January 01, 1970
Let \((M,g)\) be a closed \(n\)-dimensional Riemannian manifold with nonpositive sectional curvature. We prove sharp, logarithmically improved spectral cluster bounds for orthonormal systems in the supercritical range. More precisely, for spectral windows of size \((\log \lambda)^{-1}\), we obtain the orthonormal analogue of the logarithmically improved \(L^q\) estimates of Hassell–Tacy. Our argument combines the universal orthonormal spectral cluster bounds of Frank–Sabin with Bérard-type kernel estimates and a generalization of the Bourgain–Shao–Sogge–Yao multiplier estimate to the orthonormal setting.
Let \((M, {\rm g})\) be a closed \(n\)-dimensional manifold and \(-\Delta_{\rm g}\) be the associated Laplace–Beltrami operator, for \(n\geq 2\). For \(\lambda\geq 2\) and \(\varepsilon(\lambda)\in (0,1]\), the spectral projector with window size \(\varepsilon\) and its corresponding eigenspace are defined, using the spectral theorem, as \[\Pi_{\lambda,\varepsilon(\lambda)}\relax \mathbf{1}(\sqrt{-\Delta_{\rm g}}\in[\lambda,\lambda+\varepsilon(\lambda))), \quad E_{\lambda,\varepsilon(\lambda)}\relax \mathop{\rm Ran}\Pi_{\lambda,\varepsilon(\lambda)} .\] For \(\varepsilon(\lambda)=1\), Sogge [1] proved the universal bounds \[\begin{align} \label{ineq:Sogge-uni-bd} \|u\|_{L^q(M)} \lesssim\lambda^{\delta(q)}\|u\|_{L^{2}(M)} ,\quad u\in E_{\lambda,1} \end{align}\tag{1}\] where, for \(2\leq q\leq\infty\), \[\begin{align} \label{eq:delta40q41} \delta(q)\relax \begin{cases} \frac{n-1}{2}(\frac{1}{2}-\frac{1}{q}), & 2\leq q\leq q_c ,\\ n(\frac{1}{2} -\frac{1}{q})-\frac{1}{2}, & q_c\leq q \leq \infty , \end{cases} \quad \text{and} \quad q_c\relax \frac{2(n+1)}{n-1} . \end{align}\tag{2}\] As was shown in [2], the universal bounds 1 are sharp on any closed manifold.
More recently, Frank–Sabin [3] proved the following generalization of 1 : for any orthonormal system \((u_j)_{j\in J}\subset E_{\lambda,1}\) and coefficients \(\nu=(\nu_j)_{j\in J}\subset\mathbb{C}\), \[\begin{align} \label{ineq:FrankSabin-uni-bd} \bigg\|\sum_{j\in J}\nu_j |u_j|^2\bigg\|_{L^{q/2}(M)}\lesssim \lambda^{2\delta(q)} \|\nu\|_{\ell^{\alpha(q)}} , \end{align}\tag{3}\] where the implicit constant is independent of \(\lambda\), \(J\), \(\nu\) and the orthonormal system, and \[\begin{align} \label{eq:alpha-q} \alpha(q)\relax \begin{cases} \frac{2q}{q+2}& 2\leq q\leq q_c ,\\ \frac{q(n-1)}{2n}, & q_c\leq q \leq \infty . \end{cases} \end{align}\tag{4}\] Here and in the following, \(J\) denotes any countable index set. By contrast, combining 1 with the triangle inequality yields 3 with \(\alpha = 1\), even without the orthogonality assumption. The key point is that, for \(q>2\), Frank–Sabin’s inequality holds with some \(\alpha > 1\). This is in line with the Lieb–Thirring philosophy that orthogonality needs space.
We now assume that \((M,g)\) has nonpositive curvature, in the sense that all the sectional curvatures are nonpositive. In this setting, for \(\varepsilon(\lambda)=(\log \lambda)^{-1}\), Hassell and Tacy [4] obtained the sharp bounds \[\begin{align} \label{eq:Hassell-Tacy} \|u\|_{L^{q}(M)} \lesssim_q (\log \lambda)^{-1/2} \lambda^{n(\frac{1}{2}-\frac{1}{q})-\frac{1}{2}}\|u\|_{L^2(M)},\quad u\in E_{\lambda, (\log \lambda)^{-1}} \end{align}\tag{5}\] for all supercritical exponents \(q>q_c\). A well-known orthogonality argument shows that 5 cannot be improved.
The case \(p=\infty\) in 5 follows from earlier results of Bérard [5] and is related to improved remainder estimates in the Weyl law, \[\begin{align} \label{eq:weyl-law} N(\lambda)\relax \operatorname{Tr}(\mathbf{1}({-\Delta_{\rm g}}<\lambda^2))=|M|\frac{|B^n|}{(2\pi)^n}\lambda^{n}+\mathcal{O}\left(\frac{\lambda^{n-1}}{\log\lambda}\right). \end{align}\tag{6}\] The whole range of supercritical \(L^q\) bounds was later established under a weaker dynamical assumption in the work of Canzani and Galkowski [6].
The aim of this paper is to generalize 5 to systems of orthonormal functions. The following theorem is our main result.
Theorem 1. Let \((M,g)\) be a closed Riemannian manifold of dimension \(n\), all of whose sectional curvatures are nonpositive, and let \(q>q_c\) and \(1\leq \beta<\frac{q(n-1)}{2n}\). Then for any orthonormal system \((u_j)_{j\in J}\subset E_{\lambda,(\log\lambda)^{-1}}\) and coefficients \(\nu=(\nu_j)_{j\in J}\subset{\mathbb{C}}\), \[\begin{align} \label{ineq:main} \bigg\|\sum_{j\in J}\nu_j |u_j|^2\bigg\|_{L^{q/2}(M)}\lesssim_{q,\beta} (\log\lambda)^{-1} \lambda^{n(1-\frac{2}{q})-1} \|\nu\|_{\ell^{\beta}}. \end{align}\qquad{(1)}\]
The theorem is optimal in the following two senses: first, the factor \((\log\lambda)^{-1} \lambda^{n(1-\frac{2}{q})-1}\) cannot be improved. This follows from the optimality of 5 by taking \(\# J=1\). Second, if \(q<\infty\), the inequality fails for all \(\beta\ge \frac{q(n-1)}{2n}\). This is seen by taking an orthonormal basis \((u_j)_{j\in J}\) of the eigenspace \(E_{\lambda,(\log\lambda)^{-1}}\) and \(\nu_j=1\) for all \(j\in J\). Indeed, by Hölder’s inequality, \[\begin{align} &|M|^{1-\frac{2}{q}}\limsup_{\lambda \to \infty} \lambda^{1-n} (\log \lambda) \bigg\|\sum_{j\in J} |u_j|^2\bigg\|_{L^{q/2}(M)} \geq \limsup_{\lambda \to \infty} \lambda^{1-n} (\log \lambda) \bigg\lVert\sum_{j\in J} |u_j|^2\bigg\rVert_{L^{1}(M)} \\ &=\limsup_{\lambda \to \infty} \lambda^{1-n} (\log \lambda)\dim E_{\lambda,(\log\lambda)^{-1}} =\limsup_{\lambda \to \infty} \lambda^{1-n} (\log \lambda)(N(\lambda+(\log\lambda)^{-1})-N(\lambda))>0 \end{align}\] where the final inequality follows from the elementary fact that \(\limsup_{\lambda \to \infty} \lambda^{-n}N(\lambda)>0\). On the other hand, \(\|\nu\|_{\ell^{\beta}}=(\# J)^{1/\beta}\). Combining these observations, we see that ?? can only hold if \[\begin{align} \frac{\lambda^{n-1}}{\log\lambda}\lesssim\lambda^{n(1-\frac{2}{q})-1}(\log\lambda)^{-1 }\Big(\frac{\lambda^{n-1}}{\log\lambda}\Big)^{\frac{1}{\beta}}. \end{align}\] It follows from the relation \(n(1-\frac{2}{q})-1 + (n-1)\frac{2n}{q(n-1)} = n-1\) that we must have \(\beta < \frac{q(n-1)}{2n}\) if \(q<\infty\). For \(q=\infty\), ?? holds with \(\beta=\infty\). This was proved by Ren and Zhang [7]. Alternatively, it follows from 5 and, e.g., [8].
The notation \(A\lesssim B\) means \(A\leq CB\), for some unspecified constant \(C\) that may depend on fixed quantities such as \(M,q\) and \(\beta\) but not on \(\lambda,J,\nu\) or the orthonormal system (or on \(W,W_1,W_2\) in the dual estimates). To emphasize the dependence on parameters, we sometimes use subscripts, e.g. \(A\lesssim_{q,\beta} B\). When we write \(A\lesssim_N B_N\), we mean that for every \(N>0\) there exists a constant \(C_N\) such that \(A\leq C_N B_N\).
For an operator \(S:L^2(M)\to L^2(M)\), we denote its operator norm by \(\|S\|\) and its \(p\)-Schatten norm by \(\|S\|_{\mathfrak{S}^p}\mathrel{\vcenter{:}}= (\mathop{\mathrm{Tr}}(S^*S)^{\frac{p}{2}})^{\frac{1}{p}}\) for \(1\leq p<\infty\). We often omit \(M\) from the notation, e.g. we write \(L^p\) instead of \(L^p(M)\).
In this section, we prove a spectral multiplier estimate that only uses the universal bounds of Frank–Sabin as input. This is a generalization of the Bourgain–Shao–Sogge–Yao multiplier estimate [9] to the orthonormal setting and may be of independent interest.
We set \[\begin{align} P=\sqrt{-\Delta_{\rm g}}\quad and\quad \Pi_{k,1}\relax \mathbf{1}(P\in [k,k+1))\;for k\in {\mathbb{N}}\cup\{0\}. \end{align}\]
Theorem 2. Let \((M,{\rm g})\) be a closed Riemannian manifold of dimension \(n\geq 2\), let \(m:[0,\infty)\to {\mathbb{C}}\) be a bounded Borel function and \(2\leq q\leq\infty\). Let \(\delta(q)\) and \(\alpha(q)\) be given by 2 and 4 , respectively. Assume that \[\begin{align} K_q(m)\relax \bigg(\sum_{k=0}^{\infty}\sup_{\tau\in [k,k+1)}|m(\tau)|^2(1+k)^{2\delta(q)}\bigg)^{1/2}<\infty . \end{align}\] Then for any orthonormal system \((u_j)_{j\in J}\subset L^2(M)\) and coefficients \(\nu=(\nu_j)_{j\in J}\subset{\mathbb{C}}\), \[\begin{align} \label{eq46multiplier321} \bigg\|\sum_{j\in J}\nu_j |m(P)u_j|^2\bigg\|_{L^{q/2}(M)}\lesssim K_q(m)^2\|\nu\|_{\ell^{\alpha(q)}}. \end{align}\qquad{(2)}\] Moreover, for all \(W_1,W_2\in L^{2(q/2)'}(M)\), \[\begin{align} \label{eq46multiplier322} \|W_1m(P)W_2\|_{\mathfrak{S}^{\alpha(q)'}(L^2(M))}\lesssim K_q(|m|^{1/2})^2 \|W_1\|_{L^{2(q/2)'}(M)}\|W_2\|_{L^{2(q/2)'}(M)}. \end{align}\qquad{(3)}\]
Proof. By Frank–Sabin’s duality principle [10], ?? is equivalent to \[\begin{align} \label{spectral32multiplier32bound32dual32version} \|Wm(P)\|^2_{\mathfrak{S}^{2\alpha(q)'}}\lesssim K_q(m)^2\|W\|^2_{L^{2(q/2)'}} ,\quad W\in L^{2(q/2)'}. \end{align}\tag{7}\] Assume first that \(\mathop{\mathrm{supp}}m\subset [0,N+1]\) for some \(N\in{\mathbb{N}}\). For \(\varepsilon>0\), we define \[\begin{align} S_\varepsilon\relax \sum_{k=0}^Nc_{k,\varepsilon}^{-1}\Pi_{k,1}m(P),\quad c_{k,\varepsilon}\relax \sup_{\tau\in [k,k+1)}|m(\tau)|+\varepsilon. \end{align}\] Using orthogonality, \[\begin{align} \label{orthogonality} \Pi_{k,1}\Pi_{k',1}=\delta_{k,k'}\Pi_{k,1} ,\quad k,k'\in{\mathbb{N}}, \end{align}\tag{8}\] and the support assumption on \(m\), we observe that \[\begin{align} \sum_{k=0}^Nc_{k,\varepsilon} \Pi_{k,1} S_\varepsilon=\sum_{k=0}^N\Pi_{k,1}m(P)=m(P). \end{align}\] By the spectral theorem, \[\begin{align} \label{eq:32norm32S32leq321} \|S_\varepsilon\|_{L^2\to L^2}\leq 1. \end{align}\tag{9}\] Thus \[\begin{align} \|Wm(P)\|^2_{\mathfrak{S}^{2\alpha(q)'}} &\leq \bigg\|\sum_{k=0}^Nc_{k,\varepsilon} W\Pi_{k,1}\bigg\|^2_{\mathfrak{S}^{2\alpha(q)'}} =\bigg\|\sum_{k=0}^Nc_{k,\varepsilon}^2 W\Pi_{k,1}\overline{W}\bigg\|_{\mathfrak{S}^{\alpha(q)'}} \leq \sum_{k=0}^{\infty}c_{k,\varepsilon}^2\|W\Pi_{k,1}\overline{W}\|_{\mathfrak{S}^{\alpha(q)'}}\\ &\lesssim \Big( \sum_{k=0}^{\infty}\sup_{\tau\in[k,k+1)}|m(\tau)|^2(1+k)^{2\delta(q)}+ \varepsilon^2 N^{2\delta(q)+1} \Big) \|W\|^2_{L^{2(q/2)'}} \end{align}\] where we used \(\|AB\|_{\mathfrak S^p}\leq \|A\|_{\mathfrak S^p}\|B\|\) and 9 in the first inequality, followed by \(\|A\|_{\mathfrak S^{2p}}^2=\|AA^*\|_{\mathfrak S^p}\) and orthogonality of spectral projectors to eliminate cross terms. The final inequality uses the Frank–Sabin bound 3 for spectral projectors, or more precisely, the dual inequality \[\|W\Pi_{k,1}\overline{W}\|_{\mathfrak{S}^{\alpha(q)'}}\lesssim (1+k)^{2\delta(q)}\|W\|^2_{L^{2(q/2)'}},\] see [3]. Passing to the limit \(\varepsilon\to 0\) yields 7 .
To remove the support assumption, observe that, since \(P\) is a self-adjoint operator with discrete spectrum, \(\mathbf{1}_{[0,N+1]}(P)\) is a sequence of monotonically increasing finite-dimensional orthogonal projections that converges strongly to the identity operator on \(L^2(M)\), as \(N\to\infty.\) Setting \(m_N(P):=m(P)\mathbf{1}_{[0,N+1]}(P)\), it follows by [11] that \[\|Wm(P)\|^2_{\mathfrak{S}^{2\alpha(q)'}}\leq \sup_N\|Wm_N(P)\|^2_{\mathfrak{S}^{2\alpha(q)'}} \lesssim K_q(m_N)^2\|W\|^2_{L^{2(q/2)'}}\leq K_q(m)^2\|W\|^2_{L^{2(q/2)'}}.\]
The bound ?? follows from 7 and its dual, applied to \(|m|^{1/2}\), since \[\begin{align} \|W_1m(P)W_2\|_{\mathfrak{S}^{\alpha(q)'}} &\leq \left\lVert W_1|m|^{1/2}(P)\right\rVert_{\mathfrak{S}^{2\alpha(q)'}} \bigg \lVert\frac{m}{|m|}(P)\bigg \rVert \left\lVert |m|^{1/2}(P)W_2\right\lVert_{\mathfrak{S}^{2\alpha(q)'}} \end{align}\] and \(\frac{m}{|m|}(P)\) is a partial isometry. ◻
We record the following application of the multiplier theorem to resolvent powers.
Corollary 1. Let \((M,{\rm g})\) be a closed Riemannian manifold of dimension \(n\geq 2\). Let \(2\leq q\leq\infty\) and \(\gamma>1/2\). Then for any orthonormal system \((u_j)_{j\in J}\subset \mathop{\rm Ran}\mathbf{1}_{[0,2\lambda]}(P)\) and coefficients \(\nu=(\nu_j)_{j\in J}\subset{\mathbb{C}}\), \[\begin{align} \bigg\|\sum_{j\in J}\nu_j |(-\Delta_{{\rm g}}-(\lambda+i)^2)^{-\gamma}u_j|^2\bigg\|_{L^{q/2}(M)}\lesssim_{\gamma} \lambda^{2\delta(q)-2\gamma}\|\nu\|_{\ell^{\alpha(q)}}. \end{align}\]
Proof. Consider the function \(m_{\gamma}(\tau)\mathrel{\vcenter{:}}= (\tau^2-(\lambda+i)^2)^{-\gamma}\mathbf{1}_{[0,2\lambda]}(\tau).\) Then, since \(\gamma>1/2\), \[K_q(m_{\gamma})\lesssim \lambda^{\delta(q)-\gamma} \bigg(\sum_{k\leq 2\lambda}(1+|k-\lambda|)^{-2\gamma}\bigg)^{1/2} \lesssim_{\gamma} \lambda^{\delta(q)-\gamma}.\qedhere\] ◻
Remark 3. The case \(\gamma=1\) also follows from [8]. The elliptic part of resolvent powers, corresponding to the projection onto \(\mathop{\rm Ran}\mathbf{1}_{[2\lambda,\infty)}(P)\), can be handled by [3], combined with [8].
In the following, let \(T=c_0\log\lambda\), where \(c_0\) is a small constant to be fixed later. We adapt an argument of Bourgain–Shao–Sogge–Yao [9] to the case of orthonormal systems. Let \(q>q_c,\) and \(1\leq\beta<\frac{q(n-1)}{2n}\). By duality, to prove ?? it suffices to show that \[\begin{align} \label{eq:dual32main32estimate} \|W\Pi_{\lambda,T^{-1}}\overline{W}\|_{\mathfrak{S}^{\beta'}}\lesssim_{q,\beta} \lambda^{n(1-\frac{2}{q})-1}(\log\lambda)^{-1}\|W\|^2_{L^{2(q/2)'}}. \end{align}\tag{10}\] Let \(a\) be a nonnegative Schwartz function on \({\mathbb{R}}\) such that \[\begin{align} a\geq \mathbf{1}_{[-1,1]} \quad\text{and}\quad \mathop{\mathrm{supp}}(\widehat{a})\subset[-1,1]. \end{align}\] Since \(a\geq \mathbf{1}_{[-1,1]}\), it follows that \[\begin{align} Wa(T(P-\lambda))\overline{W}\geq W\Pi_{\lambda,T^{-1}} \overline{W} \end{align}\] in the quadratic form sense. Since for positive compact operators \(0\le B\le A\), one has \(\|B\|_{\mathfrak S^p}\le \|A\|_{\mathfrak S^p}\), 10 would thus follow from \[\begin{align} \label{eq:dual32main32estimate32smooth} \|Wa(T(P-\lambda))\overline{W}\|_{\mathfrak{S}^{\beta'}}\lesssim \lambda^{n(1-\frac{2}{q})-1}(\log\lambda)^{-1}\|W\|^2_{L^{2(q/2)'}}. \end{align}\tag{11}\] By Fourier inversion, we can write \[\begin{align} a(T(P-\lambda))=\frac{1}{2\pi T}\int_{-\infty}^{\infty}\widehat{a}(t/T){\rm e}^{-\mathrm{i}t\lambda}{\rm e}^{\mathrm{i}tP}{\rm d}t . \end{align}\] Using that \(2\cos (tP) = e^{i t P} + e^{-i t P}\) and discarding the smoothing operator \(a(-T(P+\lambda))\) arising from the \(e^{-i t P}\) term (see Remark 4 below), it suffices to consider \[\begin{align} A\relax \frac{1}{T}\int_{-\infty}^{\infty}\widehat{a}(t/T){\rm e}^{-\mathrm{i}t\lambda}\cos(tP) {\rm d}t \end{align}\] and to prove \[\begin{align} \label{eq:dual32main32estimate32smooth32cos} \|WA\overline{W}\|_{\mathfrak{S}^{\beta'}}\lesssim \lambda^{n(1-\frac{2}{q})-1}(\log\lambda)^{-1}\|W\|^2_{L^{2(q/2)'}},\quad q>q_c,\quad \beta<\frac{q(n-1)}{2n} . \end{align}\tag{12}\]
We now split \(A\) into a local and global part, \(A=A_{\rm loc}+A_{\rm glo}\), where \[\begin{align} A_{\rm loc}\relax \frac{1}{T}\int_{-\infty}^{\infty}b(t)\widehat{a}(t/T){\rm e}^{-\mathrm{i}t\lambda}\cos(tP) {\rm d}t,\quad A_{\rm glo}\relax \frac{1}{T}\int_{-\infty}^{\infty}(1-b(t))\widehat{a}(t/T){\rm e}^{-\mathrm{i}t\lambda}\cos(tP) {\rm d}t \end{align}\] and \(b\in C_c^{\infty}([-2,2])\) is such that \(b=1\) on \([-1,1]\).
Lemma 1 (Estimate for the local part). For \(2\leq q\leq\infty\), \[\begin{align} \label{eq:Aloc32at32q} \|WA_{\rm loc}\overline{W}\|_{\mathfrak{S}^{\alpha(q)'}}\lesssim_q \lambda^{2\delta(q)}(\log\lambda)^{-1}\|W\|^2_{L^{2(q/2)'}} ,\quad W\in L^{2(q/2)'}(M), \end{align}\qquad{(4)}\] where \(\delta(q)\) and \(\alpha(q)\) are given by 2 and 4 , respectively.
Proof. We write \(A_{\rm loc}\) as a spectral multiplier, \[\begin{align} A_{\rm loc}=m_{\rm loc}(P),\quad m_{\rm loc}(\tau)\relax \frac{1}{T}\int_{-\infty}^{\infty}b(t)\widehat{a}(t/T){\rm e}^{-\mathrm{i}t\lambda}\cos(t\tau) {\rm d}t. \end{align}\] Integration by parts yields \[\begin{align} \label{eq:bound32mloc} |m_{\rm loc}(\tau)|\lesssim_N T^{-1}((1+|\lambda-\tau|)^{-N}+(1+|\lambda+\tau|)^{-N}) . \end{align}\tag{13}\] Using Theorem 2 and observing that \[\begin{align} K_q(|m_{\rm loc}|^{1/2}) \lesssim_N T^{-1/2}\lambda^{\delta(q)}+\lambda^{-N} , \end{align}\] yields the claimed bound. ◻
Remark 4. Since \[|a(-T(\tau+\lambda)|\lesssim_N (1+|\lambda+\tau|)^{-N},\] the same proof yields \[\|W a(-T(P+\lambda)) \overline{W}\|_{\mathfrak S^{\alpha(q)'}} \lesssim_N \lambda^{-N}\|W\|^2_{L^{2(q/2)'}}.\]
Lemma 2 (Estimate for the global part). Under the nonpositive curvature assumption and for \(q>q_c\), there exists \(\epsilon(q)>0\) such that \[\begin{align} \label{Aglo32at32s3932with32power32gain} \|WA_{\rm glo}\overline{W}\|_{\mathfrak{S}^{2(q/2)'}}&\lesssim_{q}\lambda^{n(1-\frac{2}{q})-1-\epsilon(q)}\|W\|^2_{L^{2(q/2)'}} ,\quad W\in L^{2(q/2)'}(M). \end{align}\qquad{(5)}\] Moreover, for any \(\beta<\frac{q(n-1)}{2n}\) there exists \(\epsilon(q,\beta)>0\) such that \[\begin{align} \label{Aglo32final32at32alphaq} \|WA_{\rm glo}\overline{W}\|_{\mathfrak{S}^{\beta'}} \lesssim_{q,\beta} \lambda^{n(1-\frac{2}{q})-1-\epsilon(q,\beta)} \|W\|^2_{L^{2(q/2)'}} ,\quad W\in L^{2(q/2)'}(M). \end{align}\qquad{(6)}\]
Proof of ?? . Again we write \(A_{\rm glo}\) as a spectral multiplier, \[\begin{align} A_{\rm glo}=m_{\rm glo}(P),\quad m_{\rm glo}(\tau)\relax \frac{1}{T}\int_{-\infty}^{\infty}(1-b(t))\widehat{a}(t/T){\rm e}^{-\mathrm{i}t\lambda}\cos(t\tau) {\rm d}t . \end{align}\] Integration by parts yields \[\begin{align} \label{eq:bound32mglo} |m_{\rm glo}(\tau)|\lesssim_N (1+|\lambda-\tau|)^{-N}+(1+|\lambda+\tau|)^{-N}. \end{align}\tag{14}\] Using Theorem 2 and the estimate \[\begin{align} K_{q_c}(|m_{\rm glo}|^{1/2}) \lesssim\lambda^{\frac{1}{q_c}} \end{align}\] yields \[\begin{align} \label{eq:A95glo32at32q61q95c} \|W_1A_{\rm glo}W_2\|_{\mathfrak{S}^{n+1}}\lesssim \lambda^{\frac{n-1}{n+1}}\|W_1\|_{L^{n+1}}\|W_2\|_{L^{n+1}}. \end{align}\tag{15}\]
We recall that \(A_{\rm glo}\) depends on \(T=c_0\log\lambda\). Given \(\eta>0\) we may now fix \(c_0\) such that the pointwise kernel estimate \[\begin{align} \label{L1toLinfty32bound32Aglo32Bourgain--Shao--Sogge--Yao} \|A_{\rm glo}\|_{L^{\infty}(M\times M)}\lesssim_{\eta} \lambda^{\frac{n-1}{2}+\eta} \end{align}\tag{16}\] holds, where we denoted the kernel of \(A_{\rm glo}\) by the same symbol as the operator. This follows from [9] and is also implicit in earlier work of Bérard [5] and Hassell–Tacy [4]. As an immediate consequence of 16 , we have the Hilbert–Schmidt bound \[\begin{align} \label{eq:A95glo32at32q61infinity} \|W_1A_{\rm glo}W_2\|_{\mathfrak{S}^{2}}\lesssim_{\eta} \lambda^{\frac{n-1}{2}+\eta}\|W_1\|_{L^{2}}\|W_2\|_{L^{2}}. \end{align}\tag{17}\] We interpolate the bilinear map \[\mathcal{T}(W_1,W_2)\relax W_1A_{\rm glo}W_2 .\] By 15 , we have \[\|\mathcal{T}(W_1,W_2)\|_{\mathfrak S^{n+1}} \lesssim \lambda^{\frac{n-1}{n+1}} \|W_1\|_{L^{n+1}}\|W_2\|_{L^{n+1}},\] while 17 implies \[\|\mathcal{T}(W_1,W_2)\|_{\mathfrak S^{2}} \lesssim_{\eta} \lambda^{\frac{n-1}{2}+\eta} \|W_1\|_{L^{2}}\|W_2\|_{L^{2}} .\] Hence, by bilinear complex interpolation (see, e.g., [12]), for every \(0\le \theta\le 1\), \[\begin{align} \label{eq4632TW1W2} \|\mathcal{T}(W_1,W_2)\|_{\mathfrak S^{r_\theta}} \lesssim_{\eta} \lambda^{(1-\theta)\frac{n-1}{n+1} +\theta(\frac{n-1}{2}+\eta)} \|W_1\|_{L^{p_\theta}}\|W_2\|_{L^{p_\theta}}, \end{align}\tag{18}\] where \[\begin{align} \label{eq4632ptheta32rtheta} \frac{1}{p_\theta}=\frac{1-\theta}{n+1}+\frac{\theta}{2}, \qquad \frac{1}{r_\theta}=\frac{1-\theta}{n+1}+\frac{\theta}{2}. \end{align}\tag{19}\] Here we used [12] with \(A_0^{(1)}=A_0^{(2)}=L^{n+1}\), \(A_1^{(1)}=A_1^{(2)}=L^{2}\), \(B_0=\mathfrak{S}^{n+1}\), \(B_1=\mathfrak{S}^2\) and the fact that \(L^p\) and Schatten spaces are interpolation spaces, with \((L^{p_1},L^{p_2})_{\theta}=L^{p_{\theta}}\) and \((\mathfrak{S}^{p_1},\mathfrak{S}^{p_2})_{\theta}=\mathfrak{S}^{p_{\theta}}\) for \(\frac{1}{p_{\theta}}=\frac{1-\theta}{p_1}+\frac{\theta}{p_2}\) and \(1\leq p_1,p_2\leq\infty\).
Choosing \(\theta=1-\frac{q_c}{q},\) we obtain \(p_\theta=r_\theta=2(q/2)'\) in 19 . Therefore, setting \(W_1=W\) and \(W_2=\overline{W}\) in 18 , \[\begin{align} \label{eq:A95glo32interpolated} \|WA_{\rm glo}\overline{W}\|_{\mathfrak S^{2(q/2)'}} \lesssim_{\eta} \lambda^{(n-1)(\frac{1}{2}-\frac{1}{q})+\eta(1-\frac{q_c}{q})} \|W\|_{L^{2(q/2)'}}^2. \end{align}\tag{20}\] Fixing \(\eta:=\frac{n-1}{4}\), this shows that ?? holds with \[\epsilon(q)=\frac{n-1}{4}\Big(1-\frac{q_c}{q}\Big).\qedhere\] ◻
Proof of ?? . Let \(\chi\in C_c^\infty({\mathbb{R}})\) such that \(0\leq \chi\leq 1\), \(\chi=1\) on \([-1,1]\), and \(\mathop{\mathrm{supp}}\chi\subset [-2,2]\). For \(R\in [1,\lambda]\), to be fixed later, define \[m_{\rm near}^{(R)}(\tau)\relax \chi\Big(\frac{\tau-\lambda}{R}\Big)m_{\rm glo}(\tau), \qquad m_{\rm far}^{(R)}(\tau)\relax \Big(1-\chi\Big(\frac{\tau-\lambda}{R}\Big)\Big)m_{\rm glo}(\tau),\] and \[\label{def:Anear-far} A_{\rm near}^{(R)}\relax m_{\rm near}^{(R)}(P), \qquad A_{\rm far}^{(R)}\relax m_{\rm far}^{(R)}(P).\tag{21}\] Then \(A_{\rm glo}=A_{\rm near}^{(R)}+A_{\rm far}^{(R)}\). We will collect some properties of these operators in Lemma 3 below.
We first estimate the near part. In view of ?? , we may assume that \(\beta'<2(q/2)'\). Using ?? with \(s'=2(q/2)'\), we have \[\begin{align} \|WA_{\rm near}^{(R)}\overline{W}\|_{\mathfrak{S}^{\beta'}} \lesssim (R\lambda^{n-1})^{\frac{1}{\beta'}-\frac{1}{2(q/2)'}} \|WA_{\rm near}^{(R)}\overline{W}\|_{\mathfrak{S}^{2(q/2)'}}. \end{align}\] Since \(A_{\rm near}^{(R)}\) is a spectral multiplier satisfying the same pointwise bound 14 as \(A_{\rm glo}\), the estimate 20 applies to \(A_{\rm near}^{(R)}\) as well. Thus, after some straightforward arithmetic simplifications, using \(\alpha(q)=\frac{q(n-1)}{2n}\) for \(q\geq q_c\), \[\begin{align} \|WA_{\rm near}^{(R)}\overline{W}\|_{\mathfrak{S}^{\beta'}} &\lesssim R^{\frac{1}{\beta'}-\frac{1}{2(q/2)'}} \lambda^{n(1-\frac{2}{q})-1+\eta(1-\frac{q_c}{q})-(n-1)(\frac{1}{\beta}-\frac{2n}{q(n-1)})} \|W\|^2_{L^{2(q/2)'}}. \end{align}\] We now set \[\begin{align} \epsilon(q,\beta)\mathrel{\vcenter{:}}= \frac{n-1}{4}\Big(\frac{1}{\beta}-\frac{2n}{q(n-1)}\Big),\quad \eta(q,\beta)\mathrel{\vcenter{:}}= \frac{2\epsilon(q,\beta)}{1-\frac{q_c}{q}},\quad R(\lambda,q,\beta)\mathrel{\vcenter{:}}= \lambda^{\frac{\epsilon(q,\beta)}{\frac{1}{\beta'}-\frac{1}{2(q/2)'}}} \end{align}\] and fix \(\eta=\eta(q,\beta)\), \(R=R(\lambda,q,\beta)\) to obtain \[\begin{align} \label{estimate32near32part} \|WA_{\rm near}^{(R)}\overline{W}\|_{\mathfrak{S}^{\beta'}} &\lesssim \lambda^{n(1-\frac{2}{q})-1-\epsilon(q,\beta)} \|W\|^2_{L^{2(q/2)'}}. \end{align}\tag{22}\]
Using ?? with \(N\) sufficiently large yields a better estimate than 22 for the far part \(A_{\rm far}^{(R)}\). Combining the near and far bounds, we obtain ?? . ◻
Lemma 3. With the above notation, the following properties hold.
(i) We have \[\begin{align} \label{eq:rank-Anear-R} \operatorname{rank} (WA_{\rm near}^{(R)}\overline{W})\lesssim R\,\lambda^{n-1} \end{align}\qquad{(7)}\] and if \(1\leq \beta'\leq s'\leq \infty\), then \[\begin{align} \label{eq:rank-improvement-Anear-R} \|WA_{\rm near}^{(R)}\overline{W}\|_{\mathfrak{S}^{\beta'}} \lesssim (R\lambda^{n-1})^{\frac{1}{\beta'}-\frac{1}{s'}} \|WA_{\rm near}^{(R)}\overline{W}\|_{\mathfrak{S}^{s'}}. \end{align}\qquad{(8)}\]
(ii) For every \(q\geq 2\) and every \(N>0\), \[\begin{align} \label{Kq-mfar-R} K_q\big(|m_{\rm far}^{(R)}|^{1/2}\big) \lesssim_N R^{-N}\lambda^{\delta(q)}. \end{align}\qquad{(9)}\] Consequently, Theorem 2 implies \[\begin{align} \label{eq:Afar-negligible-R} \|WA_{\rm far}^{(R)}\overline{W}\|_{\mathfrak{S}^{\alpha(q)'}} \lesssim_N R^{-N}\lambda^{2\delta(q)} \|W\|_{L^{2(q/2)'}(M)}^2. \end{align}\qquad{(10)}\]
Proof. Part (i) follows by definition immediately, since \(m_{\rm near}^{(R)}\) is supported where \(|\tau-\lambda|\leq 2R\), so \[\operatorname{rank}A_{\rm near}^{(R)} \leq \operatorname{rank}\mathbf{1}_{[\lambda-2R,\lambda+2R]}(P) \lesssim R\,\lambda^{n-1},\] where the last inequality follows from the Weyl law 6 ; the logarithmic improvement is irrelevant here.
By ?? and Hölder, \[\|WA_{\rm near}^{(R)}\overline{W}\|_{\mathfrak{S}^{\beta'}} \lesssim (R\lambda^{n-1})^{\frac{1}{\beta'}-\frac{1}{s'}} \|WA_{\rm near}^{(R)}\overline{W}\|_{\mathfrak{S}^{s'}}.\]
For (ii), on the support of \(m_{\rm far}^{(R)}\) one has \(|\tau-\lambda|\geq R\). Therefore, by the assumption 14 , \[|m_{\rm far}^{(R)}(\tau)| \lesssim_N R^{-N/2}(1+|\lambda-\tau|)^{-N/2}+(1+|\lambda+\tau|)^{-N}.\] Hence \[\begin{align} K_q\big(|m_{\rm far}^{(R)}|^{1/2}\big)^2 = \sum_{k=0}^\infty \sup_{\tau\in[k,k+1)}|m_{\rm far}^{(R)}(\tau)| (1+k)^{2\delta(q)} \lesssim_N R^{-N/2}\lambda^{2\delta(q)}+\lambda^{-N}. \end{align}\] After replacing \(N\) by \(2N\), this gives ?? . Inequality ?? follows from Theorem 2. ◻
J.-C. C. and X. S. acknowledge support through the Engineering & Physical Sciences Research Council (EP/X011488/1). N. N. N. acknowledges partial support through the European Union through the European Research Council’s Starting Grant FermiMath, grant agreement nr. 101040991.
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