+(P/\lambda)\mathcal{K}_{\omega,\lambda}(z)\chi_-(P/\lambda)\|&\lesssim \lambda^{-\frac{d}{2}-1}(\log\lambda)^{5/2}\delta(z)^{-\frac{1}{2}} \langle z\rangle^{\frac{d-1}{4}}\|v(\lambda)\|_{\ell^{2q}},\tag{35}\\ \mathbf{E}\|\chi_+(P/\lambda)\mathcal{K}_{\omega,\lambda}(z)\chi_+(P/\lambda)\|&\lesssim \lambda^{-2}(\log\lambda)^{5/2}\|v(\lambda)\|_{\ell^{2q}}\tag{36}. \end{align}\]
Proof. Let \(\chi\in C_c^{\infty}(\mathbb{R})\) be such that \(\mathop{\mathrm{supp}}\chi\subset (1/4,4)\) and \(\chi=1\) on \((1/2,2)\). Then \[\begin{align} \mathbf{E}\|\chi_+(P/\lambda)\mathcal{K}_{\omega,\lambda}(z)\chi_-(P/\lambda)\|\leq \sum_{2^{k}\geq 8\lambda}\mathbf{E}\|\chi(P/2^k)\mathcal{K}_{\omega,\lambda}(z)\chi_-(P/\lambda)\|, \end{align}\] and 35 follows from 33 since \(|z|\leq \lambda^2\leq 2^{2k-6}\) in the range of summation. Similarly, \[\begin{align} \mathbf{E}\|\chi_+(P/\lambda)\mathcal{K}_{\omega,\lambda}(z)\chi_+(P/\lambda)\|\leq \sum_{2^{k}\geq 8\lambda}\sum_{2^{\ell}\geq 8\lambda}\mathbf{E}\|\chi(P/2^\ell)\mathcal{K}_{\omega,\lambda}(z)\chi(P/2^k)\|, \end{align}\] and 34 implies that the part of the sum with \(\ell\leq k\) is bounded by the right-hand side of 36 . Replacing \(z\) by \(\overline{z}\) in 35 and using the bound for the adjoint, the same argument applies to the part of the sum with \(\ell>k\). ◻
Proof of Proposition 17. Let \(\chi_-,\chi_+\) be such that \(\chi_-+\chi_+=1\) and with support as in Corollary 3. Then by 32 , 35 and its dual (with \(z\) replaced by \(\overline{z}\)), and 36 , \[\begin{align} \mathbf{E}\|\mathcal{K}_{\omega,\lambda}(z)\| &\leq \sum_{\sigma_1,\sigma_2\in\{+,-\}}\mathbf{E}\|\chi_{\sigma_1}(P/\lambda)\mathcal{K}_{\omega,\lambda}(z)\chi_{\sigma_2}(P/\lambda)\| \\ &\lesssim (\lambda^{-\frac{d}{p}} \delta(z)^{-1} \langle z\rangle^{\frac{d}{2p}-\frac{\mu(q)}{2}} +\lambda^{-\frac{d}{2}-1}\delta(z)^{-\frac{1}{2}} \langle z\rangle^{\frac{d-1}{4}}+\lambda^{-2}) (\log\lambda)^{5/2}\|v(\lambda)\|_{\ell^{2q}}. \end{align}\] Using \(\delta(z)\leq \langle z\rangle^{\frac{1}{2}}\lesssim\lambda\) and \(q\geq \frac{d}{2}\), we may bound the second and third term in the last expression by the first. This proves ?? , which together with Lemma 1 yields ?? . ◻
Remark 19. The above proof is the only place where we used the assumption \(q\geq d/2\).
We cannot directly apply Proposition 17 to prove Theorem 1. The problem is that the events on which the corresponding inequalities hold depend on the spectral parameter \(z\). Since \(z\) ranges over a continuum, one cannot directly apply a union bound over all \(z\). To overcome this, we will introduce a discretisation scheme.
To carry out the discretisation, let \[\begin{align} E(z,q,\lambda,K)&:=\{\omega:\|\mathcal{K}_{\omega,\lambda}(z)\|\lesssim K A_{\lambda}B(z)^{-1}\},\label{def:E40z41} \end{align}\tag{37}\] where \(A_{\lambda}\), \(B(z)\) were defined in 30 , 31 , respectively. In what follows, we will abbreviate \(E(z,q,\lambda,K)\) by \(E(z)\). The result of Proposition 17 can then be stated as \[\begin{align} \label{eq:P40E40z4141} \mathbf{P}(E(z))\geq 1-\exp(-K^2). \end{align}\tag{38}\] The aim of this section is to prove the following proposition, which is a uniform version of the above bound. For technical reasons, we need to restrict \(z\) to the complement of an arbitrarily small neighborhood of \(\mathop{\mathrm{spec}}(-\Delta_g)\).
Proposition 20. Let \(N\geq 1\) be fixed. Then there exists a constant \(C_N>0\) such that for every \(K\geq 1\), \[\begin{align} \label{eq:uniform-on-truncated-region} \mathbf{P}\Bigl( \sup_{\substack{|z|\leq \lambda^2\\ d(z)\geq \lambda^{-N}}} B(z)\,\|\mathcal{K}_{\omega,\lambda}(z)\| \leq C_NK\,A_\lambda (\log\lambda)^2 \Bigr) \geq 1-\exp(-K^2). \end{align}\qquad{(7)}\]
As a preliminary step, we record the following simple Lipschitz bound for the Birman-Schwinger operator.
Lemma 6. For any \(z,z'\in \mathbb{C}\setminus \mathop{\mathrm{spec}}(-\Delta_g)\) with \(d(z),d(z')\geq \rho>0\), we have \[\|\mathcal{K}_{\omega,\lambda}(z)-\mathcal{K}_{\omega,\lambda}(z')\| \lesssim |z-z'| \, \rho^{-2}\, \|V_{\omega,\lambda}\|_{L^{\infty}(M)}.\]
Proof. Set \[R(w):=(-\Delta_g-w)^{-1/2}, \qquad w\in \mathbb{C}\setminus \mathop{\mathrm{spec}}(-\Delta_g).\] Then \[\mathcal{K}_{\omega,\lambda}(w)=|R(w)|V_{\omega,\lambda}R(w),\] and hence \[\mathcal{K}_{\omega,\lambda}(z)-\mathcal{K}_{\omega,\lambda}(z') = \bigl(|R(z)|-|R(z')|\bigr)V_{\omega,\lambda}R(z) + |R(z')|V_{\omega,\lambda}\bigl(R(z)-R(z')\bigr).\] Therefore, taking the operator norm, \[\begin{align} \|\mathcal{K}_{\omega,\lambda}(z)-\mathcal{K}_{\omega,\lambda}(z')\| &\leq \|R(z)-R(z')\|\,\|V_{\omega,\lambda}\|\,\|R(z)\| \\ &\qquad +\|R(z')\|\,\|V_{\omega,\lambda}\|\,\|R(z)-R(z')\|, \end{align}\] where \(\|V_{\omega,\lambda}\|=\|V_{\omega,\lambda}\|_{L^{\infty}(M)}\). It remains to estimate \(\|R(w)\|\) and \(\|R(z)-R(z')\|\).
By the spectral theorem, \[\begin{align} \label{eq:trivial32bound32R40w41} \|R(w)\| = \sup_{\mu\in \mathop{\mathrm{spec}}(-\Delta_g)} |\mu-w|^{-1/2} = d(w)^{-1/2}. \end{align}\tag{39}\]
For the difference, define for \(\mu\in \mathop{\mathrm{spec}}(-\Delta_g)\) \[f_w(\mu):=|\mu-w|^{-1/2}.\] Then \[\|R(z)-R(z')\| = \sup_{\mu\in \mathop{\mathrm{spec}}(-\Delta_g)} |f_z(\mu)-f_{z'}(\mu)|.\] Using \[\begin{align} \bigl||a|^{-1/2}-|b|^{-1/2}\bigr| &= \frac{\bigl||b|^{1/2}-|a|^{1/2}\bigr|}{|a|^{1/2}|b|^{1/2}} = \frac{|\,|b|-|a|\,|}{|a|^{1/2}|b|^{1/2}(|a|^{1/2}+|b|^{1/2})}\\ &\leq \frac{|a-b|}{|a|^{1/2}|b|^{1/2}(|a|^{1/2}+|b|^{1/2})}, \end{align}\] with \(a=\mu-z\) and \(b=\mu-z'\), we obtain \[|f_z(\mu)-f_{z'}(\mu)| \leq \frac{|z-z'|}{|\mu-z|^{1/2}|\mu-z'|^{1/2} \bigl(|\mu-z|^{1/2}+|\mu-z'|^{1/2}\bigr)}.\] Since \[\frac{1}{x^{1/2}y^{1/2}(x^{1/2}+y^{1/2})} \leq \min(x^{-1/2}y^{-1},x^{-1}y^{-1/2}) \leq x^{-3/2}+y^{-3/2},\] for all \(x,y>0\), it follows that \[|f_z(\mu)-f_{z'}(\mu)| \lesssim |z-z'|\bigl(|\mu-z|^{-3/2}+|\mu-z'|^{-3/2}\bigr).\] Taking the supremum over \(\mu\in \mathop{\mathrm{spec}}(-\Delta_g)\) yields \[\|R(z)-R(z')\| \lesssim |z-z'|\bigl(d(z)^{-3/2}+d(z')^{-3/2}\bigr).\]
Substituting these bounds into the previous estimate gives \[\begin{align} \|\mathcal{K}_{\omega,\lambda}(z)-\mathcal{K}_{\omega,\lambda}(z')\| &\lesssim |z-z'|\,\|V_{\omega,\lambda}\| \bigl(d(z)^{-3/2}+d(z')^{-3/2}\bigr) \bigl(d(z)^{-1/2}+d(z')^{-1/2}\bigr). \end{align}\] If \(d(z),d(z')\geq \rho\), then \[\|\mathcal{K}_{\omega,\lambda}(z)-\mathcal{K}_{\omega,\lambda}(z')\| \lesssim |z-z'|\,\rho^{-2}\,\|V_{\omega,\lambda}\|_{L^{\infty}(M)}.\] This completes the proof. ◻
We decompose the set \(\{z\in \mathbb{C}\setminus \mathop{\mathrm{spec}}(-\Delta_g):|z|\leq \lambda^2\}\) according to the size of \(B(z)\). More precisely, for \(j\in\mathbb{Z}\), define \[\begin{align} \Omega_j&:=\Bigl\{z\in \mathbb{C}\setminus\mathop{\mathrm{spec}}(-\Delta_g): |z|\leq \lambda^2,\; 2^{j}\leq B(z)<2^{j+1}\Bigr\}. \end{align}\] Then, \[\{z\in \mathbb{C}\setminus \mathop{\mathrm{spec}}(-\Delta_g):|z|\leq \lambda^2\} = \bigcup_{j\in\mathbb{Z}}\Omega_j\] and on each \(\Omega_j\), by definition@eq:def:E40z41 , \[\begin{align} \label{eq:EnormBS95j} \|\mathcal{K}_{\omega,\lambda}(z)\| \lesssim K\,2^{-j}A_\lambda, \qquad z\in \Omega_j,\quad \omega\in E(z). \end{align}\tag{40}\]
Corollary 4. Let \(j\in \mathbb{Z}\) be such that \(\Omega_j\neq \emptyset\). Then there exist \(C,c>0\) such that for any \(K\geq 1\), there is an event \(E_j'\) of probability at least \(1-\exp(-K^2)\) such that for all \(z\in\Omega_j\), \(|z-z'|\leq c2^{j}\) and \(\omega\in E_j'\), we have \[\begin{align} \label{eq:tail95bound95K40z41-K40z3941} \|\mathcal{K}_{\omega,\lambda}(z)-\mathcal{K}_{\omega,\lambda}(z')\|\leq CK2^{-j}\sqrt{\log\lambda} \|v(\lambda)\|_{\ell^{\infty}}. \end{align}\tag{41}\]
Proof. By Lemma 6, the map \[\Omega_j\ni z\mapsto \mathcal{K}_{\omega,\lambda}(z)\] is stable under perturbations of size \(c2^{j}\) for \(c\) sufficiently small. More precisely, since \[\begin{align} d(z)\geq B(z),\quad d(z')\geq d(z)-|z-z'|, \end{align}\] Lemma 6 with \(\rho=2^j\) yields \[\begin{align} \label{eq:K40z41-K40z394195for95z95in95Omega95j} z\in\Omega_j,\quad |z-z'|\leq c2^{j}\implies \|\mathcal{K}_{\omega,\lambda}(z)-\mathcal{K}_{\omega,\lambda}(z')\|\lesssim 2^{-j}\|V_{\omega,\lambda}\|_{L^{\infty}(M)}. \end{align}\tag{42}\] Hence, the claimed bound 41 follows immediately from 42 and the \(q=\infty\) case in Lemma 7 below. ◻
Lemma 7. Let \(1\leq q <\infty\). There exists \(C_q>0\) such that for any \(K\geq 1\), \[\begin{align} \mathbf{P}(\|V_{\omega,\lambda}\|_{L^{q}(M)}\leq C_qK\lambda^{-\frac{d}{q}}\|v(\lambda)\|_{\ell^{q}}) \geq 1-\exp(-K^2). \end{align}\] Moreover, for \(q=\infty\), there exists \(C\) such that \[\begin{align} \mathbf{P}(\|V_{\omega,\lambda}\|_{L^{\infty}(M)}\leq CK\sqrt{\log\lambda}\|v(\lambda)\|_{\ell^{\infty}}) \geq 1-\exp(-K^2). \end{align}\]
Proof. We first consider the case \(q<\infty\). By the bounded overlap assumption, \[|V_{\omega,\lambda}(x)|^q \lesssim_q \sum_{j=1}^{N(\lambda)} |\omega_j|^q |v_j(\lambda)|^q \mathbf{1}_{E_j(\lambda)}(x).\] Integrating over \(M\) and using that \(|E_j(\lambda)|\lesssim \lambda^{-d}\), \[\|V_{\omega,\lambda}\|_{L^q(M)}^q \lesssim_q \lambda^{-d} \sum_{j=1}^{N(\lambda)} |\omega_j|^q |v_j(\lambda)|^q.\] Taking expectations and using \(\mathbf{E}|\omega_j|^q \lesssim_q 1\) for Bernoulli or Gaussian \(\omega_j\), \[\mathbf{E}\|V_{\omega,\lambda}\|_{L^q(M)}^q \lesssim_q \lambda^{-d}\sum_{j=1}^{N(\lambda)}|v_j(\lambda)|^q = \lambda^{-d}\|v(\lambda)\|_{\ell^q}^q.\] By Jensen’s inequality, \[\mathbf{E}\|V_{\omega,\lambda}\|_{L^q(M)}\leq (\mathbf{E}\|V_{\omega,\lambda}\|_{L^q(M)}^q)^{1/q}\lesssim_q \lambda^{-d/q}\|v(\lambda)\|_{\ell^q}.\] Lemma 1 then implies the claim.
Now assume \(q=\infty\). By Dudley’s inequality ?? , \[\begin{align} \mathbf{E}\|V_{\omega,\lambda}\|_{L^{q}(M)} &\lesssim \mathbf{E}\max_{j\leq N(\lambda)}|\omega_jv_j(\lambda)|\lesssim \sqrt{\log N(\lambda)} \max_{j\leq N(\lambda)}\|\omega_jv_j(\lambda)\|_{\psi_2}. \end{align}\] Since \(N(\lambda)\lesssim\lambda^d\), it follows that \[\begin{align} \mathbf{E}\|V_{\omega,\lambda}\|_{L^{\infty}(M)} \lesssim\sqrt{\log\lambda} \|v(\lambda)\|_{\ell^{\infty}}. \end{align}\] Again, Lemma 1 implies the claim. ◻
Remark 21. In the Bernoulli case, the bounds of Lemma 7 are deterministic: for \(1\leq q<\infty\), \[\|V_{\omega,\lambda}\|_{L^{q}(M)} \leq C_q \lambda^{-d/q}\|v(\lambda)\|_{\ell^{q}},\] while for \(q=\infty\), \[\|V_{\omega,\lambda}\|_{L^{\infty}(M)} \leq C\|v(\lambda)\|_{\ell^{\infty}}.\] Hence, only the Gaussian case requires a probabilistic argument.
Continuing with the discretisation argument, for fixed \(N\geq 1\), we set \[\Omega_{j,N}:=\Omega_j\cap \{z\in\mathbb{C}:d(z)\geq \lambda^{-N}\}.\]
Lemma 8. Let \(j\in \mathbb{Z}\) be such that \(\Omega_{j,N}\neq \emptyset\). There exists \(C_N>0\) such that for any \(K\geq 1\), \[\begin{align} \label{eq:uniform-on-Omegaj} \mathbf{P}\Bigl( \sup_{z\in \Omega_{j,N}}\|\mathcal{K}_{\omega,\lambda}(z)\| \leq C_NK\,2^{-j}A_\lambda \log \lambda \Bigr) \geq 1-\exp(-K^2). \end{align}\tag{43}\]
Proof. Fix \(j\in \mathbb{Z}\) with \(\Omega_{j,N}\neq \emptyset\), and let \(\mathcal{N}_j\subset \Omega_{j,N}\) be a \(c2^j\)-net. We denote the elements of \(\mathcal{N}_j\) by \(z_\alpha\). For each \(z_\alpha\in \mathcal{N}_j\), let \[E(z_\alpha) := \Bigl\{ \omega: \|\mathcal{K}_{\omega,\lambda}(z_\alpha)\| \leq CK\,2^{-j}A_\lambda \Bigr\}.\] By 40 , we have \[\mathbf{P}(E(z_\alpha))\geq 1-\exp(-K^2).\] Next, let \(E'_j\) denote the event on which the perturbation estimate Corollary 4 holds simultaneously for all pairs \(z,z_\alpha\) with \(z\in \Omega_{j,N}\), \(z_\alpha\in\mathcal{N}_j\), and \(|z-z_\alpha|\leq c2^j\), namely \[\|\mathcal{K}_{\omega,\lambda}(z)-\mathcal{K}_{\omega,\lambda}(z_\alpha)\| \leq CK\,2^{-j}\sqrt{\log\lambda}\,\|v(\lambda)\|_{\ell^\infty}.\] Since \[\sqrt{\log\lambda}\,\|v(\lambda)\|_{\ell^\infty} \leq (\log\lambda)^{5/2}\|v(\lambda)\|_{\ell^{2q}} = A_\lambda,\] it follows that on \(E'_j\), \[\|\mathcal{K}_{\omega,\lambda}(z)-\mathcal{K}_{\omega,\lambda}(z_\alpha)\| \leq CK\,2^{-j}A_\lambda\] for every \(z\in \Omega_{j,N}\) and every net point \(z_\alpha\) satisfying \(|z-z_\alpha|\leq c2^j\).
Hence, on the event \[F_j:=\Bigl(\bigcap_{z_\alpha\in\mathcal{N}_j}E(z_\alpha)\Bigr)\cap E'_j,\] we have, for every \(z\in \Omega_{j,N}\), choosing \(z_\alpha\in \mathcal{N}_j\) with \(|z-z_\alpha|\leq c2^j\), \[\|\mathcal{K}_{\omega,\lambda}(z)\| \leq \|\mathcal{K}_{\omega,\lambda}(z_\alpha)\| + \|\mathcal{K}_{\omega,\lambda}(z)-\mathcal{K}_{\omega,\lambda}(z_\alpha)\| \leq CK\,2^{-j}A_\lambda.\] Thus, \[F_j\subseteq \Bigl\{ \omega: \sup_{z\in \Omega_{j,N}}\|\mathcal{K}_{\omega,\lambda}(z)\| \leq CK\,2^{-j}A_\lambda \Bigr\}.\]
It remains to estimate \(\mathbf{P}(F_j)\). By the union bound, \[\mathbf{P}(F_j^c) \leq \sum_{z_\alpha\in \mathcal{N}_j}\mathbf{P}(E(z_\alpha)^c) +\mathbf{P}((E_j')^c) \leq \#\mathcal{N}_j\exp(-K^2)+\mathbf{P}(E_j'^c).\] By a volume comparison argument, the cardinality of \(\mathcal{N}_j\) satisfies \[\#\mathcal{N}_j\leq \frac{\lambda^4}{2^{2j}}\] and by Corollary 4, \(\mathbf{P}(E_j'^c)\leq \exp(-K^2)\). Hence \[\mathbf{P}(F_j^c) \leq (1+C\,2^{-2j}\lambda^4) \exp(-K^2).\] Since \(\Omega_j\neq \emptyset\) implies \[\begin{align} \label{eq:2j95poly95bounded} \lambda^{-N-1}\lesssim 2^j\lesssim \lambda^3 \end{align}\tag{44}\] the factor \(2^{-2j}\lambda^4\) is bounded by a fixed power of \(\lambda\). Replacing \(K\) by \(C_NK\log\lambda\) in the preceding argument, we obtain the claimed bound 43 . ◻
Proof of Proposition 20. For each \(j\in \mathbb{Z}\), let \[F_j:= \Bigl\{ \omega: \sup_{z\in \Omega_{j,N}}\|\mathcal{K}_{\omega,\lambda}(z)\| \leq C_NK\,2^{-j}A_\lambda \log\lambda \Bigr\}.\] By Lemma 8, \[\mathbf{P}(F_j)\geq 1-\exp(-K^2)\] for every \(j\) such that \(\Omega_{j,N}\neq \emptyset\).
We first estimate the number of such \(j\). If \(\Omega_{j,N}\neq \emptyset\), then there exists \(z\in \Omega_{j,N}\) such that \[\begin{align} \label{eq:d40z4195approx952j} 2^j\leq B(z)<2^{j+1}. \end{align}\tag{45}\] In view of 44 , we have \[-N\log_2\lambda-O(1)\leq j\leq 3\log_2\lambda+O(1),\] so the number of indices \(j\) for which \(\Omega_{j,N}\neq \emptyset\) is \(O_N(\log\lambda)\).
Now define \[F:=\bigcap_{\Omega_{j,N}\neq \emptyset} F_j.\] By the union bound, \[\mathbf{P}(F^c) \leq \sum_{\Omega_{j,N}\neq \emptyset}\mathbf{P}(F_j^c) \leq C_N(\log\lambda)\exp(-K^2).\] Absorbing the factor \(\log\lambda\) by replacing \(K\) with \(C_NK\log\lambda\), we obtain \[\mathbf{P}(F)\geq 1-\exp(-K^2).\]
It remains to identify the bound on \(F\). Let \(z\) satisfy \(|z|\leq \lambda^2\) and \(d(z)\geq \lambda^{-N}\). Then \(z\in \Omega_{j,N}\) for some \(j\), and on \(\Omega_{j,N}\) we have 45 . Thus, on the event \(F\), \[\|\mathcal{K}_{\omega,\lambda}(z)\| \leq C_NK\,2^{-j}A_\lambda (\log\lambda)^2 \lesssim C_NK\,B(z)^{-1}A_\lambda (\log\lambda)^2.\] Taking the supremum over all such \(z\) after multiplying by \(B(z)\) yields ?? . ◻
Proof. By the Birman–Schwinger principle, \(z \in \mathbb{C}\setminus \mathop{\mathrm{spec}}(-\Delta_g)\) is an eigenvalue of \(H_{\omega,\lambda}\) if and only if \(-1\) is an eigenvalue of \(\mathcal{K}_{\omega,\lambda}(z)\). In particular, \[\label{eq:BSnorm} z \in \mathop{\mathrm{spec}}(H_{\omega,\lambda}) \implies \|\mathcal{K}_{\omega,\lambda}(z)\|\ge 1.\tag{46}\]
By Proposition 20, there exists an event of probability at least \(1 - \exp(-K^2)\) on which the estimate \[\label{eq:BSbound} \|\mathcal{K}_{\omega,\lambda}(z)\| \leq C_N K A_{\lambda}(\log\lambda)^2B(z)^{-1}\tag{47}\] holds uniformly for all \(z\in\mathbb{C}\setminus\mathop{\mathrm{spec}}(-\Delta_g)\) with \(|z|\le \lambda^2\) and \(d(z)\geq \lambda^{-N}\). Combining 46 and 47 and recalling the definitions of \(A_{\lambda}\), \(B(z)\), \(\delta(z)\) in 30 , 31 , 26 , respectively, we obtain that for any eigenvalue \(z \in \mathbb{C}\setminus \mathop{\mathrm{spec}}(-\Delta_g)\) of \(H_{\omega,\lambda}\) with \(|z|\le \lambda^2\) and \(d(z)\geq \lambda^{-N}\), \[\begin{align} \label{eq:keybound} \min\{d(z), |z|^{1/2}\} \leq C_N K \langle z\rangle^{\frac{d(q+1)}{4q}-\frac{\mu(q)}{2}} \, \lambda^{-\frac{d(q+1)}{2q}} (\log \lambda)^{11/2} \|v(\lambda)\|_{\ell^{2q}}. \end{align}\tag{48}\] Here we have estimated \(\log\langle z\rangle \lesssim \log \lambda\).
We now distinguish two cases.
Case 1: \(d(z) \le |z|^{1/2}\).
Let \(\lambda_k^2 \in \mathop{\mathrm{spec}}(-\Delta_g)\) be such that \(d(z) = |z - \lambda_k^2|\). Then \[\begin{align} |z|\approx (1+\lambda_k)^2. \end{align}\] Inserting this into 48 , we obtain \[\begin{align} \label{eq:d40z41leq32case321} d(z) &\leq C_N K (1+\lambda_k)^{\frac{d(q+1)}{2q}-\mu(q)} \, \lambda^{-\frac{d(q+1)}{2q}} (\log \lambda)^{11/2} \|v(\lambda)\|_{\ell^{2q}}, \end{align}\tag{49}\] which means that \[z\in \bigcup_{\lambda_k\leq \lambda} D(\lambda_k^2,C_NKr_k(\lambda,q)).\]
Case 2: \(d(z) \ge |z|^{1/2}\).
In this case, 48 yields \[|z|^{1/2}\langle z\rangle^{-\frac{d(q+1)}{4q}+\frac{\mu(q)}{2}} \leq C_NK\, \lambda^{-\frac{d(q+1)}{2q}} (\log \lambda)^{11/2} \|v(\lambda)\|_{\ell^{2q}},\] hence \(z\in \Omega\). This proves the claimed spectral inclusion. ◻
In this appendix, we verify the local constancy property for closed Riemannian manifolds that we used in the proofs of Proposition 13 and Lemma 5.
Lemma 9. Let \(P=\sqrt{-\Delta_g}\), \(0<\varepsilon<{\rm inj}(M)/10\), and \(\chi\in\mathcal{S}(\mathbb{R})\) be such that \(\mathop{\mathrm{supp}}(\widehat{\chi})\subset [-\varepsilon,\varepsilon]\) and \(\chi\geq \mathbf{1}_{[-1,1]}\). Then for all \(x\in M\), \[\begin{align} |\chi(P/\lambda)f(x)|\lesssim \lambda^d \int_M (1+\lambda d_g(x,y))^{-N} |f(y)|\mathrm{d}y+\int_M|f(y)|\mathrm{d}y. \end{align}\]
Remark 22. This shows that \(\mathop{\rm Ran}\chi(P/\lambda)\) has the local constancy property for any scale \(h\leq \lambda^{-1}\).
Proof. We use the Fourier inversion formula to write \[\begin{align} \chi(P/\lambda)=\frac{\lambda}{2\pi}\int_{-\infty}^{\infty}\widehat{\chi}(\lambda t){\rm e}^{\mathrm{i}tP}\mathrm{d}t. \end{align}\] By [27], for \(|t|<\varepsilon\), \[\begin{align} \label{eq:32Q40t4143R40t41} {\rm e}^{\mathrm{i}tP}=Q(t)+R(t), \end{align}\tag{50}\] where \(R(t)\) has a smooth kernel \(R(t,x,y)\), the kernel \(Q(t,x,y)\) of \(Q(t)\) is supported in a small neighborhood of the diagonal in \(M\times M\), and in local coordinates takes the form \[Q(t,x,y) = (2\pi)^{-d}\int_{\mathbb{R}^d}{\rm e}^{\mathrm{i}[\varphi(x,y,\xi)+tp(y,\xi)]}q(t,x,y,\xi)\mathrm{d}\xi.\] Here, \(p(x,\xi)=|\xi|_{g(x)}\) is the principal symbol of \(P\), \(q\in S^0\) is a symbol of order zero, and \(\varphi\) is homogeneous of degree one in \(\xi\) and satisfies \[\begin{align} \label{Taylor32expansion32of32phase32function} \varphi(x,y,\xi)=(x-y)\cdot\xi+\mathcal{O}(|x-y|^2|\xi|). \end{align}\tag{51}\] Thus, for \(f\) supported in a relatively compact subset of a given coordinate patch, \[\begin{align} \chi(P/\lambda)f(x)&=\frac{\lambda}{(2\pi)^{d+1}}\int_{\mathbb{R}^d}\int_{\mathbb{R}^d}\int_{-\infty}^{\infty}\widehat{\chi}(\lambda t){\rm e}^{\mathrm{i}[\varphi(x,y,\xi)+tp(y,\xi)]}q(t,x,y,\xi)f(y)\mathrm{d}t\mathrm{d}\xi \mathrm{d}y \\ & + \frac{\lambda}{2\pi}\int_{\mathbb{R}^d}\int_{-\infty}^{\infty}\widehat{\chi}(\lambda t)R(t,x,y)f(y)\mathrm{d}t \mathrm{d}y. \end{align}\] The modulus of the second term is bounded by a multiple of \(\|f\|_{L^1(M)}\). By the change of variables \((t,\xi)\mapsto(\lambda^{-1}t,\lambda\xi)\), the kernel of the operator appearing in the first term is given by \[\begin{align} K_{\lambda}(x,y)=\frac{\lambda^d}{(2\pi)^{d+1}}\int_{\mathbb{R}^d}\int_{-\infty}^{\infty}\widehat{\chi}(t){\rm e}^{\mathrm{i}\lambda[\varphi(x,y,\xi)+t p(y,\xi)/\lambda]}q(t/\lambda,x,y,\lambda\xi)\mathrm{d}t\mathrm{d}\xi. \end{align}\] Since \(q\in S^0\), all \(t\) and \(\xi\) derivatives of \(q(t/\lambda,x,y,\lambda\xi)\) are uniformly bounded in \(\lambda\). By 51 , \[\begin{align} |\partial_{\xi}[\varphi(x,y,\xi)+t p(y,\xi)/\lambda]|\gtrsim |x-y|-\mathcal{O}(\lambda^{-1}) \end{align}\] on the support of the integrand. Therefore, if \(|x-y|\geq C\lambda^{-1}\) with \(C\) sufficiently large, nonstationary phase yields \[\begin{align} |K_{\lambda}(x,y)|\lesssim_N \lambda^d (1+\lambda|x-y|)^{-N}. \end{align}\] The claim follows. ◻
Consider a cover \(\{B_j\}\) of \(M\) by geodesic balls of radius \(\lambda^{-1}\), and let \(\{\psi_j\}\) be a partition of unity subordinate to this cover. We define the weight functions \[\begin{align} \label{def4632w95j} w_{j}(x)=\lambda^d(1+\lambda\mathop{\mathrm{dist}}(x,B_j))^{-100d},\quad x\in M. \end{align}\tag{52}\]
Lemma 10. Let \(f\in L^2(M)\), and assume that \(f\) is spectrally localised to frequencies at most \(\lambda\), with respect to \(P\). Then, for every \(j\), \[\begin{align} \|f\|_{L^{\infty}(B_j)}\lesssim \|f\|_{L^1(w_j)}+\|f\|_{L^1(M)}. \end{align}\]
Proof. This is an immediate consequence of Lemma 9. ◻
Lemma 11. Let \(f\in L^2(M)\), and assume that \(f\) is spectrally localised to frequencies at most \(\lambda\geq1\), with respect to \(P\). Let \(\Lambda\subset M\) be a set of \(\lambda^{-1}\)-separated points. Then for any \(p\in [1,\infty]\), \[\begin{align} \|f\|_{\ell^p(\Lambda)}\lesssim \lambda^{d/p}\|f\|_{L^p(M)}. \end{align}\]
Proof. Using Lemma 10, the estimate for the first term follows as in the proof of Proposition 9. The estimate for the second term is trivial since \(\#\Lambda\lesssim\lambda^d\) and \(\|f\|_{L^1(M)}\lesssim \|f\|_{L^p(M)}\) by Hölder’s inequality and \(\lambda\geq1\). ◻
© 2026 by the authors. This paper may be reproduced, in its entirety, for non-commercial purposes. J.-C. C. acknowledges support through the Engineering & Physical Sciences Research Council (EP/X011488/1). E. S. acknowledges support through the Austrian Science Fund (FWF) [Grant-DOI 10.55776/P35322, 10.55776/PAT5120424, and 10.55776/PAT4632823].↩︎