Sharp Spectral-Cluster Restriction Bounds for Orthonormal Systems

Changbiao Jian Institute of Applied Physics & Computational Mathematics, Beijing, 100088, PR China bobjian1@gmail.com , Xing Wang School of Mathematics, Hunan University, Changsha, HN 410012, PR China xingwang@hnu.edu.cn and Yakun Xi School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, PR China yakunxi@zju.edu.cn

Abstract.

For a smooth $k$ -dimensional submanifold $\Sigma$ of a $d$ -dimensional compact Riemannian manifold $M$ , we extend the $L^{p}(\Sigma)$ restriction bounds of Burq-Gérard-Tzvetkov [BGT07]—originally proved for individual Laplace–Beltrami eigenfunctions—to arbitrary systems of $L^{2}(M)$ -orthonormal functions. Our bounds are essentially optimal for every triple $(k,d,p)$ with $p\geq 2$ , except possibly when $d\geq 3,\quad k=d-1,\quad 2\leq p\leq 4.$ This work is inspired by Frank and Sabin [Fra17b], who established analogous $L^{p}(M)$ bounds for $L^{2}(M)$ -orthonormal systems.

1. Introduction

Let $(M,g)$ be a compact boundaryless Riemannian manifold of dimension $d\geq 2$ , and let $\Delta_{g}$ be the corresponding Laplace–Beltrami operator on $M$ . Let $e_{\lambda}$ be an $L^{2}$ -normalized eigenfunction of $\Delta_{g}$ , that is,

-\Delta_{g}e_{\lambda}=\lambda^{2}e_{\lambda},\quad\text{and }\int_{M}|e_{\lambda}|^{2}\,dV_{g}=1.

Studying how such eigenfunctions concentrate on a given manifold is of great interest. One important way is to study the growth of the $L^{p}$ norms $\|e_{\lambda}\|_{L^{p}(M)}$ . It is a classical result of Sogge [Sog88] that the $L^{p}$ norms of $e_{\lambda}$ satisfy

(1.1)

\|e_{\lambda}\|_{L^{p}{(M)}}\leq C\lambda^{\sigma{(p,d)}}\|e_{\lambda}\|_{L^{2}(M)},

where $2\leq p\leq\infty$ and $\sigma(p,d)$ is given by

(1.2)

\displaystyle\sigma(p,d)=\begin{cases}\dfrac{d-1}{2}\bigg{(}\dfrac{1}{2}-\dfrac{1}{p}\bigg{)},&\text{if }2\leq p\leq\dfrac{2(d+1)}{d-1},\cr d\bigg{(}\dfrac{1}{2}-\dfrac{1}{p}\bigg{)}-\dfrac{1}{2},&\text{if }\dfrac{2(d+1)}{d-1}\leq p\leq\infty.\end{cases}

These estimates are the best possible for general Riemannian manifolds since they are achieved on the standard sphere $\mathbb{S}^{d}$ . These bounds of Sogge have been further improved and generalized under various geometric assumptions. We refer the reader to the articles [SZ02, BGT04, BGT05, Sog17, Bla18, BS19, CG23, BHS22] for further discussions.

In 2017, Frank and Sabin [Fra17b] generalized the bounds (1.1) to systems of orthonormal functions. To describe their results, we define the spectral projection operator

\Pi_{\lambda}:=\mathds{1}(\sqrt{-\Delta_{g}}\in I_{\lambda}),\quad\text{where }I_{\lambda}:=[\lambda,\lambda+1)

and the spectral cluster

E_{\lambda}:=\Pi_{\lambda}L^{2}(M).

Let $Q\subset E_{\lambda}$ be a linear subspace of $E_{\lambda}$ and let $\{f_{j}\}_{j\in J}$ be an orthonormal basis of $Q$ . Frank and Sabin [Fra17b] proved that

(1.3)

\Biggl{\|}\sum_{j\in J}t_{j}|f_{j}|^{2}\Biggr{\|}_{L^{p/2}(M)}\leq C\lambda^{2\sigma(p,d)}\|\{t_{j}\}\|_{l^{\alpha(d,p)}},

where

(1.4)

\displaystyle\alpha(d,p)=\begin{cases}\dfrac{2p}{p+2},&\text{if }2\leq p\leq\dfrac{2(d+1)}{d-1},\cr\dfrac{p(d-1)}{2d},&\text{if }\dfrac{2(d+1)}{d-1}\leq p\leq\infty.\end{cases}

In addition, Frank and Sabin showed that this exponent $\alpha(d,p)$ is sharp by constructing explicit examples on the two-sphere. Sogge’s bounds (1.1) corresponds to the case when $\dim Q=1$ . Furthermore, since $\alpha(d,p)>1$ for $p>2$ , these bounds improve on those obtained by simply applying triangle inequality (or Minkowski inequality) to (1.1) in this case. This gain is owing to the orthogonality of $\{f_{j}\}_{j\in J}$ over $L^{2}(M)$ .

In the work of Burq–Gérard–Tzvetkov [BGT07] (see also [Hu09]), $L^{p}$ estimates similar to (1.1) have been established for the restriction of eigenfunctions to a submanifold. In what follows, let $\Sigma\subset M$ be a smoothly embedded submanifold of dimension $k$ . Let ${\delta(k,d,p)}$ be

	$\displaystyle\delta(d-1,d,p)=\begin{cases}\dfrac{d-1}{4}-\dfrac{d-2}{2p},~{}~{}~{}~{}\text{if}~{}2\leq p<\dfrac{2d}{d-1},\\ \dfrac{d-1}{2}-\dfrac{d-1}{p},~{}~{}~{}~{}\text{if}~{}\dfrac{2d}{d-1}<p\leq\infty,\end{cases}$
	$\displaystyle\delta(d-2,d,p)=\frac{d-1}{2}-\frac{d-2}{p},\ \text{if }2\leq p\leq\infty,$
	$\displaystyle\delta(k,d,p)=\frac{d-1}{2}-\frac{k}{p},\text{ if }1\leq k\leq d-3\text{ and }~{}2\leq p\leq\infty.$

Burq–Gérard–Tzvetkov proved the following sharp restriction bounds

(1.5)

\|e_{\lambda}\|_{L^{p}(\Sigma)}\leq C\lambda^{{\delta(k,d,p)}}\|e_{\lambda}\|_{L^{2}(M)},

except a $(\log\lambda)^{\frac{1}{2}}$ loss for $(p,k)=(\frac{2d}{d-1},d-1)$ and $(p,k)=(2,d-2)$ .

A couple of years later, Hu [Hu09] gave another proof of (1.5) and removed the $\log$ loss for the case $(p,k)=(\frac{2d}{d-1},d-1)$ . Recently, the second author and Zhang [WZ21] removed the $\log$ loss for totally geodesic submanifolds and curves with non-vanishing curvature in the $(p,k)=(2,d-2)$ case. The bounds (1.5) have been improved and generalized under various geometric assumptions. See, e.g., [CS14, XZ17, Zha17, Bla18, Xi19, GMX24]. Moreover, certain $L^{p}(\Sigma)$ restriction bounds have been shown to be connected to the $L^{p}(M)$ norm of eigenfunctions via Kakeya–Nikodym norms. See, e.g., [Bou09, Sog11, BS15b, BS15a, MSXY16, BS17].

In this article, inspired by (1.3), we generalize (1.5) to systems of orthonormal eigenfunctions $\{f_{j}\}_{j\in J}$ . To be more precise, let $\Lambda=\Lambda(k,d,p)$ denote the square of the current known state-of-the-art bounds for a single eigenfunction, i.e.

\Lambda(k,d,p)=\begin{cases}\lambda^{2{\delta(k,d,p)}},~{}~{}~{}~{}\text{if}~{}(p,k)\neq(2,d-2),\\ \lambda^{2{\delta(k,d,p)}}\log\lambda,~{}~{}~{}~{}\text{if}~{}(p,k)=(2,d-2).\end{cases}

We are concerned with estimates of the form

(1.6)

\Biggl{\|}\sum_{j\in J}t_{j}\,|f_{j}|^{2}\Biggr{\|}_{L^{p/2}(\Sigma)}\leq C\,\Lambda\,\|\{t_{j}\}\|_{\ell^{\alpha}(J)}.

A result like (1.6) is natural in view of (1.3).

Our first result establishes optimal bounds in the case $\operatorname{codim}\Sigma\geq 2$ .

Theorem 1.1.

Suppose that $d\geq 3$ and $1\leq k\leq d-2$ . For any $p\in[2,\infty],$ $1\leq\alpha\leq\frac{p}{2}$ , there exists a constant $C>0$ such that

\Biggl{\|}\sum_{j\in J}t_{j}\,|f_{j}|^{2}\Biggr{\|}_{L^{p/2}(\Sigma)}\leq C\,\Lambda\,\|\{t_{j}\}\|_{\ell^{\alpha}(J)}.

The range of $\alpha$ is sharp.

The situation is far more delicate in codimension $1$ . Nonetheless, we are able to prove essentially sharp bounds when $\dim M=2$ . Let

(\alpha(k,d,p),h(k,d,p)):=\begin{cases}(\max\{2,\frac{p}{2}\},\min\{\frac{1}{2},\frac{2}{p}\})&(k,d)=(1,2),\\[4.0pt] (\tfrac{2p}{p+2},0)&k=d-1,\;2\leq p<\tfrac{2d}{d-1},\\[4.0pt] (\tfrac{2p(d-2)}{4d-p-4},\tfrac{2d-p}{p(d-2)})&k=d-1,\;\tfrac{2d}{d-1}\leq p\leq 4,\\[4.0pt] (\tfrac{p}{2},\tfrac{2}{p})&k=d-1,\;4\leq p\leq\infty.\end{cases}

Theorem 1.2.

Sppose that $d=2$ and $k=1.$

(i)

For any $p\in[2,4)\cup(4,\infty]$ and $\alpha\in[1,\alpha(1,2,p))$ there exists $C_{\alpha}>0$ such that

(1.7)

\Biggl{\|}\sum_{j\in J}t_{j}\,|f_{j}|^{2}\Biggr{\|}_{L^{p/2}(\Sigma)}\leq C_{\alpha}\,\Lambda\,\|\{t_{j}\}\|_{\ell^{\alpha}(J)}.

(ii)

For any $p\in[2,\infty]$ there exists $C>0$ such that

(1.8)

\Biggl{\|}\sum_{j\in J}t_{j}\,|f_{j}|^{2}\Biggr{\|}_{L^{p/2}(\Sigma)}\leq C\,(\log\lambda)^{h(1,2,p)}\Lambda\,\,\|\{t_{j}\}\|_{\ell^{\alpha(1,2,p)}(J)}.

(iii)

When $\Sigma$ is a great circle on the standard two-sphere $M$ , the exponent $\alpha(1,2,p)$ are essentially sharp.

In higher dimensions, for submanifolds of codimension $1$ we obtain the following result. The exponent $\alpha(d-1,d,p)$ is the best possible for $4\leq p\leq\infty$ .

Theorem 1.3.

Suppose that $d\geq 3$ and $k=d-1$ .

(i)

For any $p\in(4,\infty]$ and $\alpha\in[1,\alpha(d-1,d,p))$ there exists $C_{\alpha}>0$ such that

(1.9)

\Biggl{\|}\sum_{j\in J}t_{j}\,|f_{j}|^{2}\Biggr{\|}_{L^{p/2}(\Sigma)}\leq C_{\alpha}\,\Lambda\bigl{\|}\{t_{j}\}\bigr{\|}_{\ell^{\alpha}(J)}.

(ii)

For any $p\in[2,\infty]$ there exists $C>0$ such that

(1.10)

\Biggl{\|}\sum_{j\in J}t_{j}\,|f_{j}|^{2}\Biggr{\|}_{L^{p/2}(\Sigma)}\leq C\,(\log\lambda)^{h(d-1,d,p)}\Lambda\bigl{\|}\{t_{j}\}\bigr{\|}_{\ell^{\alpha(d-1,d,p)}(J)}.

(iii)

When $M$ is the standard $d$ -sphere, the exponent $\alpha(d-1,d,p)$ is essentially sharp for $4\leq p\leq\infty$ .

Figure 1.

\delta(1,2,p)

and

\alpha(1,2,p)

Remark 1.4.

As we shall see in Section 6, Theorem 1.2 is sharp for all possible $\dim Q$ of the orthonormal system. However, in the final section, we will show that the sharpness of Theorem 1.1 and Theorem 1.3 holds only for a certain, although wide, range of $\dim Q$ , which in itself is an interesting phenomenon. On the other hand, as stated in Theorem 1.3, we do not know whether $\alpha(d-1,d,p)$ is sharp for $2<p<4$ . Compared to Theorem 1.2, it seems that our bounds might not be the best possible.

Similar to [Fra17b], a crucial point of our result is that the exponent $1/\alpha(k,d,p)<1$ for all $p>2$ . Indeed, applying the triangle inequality to the right‐hand side of (1.6) and estimating each $f_{j}$ using (1.5) yields

(1.11)

\Biggl{\|}\sum_{j\in J}|f_{j}|^{2}\Biggr{\|}_{L^{p/2}(\Sigma)}\leq C\,\lambda^{2\delta(k,d,p)}\,\dim Q,

which is far from optimal in light of our results.

Moreover, the exponent $\alpha(k,d,p)=\tfrac{p}{2}$ that appear mutiple times in our theorems is natural. To see this, consider any orthogonal (but not necessarily normalized) subset $\{f_{j}\}_{j\in J}\subset E_{\lambda}$ . Then by our reults ¹¹1Possibly modulo a $\log\lambda$ –loss.,

(1.12)

\Big{\|}\|\{f_{j}\}\|_{l^{2}(J)}\Big{\|}_{L^{p}(\Sigma)}\leq C\,\lambda^{\delta(k,d,p)}\,\Big{\|}\{\|f_{j}\|_{L^{2}(M)}\}\Big{\|}_{l^{p}(J)}.

This shows that Theorems 1.1, 1.2 and 1.3 allow us to interchange the underlying space in the Lebesgue norm. On the other hand, a direct application of Minkowski’s inequality together with (1.5) gives

(1.13)

\Big{\|}\|\{f_{j}\}\|_{l^{2}(J)}\Big{\|}_{L^{p}(\Sigma)}\leq C\,\lambda^{\delta(k,d,p)}\,\Big{\|}\{\|f_{j}\|_{L^{2}(M)}\}\Big{\|}_{l^{2}(J)}.

Clearly (1.12) improves (1.13) for all $p>2$ .

The optimality of our results can be understood from a different point of view. Let us illustrate this in the surface case. When $\dim M=2$ , we have the following corollary.

Corollary 1.5.

Let $d=2$ , $k=1$ . For any orthonormal system $\{f_{j}\}_{j\in J}\subset E_{\lambda}$ with $\#J\sim\lambda^{\beta}$ for some $0\leq\beta\leq 1$ , we have

(1.14)

\displaystyle\Biggl{\|}\sum_{j\in J}|f_{j}|^{2}\Biggr{\|}_{L^{p/2}(\Sigma)}\leq C\,(\log\lambda)^{h(1,2,p)}\,\lambda^{\gamma(p)},

where

\displaystyle\gamma(p)=\begin{cases}\dfrac{\beta+1}{2},&2\leq p\leq 4,\\ 1-\dfrac{2(1-\beta)}{p},&4\leq p\leq\infty.\end{cases}

Remark 1.6.

The work of Burq–Gérard–Tzvetkov shows that (1.14) is sharp when $\beta=0$ (equivalently $\#J\sim 1$ ), up to a single $\log\lambda$ loss. For $\beta=1$ —that is, for $Q=E_{\lambda}$ —we have $\gamma(p)=1$ , so (1.14) is already optimally sharp by the pointwise Weyl law on the sphere. Moreover, as noted earlier, Section 6 proves the stronger fact that (1.14) is saturated on the two-sphere for every $\beta\in(0,1)$ .

The proof of our result rests on three key ingredients: a duality principle in Schatten spaces, a parametrix for the wave equation, and certain oscillatory integral estimates from [BGT07]. As in Frank–Sabin, we use the duality principle to reduce the estimation of $L^{p}$ norms for an orthonormal system of eigenfunctions to appropriate Schatten‐type bounds. We then follow the strategy of Burq–Gérard–Tzvetkov [BGT07]: they employ Sogge’s trick to rewrite the spectral projection operator via the half‐wave operator and use a parametrix to convert it into an oscillatory integral operator. A standard $TT^{*}$ argument and a dyadic decomposition of the resulting kernel

K(z,z^{\prime})=\sum_{j}K_{j}(z,z^{\prime}),\qquad|z-z^{\prime}|\sim 2^{j},

together with a formula for the Schatten‐2 norm of the corresponding operator, then completes the proof of Theorem 1.1. Unlike in [Fra17b], we do not employ complex interpolation.

This paper is organized as follows. Section 2 recalls the definition of Schatten classes and the duality principle from [Fra17a]. In Section 3 we carry out the standard reduction. Sections 4 and 5 assemble these ingredients to prove our main theorems. Section 6 shows that (1.14) is essentially sharp on the two-sphere for every $\beta\in(0,1)$ . Finally, Section 7 establishes the sharpness parts of Theorems 1.1 and 1.3.

Notation.

Throughout this paper, the symbol $C$ denotes a (positive) constant that depends only on the fixed data $(M,\Sigma)$ ; its value may vary from line to line. We shall write $A\lesssim B$ , if there exists an absolute constant $C>0$ , so that $A<CB$ and write $A\sim B$ if one has $A\lesssim B$ and $B\lesssim A$ . For brevity, whenever the dimensions $k$ and $d$ are clear from the context, we will abbreviate $\delta(k,d,p)$ , $\alpha(k,d,p)$ , and $h(k,d,p)$ by writing simply $\delta(p)$ , $\alpha(p)$ , and $h(p)$ , respectively.

Acknowledgements

This project is supported by the National Key Research and Development Program of China No. 2022YFA1007200. X. W. is partially supported by the Fundamental Research Funds for the Central Universities Grant No. 531118010864. Y. X. is partially supported by Zhejiang Provincial Natural Science Foundation of China under Grant No. LR23A010002, and NSF China Grant No. 12171424. The authors would like to thank Rupert Frank and Julien Sabin for their kind communications and for pointing out a gap in an earlier version of the paper.

2. Schatten classes

In this section, we recall some fundamental properties of Schatten class operators and review a duality principle established by Frank and Sabin. Let $\mathfrak{H}$ and $\mathfrak{K}$ be complex, separable Hilbert spaces, and denote by $\mathfrak{B}_{0}(\mathfrak{H,\mathfrak{K}})$ the space of compact linear operators from $\mathfrak{H}$ to $\mathfrak{K}$ . For $T\in\mathfrak{B}_{0}(\mathfrak{H,\mathfrak{K}})$ , the operator $|T|:=(T^{*}T)^{1/2}$ is compact and positive. Its eigenvalues, which are non-negative, are called the singular values of $T$ , and are arranged in decreasing order as $\sigma_{1}\geq\sigma_{2}\geq\cdots\geq 0$ . For $1\leq p\leq\infty$ , the Schatten class $\mathfrak{S}^{p}(\mathfrak{H},\mathfrak{K})$ consists of all compact operators $T$ whose singular value(counted according to multiplicity) form a sequence in $l^{p}$ . Naturally, the Schatten norm of $T$ is defined by the $l^{p}$ norm of the singular value sequence, that is,

\big{\|}T\big{\|}_{\mathfrak{S}^{p}(\mathfrak{H},\mathfrak{K})}:=\Bigl{(}\sum_{j\geq 0}\sigma^{p}_{j}\Bigr{)}^{1/p},

with the standard modification when $p=\infty.$ When $\mathfrak{H}=\mathfrak{K}$ , we write $\mathfrak{S}^{p}(\mathfrak{H}):=\mathfrak{S}^{p}(\mathfrak{H},\mathfrak{H})$ and $\mathfrak{B}_{0}(\mathfrak{H}):=\mathfrak{B}_{0}(\mathfrak{H},\mathfrak{H})$ . For further details, we refer the reader to Simon’s monograph [Sim05].

We will require the following lemma from [Sim05, Theorem 2.7].

Lemma 2.1 (Theorem 2.7 [Sim05]).

Let $A$ and $C$ be bounded operators on $\mathfrak{H}.$ Then for all $B\in\mathfrak{B}_{0}(\mathfrak{H})$ and $1\leq p\leq\infty$ , we have

\|ABC\|_{\mathfrak{S}^{p}(\mathfrak{H})}\leq\|A\|\|C\|\|B\|_{\mathfrak{S}^{p}(\mathfrak{H})}.

Here $\|{}\cdot{}\|$ denotes the operator norm on $\mathfrak{H}.$

We now review a well-known duality principle for Schatten class operators, as presented in [Fra17a, Lemma 3].

Lemma 2.2 (Duality principle).

Let $\mathfrak{H}$ be a separable Hilbert space. For $2\leq p\leq\infty$ , $\alpha\geq 1$ , with $1/\alpha+1/\alpha^{\prime}=1/p+1/p^{\prime}=1$ , suppose that $T$ is a bounded operator from $\mathfrak{H}$ to $L^{p}(\mathbb{R}^{d})$ . Then the following are equivalent.

(1)

There exists a constant $C>0$ such that

(2.1)

\Big{\|}WTT^{*}\overline{W}\Big{\|}_{\mathfrak{S}^{\alpha^{\prime}}(L^{2}(\mathbb{R}^{d}))}\leq C\|W\|^{2}_{L^{2p/(p-2)}(\mathbb{R}^{d})},~{}~{}~{}\text{for all }W\in L^{2p/(p-2)}(\mathbb{R}^{d},\mathbb{C}).

(2)

For any orthonormal system $\{f_{j}\}_{j\in J}$ in $\mathfrak{H}$ and any sequence $\{t_{j}\}_{j\in J}\subset\mathbb{C}$ , there is a constant $C^{\prime}$ such that

(2.2)

\Biggl{\|}\sum_{j\in J}t_{j}|Tf_{j}|^{2}\Biggr{\|}_{L^{p/2}(\mathbb{R}^{d})}\leq C^{\prime}\Biggl{(}\sum_{j\in J}|t_{j}|^{\alpha}\Biggr{)}^{1/\alpha},

Moreover, the values of the optimal constants $C$ and $C^{\prime}$ coincide.

If one examines the proof of the above lemma in [Fra17a], one can see that this duality principle is a direct consequence of the duality between Lebesgue $L^{p}$ spaces.

3. Approximate projection operators

In this section, we perform the standard reduction introduced by Sogge [Sog17] and collect a few key oscillatory‐integral estimates from [BGT07]. Sogge’s idea is to consider an operator that reproduces eigenfunctions. Fix a small constant $\epsilon_{0}>0$ and let $\lambda\geq 1$ . Choose a Schwartz function $\chi$ such that

\chi(0)=1,\quad\chi(t)>\tfrac{1}{2}\text{ for }t\in[0,1],\quad\text{and}\quad\widehat{\chi}(t)=0\quad\text{unless }|t|\in[\epsilon_{0},2\epsilon_{0}].

Define

\chi_{\lambda}f:=\chi\bigl{(}\sqrt{-\Delta_{g}}-\lambda\bigr{)}f=\frac{1}{2\pi}\int_{\epsilon_{0}}^{2\epsilon_{0}}\bigl{(}e^{it\sqrt{-\Delta_{g}}}f\bigr{)}\,e^{-it\lambda}\,\widehat{\chi}(t)\,dt.

For $\epsilon_{0}\ll 1$ and $|t|\leq 2\epsilon_{0}$ , a local‐coordinate parametrix shows that $e^{it\sqrt{-\Delta_{g}}}$ is a Fourier integral operator. A stationary‐phase argument then gives the following slight variant of [Sog17, Lemma 5.1.3], as in [BGT07, Sog11].

Lemma 3.1.

Let $\epsilon_{0}>0$ be smaller than one-tenth of the injectivity radius of $(M,g)$ . In local coordinates,

\chi_{\lambda}f(x)=\lambda^{\frac{d-1}{2}}\int_{M}e^{\,i\lambda\psi(x,y)}\,a_{\lambda}(x,y)\,f(y)\,dy+R_{\lambda}f(x),

where

{\rm supp\,}a_{\lambda}\subset\bigl{\{}(x,y)\colon\tfrac{\epsilon_{0}}{2}\leq d_{g}(x,y)\leq\epsilon_{0}\bigr{\}},\qquad\psi(x,y)=-d_{g}(x,y),

and $a_{\lambda}\in C^{\infty}_{0}$ satisfies

|\partial_{x,y}^{\alpha}a_{\lambda}(x,y)|\leq C_{\alpha}\quad\text{for all multi‐indices }\alpha.

Moreover, for every $N\in\mathbb{Z}_{+}$ and $2\leq p\leq\infty$ there is $C_{N,p}$ so that

\|R_{\lambda}\|_{L^{2}\to L^{p}}\leq C_{N,p}\,\lambda^{-N}.

Remark 3.2.

By a partition of unity we may also assume $a_{\lambda}(x,y)$ is supported in small coordinate‐chart neighborhoods of fixed points $x_{0},y_{0}\in M$ with $d_{g}(x,y)\in[\epsilon_{0}/2,\epsilon_{0}]$ , both lying inside the geodesic ball $B(x_{0},10\epsilon_{0})$ . We can always take $\epsilon_{0}>0$ smaller if needed.

Define

T_{\lambda}f(x)=\int_{M}e^{\,i\lambda\psi(x,y)}\,a_{\lambda}(x,y)\,f(y)\,dy.

The approximate projection operator $\chi_{\lambda}$ is usually called a reproducing operator since it reproduces eigenfunctions by $\chi_{\lambda}e_{\lambda}=e_{\lambda}$ . So it suffices to consider the operator norm estimates of $T_{\lambda}$ to get the eigenfunction estimates, as $R_{\lambda}$ satisfies much better bounds than we want to prove.

4. Proof of Theorem 1.2

We are now ready to prove Theorem 1.2.

Proof of Theorem 1.2.

We begin by establishing the second part of the theorem, namely, the bound stated in inequality (1.8). Recall that the operator $\chi_{\lambda}$ is defined by

\chi_{\lambda}f:=\chi(\sqrt{-\Delta_{g}}-\lambda)f=\frac{1}{2\pi}\int^{2\epsilon_{0}}_{\epsilon_{0}}(e^{it\sqrt{-\Delta_{g}}}f)e^{-it\lambda}\hat{\chi}(t)\,dt.

Since $\lambda_{j}\in[\lambda,\lambda+1]$ for $j\in J$ , and $\chi>1/2$ on $[0,1]$ , it follows that

\sum_{j\in J}t_{j}|f_{j}|^{2}\leq 2\sum_{j\in J}t_{j}|\chi_{\lambda}f_{j}|^{2}.

Therefore, establishing the second part of Theorem 1.2 reduces to proving the following estimate involving $\chi_{\lambda}f_{j}$ :

(4.1)

\Biggl{\|}\sum_{j\in J}t_{j}|\chi_{\lambda}f_{j}|^{2}\Biggr{\|}_{L^{p/2}(\Sigma)}\leq C\lambda^{2\delta{(p)}}(\log\lambda)^{h(p)}\Big{\|}\{t_{j}\}\Big{\|}_{l^{\alpha(p)}(J)}.

By Lemma 2.2, this inequality is equivalent to the Schatten norm estimate

(4.2)

\Big{\|}W{\chi}_{\lambda}\chi^{*}_{\lambda}\overline{W}\Big{\|}_{\mathfrak{S}^{(\alpha(p))^{\prime}}(L^{2}(\Sigma))}\leq C\lambda^{2\delta(p)}(\log\lambda)^{h(p)}\|W\|^{2}_{L^{2p/(p-2)}(\Sigma)},~{}~{}~{}{\text{for all}~{}}W\in L^{2p/(p-2)}(\Sigma),

where $C$ is independent of $W$ and $\lambda$ .

Note that, by Lemma 3.1, $R_{\lambda}$ is an operator that always contributes a term rapidly decaying in $\lambda$ , it suffices to prove the bound

(4.3)

\Big{\|}WT_{\lambda}T^{*}_{\lambda}\overline{W}\Big{\|}_{\mathfrak{S}^{(\alpha(p))^{\prime}}(L^{2}(\Sigma))}\leq C\lambda^{2\delta(p)-1}(\log\lambda)^{h(p)}\|W\|^{2}_{L^{2p/(p-2)}(\Sigma)}.

First, consider the case $p=\infty$ , which corresponds to $(\alpha(p))^{\prime}=1$ . Then we have

(4.4)	$\displaystyle\Big{\\|}WT_{\lambda}T^{*}_{\lambda}\overline{W}\Big{\\|}_{\mathfrak{S}^{1}(L^{2}(\Sigma))}$	$\displaystyle=\\|WT_{\lambda}\\|^{2}_{\mathfrak{S}^{2}(L^{2}(M)\to L^{2}(\Sigma))}$
		$\displaystyle=\int_{\Sigma}\int_{M}\|W(x)\|^{2}\|T_{\lambda}(x,y)\|^{2}\,dy\,dx$
		$\displaystyle\leq C\\|W\\|^{2}_{L^{2}(\Sigma)},$

where $T_{\lambda}(x,y)=e^{-i\lambda d_{g}(x,y)}a_{\lambda}(x,y)$ is the kernel function associated with $T_{\lambda}$ . Here we have used the fact that if

S:L^{2}(M)\longrightarrow L^{2}(\Sigma)

is a Hilbert–Schmidt operator with integral kernel $S(x,y)$ , then

\big{\|}S\big{\|}_{\mathfrak{S}^{2}(L^{2}(M)\to L^{2}(\Sigma))}^{2}=\int_{\Sigma}\!\!\int_{M}\bigl{|}S(x,y)\bigr{|}^{2}\,dy\,dx.

Next we prove (4.3) for $p=4$ . Once this case is established, the range $4\leq p\leq\infty$ follows immediately by interpolation. Because $\Sigma$ is compact, Hölder’s inequality implies that for every $2\leq p<4$ ,

\Biggl{\|}\sum_{j\in J}t_{j}\,|\chi_{\lambda}f_{j}|^{2}\Biggr{\|}_{L^{p/2}(\Sigma)}\;\lesssim\;\Biggl{\|}\sum_{j\in J}t_{j}\,|\chi_{\lambda}f_{j}|^{2}\Biggr{\|}_{L^{2}(\Sigma)}.

Hence (1.8) is a consequence of (4.3) with $p=4$ .

Assume that we are in the geodesic normal coordinate system about $x_{0}\in M$ , and $\Sigma:[0,1]\to M$ parameterized by the arc length $s$ and passes through $x_{0}$ . Partition of unity allows us to assume that $\Sigma$ is contained in the geodesic ball centered at $x_{0}$ with radius $\epsilon_{0}/10$ , i.e., $|x(s)|\leqslant\epsilon_{0}/10$ and $x(0)=0$ , which forces $\epsilon_{0}/3\leq|y|\leq 4\epsilon_{0}/3$ . We write

\mathcal{T}_{\lambda}(f)(s):={T}_{\lambda}(f)(x(s))=\int e^{i\lambda\psi(x(s),y)}a(x(s),y)f(y)\,dy,

and denote by $\mathcal{K}(s,t)$ the kernel of operator $\mathcal{T}_{\lambda}\mathcal{T}_{\lambda}^{*}$ . Then we have

\mathcal{K}(s,t)=\int e^{i\lambda[\psi(x(s),y)-\psi(x(t),y)]}a(x(s),y)\overline{a(x(t),y)}\,dy.

Our goal is reduced to proving

(4.5)

\Big{\|}W\mathcal{T}_{\lambda}\mathcal{T}^{*}_{\lambda}\overline{W}\Big{\|}_{\mathfrak{S}^{2}(L^{2}(\Sigma))}\lesssim\lambda^{-1/2}(\log\lambda)^{1/2}\|W\|^{2}_{L^{4}(\Sigma)},~{}~{}~{}{\text{for all}}~{}W\in L^{4}(\Sigma),

which is equivalent to

(4.6)

\Big{(}\int_{0}^{1}\int_{0}^{1}|W(x(s))\mathcal{K}(s,t)W(x(t))|^{2}\,ds\,dt\Big{)}^{1/2}\lesssim\lambda^{-1/2}(\log\lambda)^{1/2}\|W\|^{2}_{L^{4}(\Sigma)}.

Since $y$ stays in the annulus with inner radius ${\epsilon_{0}}/3$ and outer radius ${4\epsilon_{0}}/3$ , we can restrict $y$ to the circle with radius $r$ . In fact, we can represent $y$ in polar coordinates as $y=r\omega$ (i.e., geodesic polar coordinates on $M$ ), $\epsilon_{0}/3\leqslant r\leqslant 4\epsilon_{0}/3$ , $\omega=(\omega_{1},\omega_{2})\in S^{1}$ and denote:

\psi_{r}(x(s),\omega)=\psi(x(s),y),~{}a_{r}(x(s),\omega)\overline{a_{r}(x(t),\omega)}=\kappa(r,\omega)a_{r}(x(s),\omega)\overline{a_{r}(x(t),\omega)}

for some smooth function $\kappa$ . Define

\mathcal{K}_{r}(s,t)=\int_{\mathbb{S}^{1}}e^{i\lambda[\psi_{r}(x(s),\omega)-\psi_{r}(x(t),\omega)]}a_{r}(x(s),\omega)\overline{a_{r}(x(t),\omega)}\,d\omega.

Then,

\mathcal{K}(s,t)=\int_{\epsilon_{0}/3}^{4\epsilon_{0}/3}\mathcal{K}_{r}(s,t)\,dr.

By applying [BGT07, Lemma 3.2], we have that $K_{r}(s,t)$ is bounded by

C(1+\lambda|s-t|)^{-1/2},

and then

\mathcal{K}(s,t)\lesssim(1+\lambda|s-t|)^{-1/2}.

Consequently, the square of the left-hand side of (4.6) is bounded by

(4.7)

\int_{0}^{1}\int_{0}^{1}|W(x(s))|^{2}(1+\lambda|s-t|)^{-1}|W(x(t))|^{2}\,dt\,ds,

Applying the Young inequality, we obtain

	$\displaystyle\eqref{gg26}$	$\displaystyle\leq\\|W\\|^{2}_{L^{4}(\Sigma)}\cdot\Big{\\|}\int^{1}_{0}(1+\lambda\|s-t\|)^{-1}\|W(x(t))\|^{2}dt\Big{\\|}_{L^{2}(0,1)}$
		$\displaystyle\leq\\|W\\|_{L^{4}(\Sigma)}^{4}\cdot\int^{1}_{0}(1+\lambda t)^{-1}dt$
		$\displaystyle\lesssim\lambda^{-1}\log\lambda\\|W\\|_{L^{4}(\Sigma)}^{4}.$

Taking square roots completes the proof of (4.5).

Now we prove the first part of Theorem 1.2 (inequality (1.7)), beginning with the case $2\leq p<4$ .

Let $1<\alpha<2$ , and thus $\alpha^{\prime}>2$ . the operator $W\,T_{\lambda}T_{\lambda}^{*}\,\overline{W}$ is self-adjoint. Invoking the Hausdorff–Young theorem for integral operators [Rus77, Theorem 1] and then Young’s inequality we obtain

	$\displaystyle\Big{\\|}WT_{\lambda}T_{\lambda}^{*}\overline{W}\Big{\\|}_{\mathfrak{S}^{\alpha^{\prime}}}$	$\displaystyle\leq\Big{(}\int^{1}_{0}\Big{(}\int^{1}_{0}\big{\|}W(x(s))\mathcal{K}(s,t)\overline{W(x(t))}\big{\|}^{\alpha}\,ds\Big{)}^{\alpha^{\prime}/\alpha}\,dt\Big{)}^{1/\alpha^{\prime}}$
		$\displaystyle\leq\\|W\\|_{L^{2\alpha^{\prime}}(\Sigma)}\Big{(}\int^{1}_{0}\Big{(}\int^{1}_{0}\|W(x(s))\|^{\alpha}(1+\lambda\|s-t\|)^{-\alpha/2}\,ds\Big{)}^{2\alpha^{\prime}/\alpha}\,dt\Big{)}^{1/2\alpha^{\prime}}$
		$\displaystyle\leq\\|W\\|^{2}_{L^{2\alpha^{\prime}}(\Sigma)}\Big{(}\int^{1}_{0}(1+\lambda\|s\|)^{-\frac{\alpha}{2}}\,ds\Big{)}^{1/\alpha}$
		$\displaystyle\lesssim\lambda^{-1/2}\\|W\\|^{2}_{L^{2\alpha^{\prime}}(\Sigma)}.$

Set $2p/(p-2)=2\alpha^{\prime}>4$ , equivalently $2\leq p<4$ . By the duality principle (Lemma 2.2), the above estimate gives us

(4.8)

\Big{\|}\sum_{j}t_{j}|T_{\lambda}f_{j}|^{2}\Big{\|}_{L^{p/2}(\Sigma)}\leq C_{p}\lambda^{-1/2}\big{\|}\{t_{j}\}\big{\|}_{l^{\alpha}(J)}.

Since the curve $\Sigma$ is compact, Hölder’s inequality implies that for any $2\leq q\leq p$ , we have

\Big{\|}\sum_{j}t_{j}|T_{\lambda}f_{j}|^{2}\Big{\|}_{L^{q/2}(\Sigma)}\leq C_{\alpha}\lambda^{-1/2}\big{\|}\{t_{j}\}\big{\|}_{l^{\alpha}(J)}.

Note that we can choose $p$ in (4.8) arbitrarily close to $4$ , thus the associated exponent $\alpha$ can be taken arbitrarily close to $2$ . This completes the proof for the range $2\leq p<4$ .

Now let $4<p\leq\infty$ . Note that for any $p_{0}\in(4,p)$ , we have

	$\displaystyle\Big{\\|}WT_{\lambda}T^{\star}_{\lambda}\overline{W}\Big{\\|}_{\mathfrak{S}^{2}(L^{2}(\Sigma))}$	$\displaystyle=\Big{(}\int^{1}_{0}\int^{1}_{0}\|W(x(s))\|^{2}\|\mathcal{K}(s,t)\|^{2}\|\overline{W(x(t))}\|^{2}\,ds\,dt\Big{)}^{1/2}$
		$\displaystyle\leq\\|W\\|_{L^{2p_{0}/(p_{0}-2)}(\Sigma)}\Big{\\|}\int^{1}_{0}\|\mathcal{K}(s,t)\|^{2}\|W(x(t))\|^{2}\,dt\Big{\\|}^{1/2}_{L^{p_{0}/2}(0,1)}$
		$\displaystyle\leq\\|(1+\lambda\|s\|)^{-1}\\|^{1/2}_{L^{p_{0}/4}(0,1)}\\|W\\|^{2}_{L^{2p_{0}/(p_{0}-2)}(\Sigma)}$
		$\displaystyle\leq C_{p_{0}}\lambda^{-2/p_{0}}\\|W\\|_{L^{2p_{0}/(p_{0}-2)}(\Sigma)}^{2}.$

Interpolating this with the trace-class bound (4.4) yields

\Big{\|}WT_{\lambda}T^{\star}_{\lambda}\overline{W}\Big{\|}_{\mathfrak{S}^{(2p/p_{0})^{\prime}}(L^{2}(\Sigma))}\leq C_{p_{0}}\lambda^{-2/p}\|W\|^{2}_{L^{2p/(p-2)}(\Sigma)}.

By duality this is equivalent to

\Biggl{\|}\sum_{j\in J}t_{j}|T_{\lambda}f_{j}|^{2}\Biggr{\|}_{L^{p/2}(\Sigma)}\leq C_{p_{0}}\lambda^{-2/p}\big{\|}\{t_{j}\}\big{\|}_{l^{2p/p_{0}}(J)}.

Since $p_{0}$ can be taken to be arbitrarily close to $4$ , we see that the following holds for any $1\leq\alpha<p/2.$

\Biggl{\|}\sum_{j\in J}t_{j}|T_{\lambda}f_{j}|^{2}\Biggr{\|}_{L^{p/2}(\Sigma)}\leq C_{\alpha}\lambda^{-2/p}\big{\|}\{t_{j}\}\big{\|}_{l^{\alpha}(J)},

as desired.

∎

5. Proof Theorem 1.1 and 1.3

In this section, we prove Theorem 1.1 and 1.3. Assume that in the geodesic normal coordinate system about $x_{0}\in M,$ $\Sigma$ is parameterized by $x(z_{1},z_{2},\cdots,z_{k}),$ $x(0)=0$ . Again by a partition of unity, we can assume that (a local piece of) $\Sigma$ is contained in a small enough geodesic ball about $x_{0}$ , i.e., $|x(z)|\leq\epsilon_{0}/10,$ which forces $\epsilon_{0}/3\leq|y|\leq 4\epsilon_{0}/3$ . We write

\mathcal{T}_{\lambda}(f)(z):=T_{\lambda}(f)(x(z))=\int_{M}e^{i\lambda\psi(x(z),y)}a_{\lambda}(x(z),y)f(y)\,dy

and denote by ${K}(x,x^{\prime})$ the kernel of the operator ${T}_{\lambda}{T}^{*}_{\lambda}$ . Let

\mathcal{K}(z,z^{\prime}):=K(x(z),x(z^{\prime}))=\int_{M}e^{i\lambda[\psi(x(z),y)-\psi(x(z^{\prime}),y)]}a_{\lambda}(x(z),y)\overline{a_{\lambda}(x(z^{\prime}),y)}\,dy

be the kernel of the operator $\mathcal{T}_{\lambda}\mathcal{T}^{*}_{\lambda}$ . We now turn to an estimate from [BGT07] that will be essential for our analysis.

Lemma 5.1 (Lemma 6.1 in [BGT07]).

For any $(x,x^{\prime})$ , $|{K}(x,x^{\prime})|\lesssim 1$ . If $|\psi(x,x^{\prime})|\gtrsim\lambda^{-1}$ , then for any $N\in\mathbb{Z}^{+}$ , there exists $\epsilon_{0}\ll 1$ so that

(5.1)

{K}(x,x^{\prime})=\sum_{\pm}\sum^{N-1}_{n=0}\frac{e^{\pm i\lambda\psi(x,x^{\prime})}}{(\lambda\psi(x,x^{\prime}))^{\frac{d-1}{2}+n}}a_{n}^{\pm}(x,x^{\prime},\lambda)+b_{N}(x,x^{\prime},\lambda),

where $a^{\pm}_{n},b_{n}\in C^{\infty}(\mathbb{R}^{d}\times\mathbb{R}^{d}\times\mathbb{R})$ and $-\psi(x,y)=d_{g}(x,y)$ is the geodesic distance between $x$ and $x^{\prime}$ . Furthermore, $a^{\pm}$ are real, have supports of size $O(\epsilon_{0})$ with respect to the first two variables and are uniformly bounded with respect to $\lambda$ . Finally

(5.2)

|b_{N}(x,x^{\prime},\lambda)|\lesssim|1+\lambda\psi(x-x^{\prime})|^{-\frac{d-1}{2}-N}.

In view of

d_{g}(x(z),x(z^{\prime}))\sim|z-z^{\prime}|,

noting that $K$ is bounded, applying Lemma 5.1 yields a rough bound on the kernel of $\mathcal{T}_{\lambda}\mathcal{T}^{*}_{\lambda}$

(5.3)

|K(x(z),x(z^{\prime}))|\leq C(1+\lambda|z-z^{\prime}|)^{-\frac{d-1}{2}}.

However, this estimate alone is insufficient to achieve our desired bounds. To improve upon it, we follow the approach in [BGT07] by exploiting the oscillatory nature of the phase in $K(x(z),x(z^{\prime}))$ . This involves dyadically decomposing the kernel based on the size of $|z-z^{\prime}|$ . We now provide a detailed description of this decomposition and collect the key estimates that will be instrumental in our proof.

We fix a compactly supported bump function $\chi_{0}\in C^{\infty}_{0}(\mathbb{R}^{k}),$ so that $\mathrm{supp}\,\chi_{0}\in\{x\in\mathbb{R}^{k}:|x|\leq C\}$ . Additionally, let $\tilde{\chi}\in C^{\infty}_{0}(\mathbb{R}^{k})$ be supported in the set $\{x\in{\mathbb{R}^{k}}:\frac{1}{2}<|x|<2\}$ such that there is a partition of unity on $\{x\in\mathbb{R}^{k}:|x|<1\}$ of the form

(5.4)

1=\chi_{0}(\lambda x)+\sum^{\log\lambda/\log 2}_{j=1}\tilde{\chi}(2^{j}x).

Using this, we decompose the kernel $\mathcal{K}(z,z^{\prime})$ dyadically:

(5.5)		$\displaystyle\mathcal{K}(z,z^{\prime})$	$\displaystyle=\mathcal{K}(z,z^{\prime})\chi_{0}(\lambda(z-z^{\prime}))+\sum^{\log\lambda/\log 2}_{j=1}\mathcal{K}(z,z^{\prime})\tilde{\chi}(2^{j}(z-z^{\prime}))$
(5.5)			$\displaystyle=\mathcal{K}_{0}(z,z^{\prime})+\sum^{\log\lambda/\log 2}_{j=1}\mathcal{K}_{j}(z,z^{\prime}).$

In the proof of Theorem 1.1, we will estimate the contribution of each dyadic piece $\mathcal{K}_{j}$ . To facilitate these estimates, we rely on the following crucial estimates from [BGT07].

Lemma 5.2 (Proposition 6.3 in [BGT07]).

For sufficiently large $j$ , let $(\mathcal{T}_{\lambda}\mathcal{T}_{\lambda}^{*})_{j}$ be the operator whose integral kernel is $\mathcal{K}_{j}(z,z^{\prime})$ . Then it satisfies the estimate

(5.6)		$\displaystyle\\|(\mathcal{T}_{\lambda}\mathcal{T}^{*}_{\lambda})_{j}f\\|_{L^{\infty}(\Sigma)}\lesssim\Big{(}\frac{2^{j}}{\lambda}\Big{)}^{\frac{d-1}{2}}\\|f\\|_{L^{1}(\Sigma)},$
(5.7)		$\displaystyle\\|(\mathcal{T}_{\lambda}\mathcal{T}^{*}_{\lambda})_{j}f\\|_{L^{2}(\Sigma)}\lesssim 2^{-jk}\Big{(}\frac{2^{j}}{\lambda}\Big{)}^{\frac{d-1}{2}+\frac{k-1}{2}}\\|f\\|_{L^{2}(\Sigma)}.$

We remark that the operator corresponding to $\mathcal{K}_{0}$ satisfies (5.6) and (5.7) with $2^{j}\sim\lambda,$ and thus we do not need to handle it separately. Furthermore, note that the operator $(\mathcal{T}_{\lambda}\mathcal{T}^{*}_{\lambda})_{j}$ in Proposition 6.3 of [BGT07] does not include the remainder term $b_{N}$ . However, one readily verifies that the same estimates hold when $b_{N}$ is included, provided $N$ is chosen sufficiently large. Indeed, by (5.2), if $N$ in Lemma 5.1 is sufficiently large, then the operator with kernel

b_{N}(z,z^{\prime},\lambda)\,\tilde{\chi}\bigl{(}2^{j}(z-z^{\prime})\bigr{)}

satisfies (5.6) and (5.7) via Young’s inequality.

5.1. Proof of Theorem 1.1

We begin by adopting the same strategy used in the two-dimensional case. By the same reasoning that led to inequality (4.3), it suffices to establish that for every $W\in L^{\frac{2p}{p-2}}(\Sigma)$ , the following holds:

(5.8)

\|W\,\mathcal{T}_{\lambda}\mathcal{T}_{\lambda}^{*}\overline{W}\|_{\mathfrak{S}^{(p/2)^{\prime}}(L^{2}(\Sigma))}\lesssim\lambda^{2\delta(d,k,p)-(d-1)}\|W\|_{L^{\frac{2p}{p-2}}(\Sigma)}^{2}.

We shall apply the dyadic decomposition (5.5) to the kernel of $\mathcal{T}_{\lambda}\mathcal{T}_{\lambda}^{*}$ . This decomposition enables us to leverage Proposition 5.2 to obtain bounds on the Schatten norms of the operators $W(\mathcal{T}_{\lambda}\mathcal{T}_{\lambda}^{*})_{j}\overline{W}$ . Summing these bounds over all dyadic levels $j$ then yields the desired Schatten bound for $W\mathcal{T}_{\lambda}\mathcal{T}_{\lambda}^{*}\overline{W}$ .

By Lemmas 2.1 and using the fact that the largest singular value of a compact operator is equal to its operator norm, we have

(5.9)		$\displaystyle\Big{\\|}W(\mathcal{T}_{\lambda}\mathcal{T}^{*}_{\lambda})_{j}\overline{W}\Big{\\|}_{\mathfrak{S}^{\infty}(L^{2}(\Sigma))}$	$\displaystyle\leq\\|W\\|^{2}_{L^{\infty}(\Sigma)}\\|(\mathcal{T}_{\lambda}\mathcal{T}^{*}_{\lambda})_{j}\\|_{\mathfrak{S}^{\infty}(L^{2}(\Sigma))}$
(5.9)			$\displaystyle=\\|W\\|^{2}_{L^{\infty}(\Sigma)}\\|(\mathcal{T}_{\lambda}\mathcal{T}^{*}_{\lambda})_{j}\\|_{L^{2}(\Sigma)\to L^{2}(\Sigma)}.$

Inserting the operator‐norm bound $\eqref{g27}$ into the above inequality gives

(5.10)

\Big{\|}W(\mathcal{T}_{\lambda}\mathcal{T}^{*}_{\lambda})_{j}\overline{W}\Big{\|}_{\mathfrak{S}^{\infty}(L^{2}(\Sigma))}\leq C_{1}2^{-jk}\Big{(}\frac{2^{j}}{\lambda}\Big{)}^{\frac{d-1}{2}+\frac{k-1}{2}}\|W\|^{2}_{L^{\infty}(\Sigma)},

where $C_{1}$ is a constant that depends only on the fixed data $(M,\Sigma)$ .

Recalling that $\tilde{\chi}$ is supported in the set $\{x\in{\mathbb{R}^{k}}:\frac{1}{2}<|x|<2\}$ and applying the kernel estimate (5.3), or directly using the bound $\eqref{g26}$ , one shows

(5.11)	$\displaystyle\Big{\\|}W(\mathcal{T}_{\lambda}\mathcal{T}^{*}_{\lambda})_{j}\overline{W}\Big{\\|}_{\mathfrak{S}^{2}(L^{2}(\Sigma))}$	$\displaystyle=\Big{(}\iint_{\Sigma\times\Sigma}\Big{\|}W\mathcal{K}_{j}(\cdot,\cdot)\overline{W}\Big{\|}^{2}\Big{)}^{1/2}$
		$\displaystyle\leq C_{2}\Big{(}\int_{B^{k}}\int_{B^{k}}\Big{\|}W(x(z))\Big{\|}^{2}(1+2^{-j}\lambda)^{-(d-1)}\Big{\|}\overline{W(x(z^{\prime}))}\Big{\|}^{2}dzdz^{\prime}\Big{)}^{1/2}$
		$\displaystyle\leq C_{2}\Big{(}\frac{2^{j}}{\lambda}\Big{)}^{\frac{d-1}{2}}\\|W\\|^{2}_{L^{2}(\Sigma)},$

where $B^{k}$ denotes the unit ball in $\mathbb{R}^{k}$ and $C_{2}$ is an absolute constant. Here we used the fact that for every operator $T$ acting on functions on $L^{2}(\Sigma)$ ,

\big{\|}T\big{\|}^{2}_{\mathfrak{S}^{2}(L^{2}(\Sigma))}=\int_{\Sigma}\int_{\Sigma}|T(x,y)|^{2}\,dx\,dy,

where $T(\cdot,\cdot)$ denotes the integral kernel of $T$ .

Interpolating between the bounds (5.10) and (5.11) in the Schatten spaces shows that for any $p_{0}\geq 2$ ,

(5.12)

\Big{\|}W(\mathcal{T}_{\lambda}\mathcal{T}^{*}_{\lambda})_{j}\overline{W}\Big{\|}_{\mathfrak{S}^{2p_{0}/(p_{0}-2)}(L^{2}(\Sigma))}\leq C_{1}^{\frac{2}{p_{0}}}C_{2}^{1-\frac{2}{p_{0}}}2^{-\frac{2jk}{p_{0}}}\Big{(}\frac{2^{j}}{\lambda}\Big{)}^{\frac{d-1}{2}+\frac{k-1}{p_{0}}}\|W\|^{2}_{L^{2p_{0}/(p_{0}-2)}(\Sigma)},

where we take $\frac{2p_{0}}{p_{0}-2}=\infty$ as $p_{0}=2$ and $\frac{2p_{0}}{p_{0}-2}=2$ as $p_{0}=\infty$ . Summing over $1\ll j\leq\log{\lambda}/\log{2}$ yields²²2Here we may require $j$ to be sufficiently large by choosing $\epsilon_{0}$ small.

(5.13)		$\displaystyle\Big{\\|}W\mathcal{T}_{\lambda}\mathcal{T}^{*}_{\lambda}\overline{W}\Big{\\|}$	$\displaystyle{}_{\mathfrak{S}^{2p_{0}/(p_{0}-2)}(L^{2}(\Sigma))}\leq C_{1}^{\frac{2}{p_{0}}}C_{2}^{1-\frac{2}{p_{0}}}\lambda^{-\frac{d-1}{2}-\frac{k-1}{p_{0}}}\sum_{j=1}^{\log\lambda/\log 2}2^{j(\frac{d-1}{2}-\frac{k+1}{p_{0}})}\\|W\\|^{2}_{L^{2p_{0}/(p_{0}-2)}(\Sigma)}$
(5.13)			$\displaystyle\leq C_{1}^{\frac{2}{p_{0}}}C_{2}^{1-\frac{2}{p_{0}}}\begin{cases}\lambda^{-\frac{d-1}{2}-\frac{k-1}{p_{0}}+\frac{d-1}{2}-\frac{k+1}{p_{0}}}\\|W\\|^{2}_{L^{2p_{0}/(p_{0}-2)}(\Sigma)},\quad\text{if}\quad p_{0}>\frac{2(k+1)}{d-1},\\ \lambda^{-\frac{d-1}{2}-\frac{k-1}{p_{0}}}\\|W\\|^{2}_{L^{2p_{0}/(p_{0}-2)}(\Sigma)},\qquad\qquad\quad\,\text{if}\quad p_{0}<\frac{2(k+1)}{d-1},\\ \lambda^{-\frac{d-1}{2}-\frac{k-1}{p_{0}}}\log\lambda\\|W\\|^{2}_{L^{2p_{0}/(p_{0}-2)}(\Sigma)},\qquad\quad\text{if}\quad p_{0}=\frac{2(k+1)}{d-1}.\end{cases}$

Note that estimate (5.13) remains valid for any $\Sigma$ with $1\leq\dim\Sigma\leq d-1$ . In particular, we will later apply (5.13) in the proof of Theorem 1.3.

Now we first solve the case where $\dim\Sigma<d-2$ . A direct calculation yields

(5.14)	$\displaystyle\Big{\\|}W\mathcal{T}_{\lambda}\mathcal{T}^{*}_{\lambda}\overline{W}\Big{\\|}_{\mathfrak{S}^{1}(L^{2}(\Sigma))}$	$\displaystyle=\\|W\mathcal{T}_{\lambda}\\|^{2}_{\mathfrak{S}^{2}(L^{2}(M)\to L^{2}(\Sigma))}$
		$\displaystyle=\int_{\Sigma}\|W(x)\|^{2}\int_{M}\|\mathcal{T}_{\lambda}(x,y)\|^{2}dx^{\prime}dz$
		$\displaystyle\leq C_{3}\\|W\\|^{2}_{L^{2}(\Sigma)},$

where $\mathcal{T}_{\lambda}(x,y)=e^{-i\lambda d_{g}(x(z),y)}a_{\lambda}(x,y)$ is the kernel of $\mathcal{T}_{\lambda}$ and $C_{3}$ is a constant that depends only on the fixed data $(M,\Sigma)$ .

For $p_{0}>\frac{2(k+1)}{d-1}$ , interpolating this estimate with the previous Schatten bounds (5.13) yields

(5.15)

\Big{\|}W\mathcal{T}_{\lambda}\mathcal{T}_{\lambda}^{*}\overline{W}\Big{\|}_{\mathfrak{S}^{\frac{2p}{2p-p_{0}-2}}(L^{2}(\Sigma))}\leq C_{1}^{\frac{2}{p}}C_{2}^{\frac{p_{0}}{p}-\frac{2}{p}}C_{3}^{1-\frac{p_{0}}{p}}\lambda^{-\frac{2k}{p}}\|W\|_{L^{2p/(p-2)}(\Sigma)}^{2}.

In particular, choosing $p_{0}=2$ , we obtain

\Big{\|}W\mathcal{T}_{\lambda}\mathcal{T}_{\lambda}^{*}\overline{W}\Big{\|}_{\mathfrak{S}^{\frac{p}{p-2}}(L^{2}(\Sigma))}\leq C_{1}^{\frac{2}{p}}C_{3}^{1-\frac{2}{p}}\lambda^{-\frac{2k}{p}}\|W\|_{L^{2p/(p-2)}(\Sigma)}^{2},

which establishes the desired bound $\eqref{g37}$ for $\dim\Sigma=k<d-2$ .

Next, when $\dim\Sigma=d-2$ , setting $p_{0}=2$ in (5.13) yields

\Big{\|}W\mathcal{T}_{\lambda}\mathcal{T}_{\lambda}^{*}\overline{W}\Big{\|}_{\mathfrak{S}^{\infty}(L^{2}(\Sigma))}\leq C_{1}\lambda^{-(d-1)+1}\log\lambda\,\|W\|_{L^{\infty}(\Sigma)}^{2},

which aligns with the desired bound $\eqref{g37}$ for the case $p=2$ .

For $2<p_{0}\leq p$ , applying the duality principle (see Lemma 2.2) shows that (5.15) is equivalent to

(5.16)

\Biggl{\|}\sum_{j\in J}t_{j}|f_{j}|^{2}\Biggr{\|}_{L^{p/2}(\Sigma)}\leq C_{1}^{\frac{2}{p}}C_{2}^{\frac{p_{0}}{p}-\frac{2}{p}}C_{3}^{1-\frac{p_{0}}{p}}\lambda^{-\frac{2(d-2)}{p}}\Big{\|}\{t_{j}\}\Big{\|}_{l^{\frac{2p}{p_{0}+2}}(J)}

Since the right-hand side norm $\left\|\{t_{j}\}\right\|_{l^{\frac{2p}{p_{0}+2}}(J)}$ is increasing in $p_{0}$ , taking the infimum over $p_{0}\in(2,p]$ gives

\Biggl{\|}\sum_{j\in J}t_{j}|f_{j}|^{2}\Biggr{\|}_{L^{p/2}(\Sigma)}\leq C_{1}^{\frac{2}{p}}C_{3}^{1-\frac{2}{p}}\lambda^{-\frac{2(d-2)}{p}}\left\|\{t_{j}\}\right\|_{l^{\frac{p}{2}}(J)}.

Using the duality principle again, we have

\Big{\|}W\mathcal{T}_{\lambda}\mathcal{T}_{\lambda}^{*}\overline{W}\Big{\|}_{\mathfrak{S}^{\frac{p}{p-2}}(L^{2}(\Sigma))}\lesssim\lambda^{-\frac{2(d-2)}{p}}\|W\|_{L^{2p/(p-2)}(\Sigma)}^{2},

completing the proof of $\eqref{g37}$ for $\dim\Sigma=d-2$ when $p>2$ . ∎

5.2. Proof of Theorem 1.3

We begin by deriving (1.10). We observe that inequality (5.13) covers (1.10) when $2\leq p\leq\frac{2d}{d-1}$ . Therefore, it suffices to consider the case $\frac{2d}{d-1}<p\leq\infty$ .

Now we consider the range $4\leq p\leq\infty$ . Recall that $\alpha(d-1,d,p)=\frac{p}{2}$ . In particular, when $p=4$ , we have $\alpha(d-1,d,4)=2$ . Applying inequality (5.3) and Young’s inequality, we obtain

(5.17)	$\displaystyle\Big{\\|}W\mathcal{T}_{\lambda}\mathcal{T}_{\lambda}^{*}\overline{W}$	$\displaystyle\Big{\\|}_{\mathfrak{S}^{2}(L^{2}(\Sigma))}=\left(\iint_{B^{d-1}\times B^{d-1}}\left\|W(x(z))K(x(z),x(z^{\prime}))\overline{W(x(z^{\prime}))}\right\|^{2}\,dz\,dz^{\prime}\right)^{1/2}$
		$\displaystyle\lesssim\left(\iint_{B^{d-1}\times B^{d-1}}\|W(x(z))\|^{2}(1+\lambda\|z-z^{\prime}\|)^{-(d-1)}\|\overline{W(x(z^{\prime}))}\|^{2}\,dz\,dz^{\prime}\right)^{1/2}$
		$\displaystyle\lesssim\\|W\\|_{L^{4}(\Sigma)}\left\\|\int_{B^{d-1}}(1+\lambda\|z-z^{\prime}\|)^{-(d-1)}\|\overline{W(x(z^{\prime}))}\|\,dz^{\prime}\right\\|_{L^{2}(\Sigma)}^{1/2}$
		$\displaystyle\leq\\|W\\|_{L^{4}(\Sigma)}^{2}\left\|\int_{B^{d-1}}(1+\lambda\|z\|)^{-(d-1)}\,dz\right\|^{1/2}$
		$\displaystyle\lesssim\lambda^{-(d-1)/2}(\log\lambda)^{1/2}\\|W\\|_{L^{4}(\Sigma)}^{2}.$

Interpolating this estimate with the Schatten bounds given in (5.14) yields

\Big{\|}W\mathcal{T}_{\lambda}\mathcal{T}_{\lambda}^{*}\overline{W}\Big{\|}_{\mathfrak{S}^{p/(p-2)}(L^{2}(\Sigma))}\lesssim\lambda^{-2(d-1)/p}(\log\lambda)^{2/p}\|W\|_{L^{2p/(p-2)}(\Sigma)}.

This completes the proof of (1.10) for the case $4\leq p\leq\infty$ .

Next, consider the range $\frac{2d}{d-1}<p<4$ . By interpolating the bound in (5.17) with inequality (5.13)—which applies when $\dim\Sigma=d-1$ and $p_{0}=\frac{2d}{d-1}$ —we obtain

\Big{\|}W\mathcal{T}_{\lambda}\mathcal{T}_{\lambda}^{*}\overline{W}\Big{\|}_{\mathfrak{S}^{\frac{2p(d-2)}{2pd-4d-3p+4}}(L^{2}(\Sigma))}\lesssim\lambda^{-2(d-1)/p}(\log\lambda)^{\frac{2d-p}{p(d-2)}}\|W\|_{L^{\frac{2p}{p-2}}(\Sigma)}.

This completes the analysis for $\frac{2d}{d-1}<p<4$ .

Finally, we turn to the proof of inequality (1.9). The approach follows a similar pattern to the two-dimensional case. For any $p_{0}\in(4,p)$ , applying the bound (5.3) and Young’s inequality gives

	$\displaystyle\Big{\\|}W\mathcal{T}_{\lambda}\mathcal{T}_{\lambda}^{*}\overline{W}$	$\displaystyle\Big{\\|}_{\mathfrak{S}^{2}(L^{2}(\Sigma))}=\left(\iint_{B^{d-1}\times B^{d-1}}\|W(x(z))\|^{2}\|K(x(z),x(z^{\prime}))\|^{2}\|\overline{W(x(z^{\prime}))}\|^{2}\,dz\,dz^{\prime}\right)^{1/2}$
		$\displaystyle\leq\\|W\\|_{L^{2p_{0}/(p_{0}-2)}(\Sigma)}\left\\|\int_{B^{d-1}}\|K(x(z),x(z^{\prime}))\|^{2}\|W(x(z^{\prime}))\|^{2}\,dz^{\prime}\right\\|_{L^{p_{0}/2}(B^{d-1})}^{1/2}$
		$\displaystyle\leq\left\\|(1+\lambda\|s\|)^{-(d-1)}\right\\|_{L^{p_{0}/4}(B^{d-1})}^{1/2}\\|W\\|_{L^{2p_{0}/(p_{0}-2)}(\Sigma)}^{2}$
		$\displaystyle\leq C_{p_{0}}\lambda^{-2(d-1)/p_{0}}\\|W\\|_{L^{2p_{0}/(p_{0}-2)}(\Sigma)}^{2}.$

Interpolating this with the trace-class bound in (5.14) yields

\Big{\|}W\mathcal{T}_{\lambda}\mathcal{T}_{\lambda}^{*}\overline{W}\Big{\|}_{\mathfrak{S}^{(2p/p_{0})^{\prime}}(L^{2}(\Sigma))}\leq C_{p_{0}}\lambda^{-2(d-1)/p}\|W\|_{L^{2p/(p-2)}(\Sigma)}^{2}.

By the duality principle (Lemma 2.2), this is equivalent to

\Biggl{\|}\sum_{j\in J}t_{j}|T_{\lambda}f_{j}|^{2}\Biggr{\|}_{L^{p/2}(\Sigma)}\leq C_{p_{0}}\lambda^{-2(d-1)/p}\left\|\{t_{j}\}\right\|_{\ell^{2p/p_{0}}(J)}.

Since $p_{0}$ can be chosen arbitrarily close to 4, it follows that for any $\alpha<p/2$ ,

\Biggl{\|}\sum_{j\in J}t_{j}|T_{\lambda}f_{j}|^{2}\Biggr{\|}_{L^{p/2}(\Sigma)}\leq C_{\alpha}\lambda^{-2(d-1)/p}\left\|\{t_{j}\}\right\|_{\ell^{\alpha}(J)}.

This concludes the proof. ∎

6. Sharpness in dimension two

In this section, we demonstrate that the estimate in Theorem 1.2 is essentially sharp for any intermediate value of $\#J$ , except for the logarithmic loss on the two-sphere $\mathbb{S}^{2}$ . To achieve this, we will utilize a key estimate from [Fra17b].

When $M$ is the standard sphere $\mathbb{S}^{2}$ and $2\leq p\leq\infty$ , consider an increasing function, $t_{\lambda}=t(\lambda),$ $\lambda>0$ , satisfying the following conditions:

\lim_{\lambda\to\infty}t_{\lambda}=+\infty,~{}~{}~{}~{}~{}~{}\lim_{\lambda\to\infty}\dfrac{t_{\lambda}}{\lambda}=0.

Our goal is to construct, for sufficiently large $\lambda$ , geodesic segments $\gamma\subset\mathbb{S}^{2}$ and orthonormal systems $\{f_{j}\}_{j\in J_{\lambda}}\subset E_{\lambda}$ and $\{g_{j}\}_{j\in J_{\lambda}}\subset E_{\lambda}$ with $\#J_{\lambda}\sim t_{\lambda}$ , such that the following estimates hold:

(6.1)		$\displaystyle\Big{\\|}\sum_{j\in J_{\lambda}}\|f_{j}\|^{2}\Big{\\|}_{L^{p/2}(\gamma)}\gtrsim\lambda^{1-\frac{2}{p}}t_{\lambda}^{\frac{2}{p}};$
(6.2)		$\displaystyle\Big{\\|}\sum_{j\in J_{\lambda}}\|g_{j}\|^{2}\Big{\\|}_{L^{p/2}(\gamma)}\gtrsim\lambda^{\frac{1}{2}}t^{\frac{1}{2}}_{\lambda}.$

For each $\lambda>0$ , we construct orthonormal systems using spherical harmonics of degree $k\sim\lambda$ . Recall that for any spherical harmonic $Y(x)$ of degree $k$ on $L^{2}(\mathbb{S}^{2})$ , the Laplace–Beltrami operator satisfies

\Delta_{\mathbb{S}^{2}}Y(x)=-k(k+1)Y(x).

Let $Y_{k}^{\alpha_{k}}$ denote an $L^{2}$ -normalized spherical harmonic of degree $k$ , where $\alpha_{k}\in[-k,k]$ . Consider a sequence $\{t_{k}\}_{k\in\mathbb{N}}$ such that

\lim_{k\to\infty}t_{k}=+\infty,\quad\text{and}\quad\lim_{k\to\infty}\frac{t_{k}}{k}=0.

Assuming that for sufficiently large $k$ , $t_{k}\leq\frac{k}{2}$ , we claim the following:

•

The orthonormal system $\{Y_{k}^{\alpha_{k}}\}_{\alpha_{k}\in[t_{k},2t_{k}]\cap\mathbb{Z}}$ satisfies the bound corresponding to (6.1).
•

The orthonormal system $\{Y_{k}^{\alpha_{k}}\}_{\alpha_{k}\in[k-2t_{k},\,k-t_{k}]\cap\mathbb{Z}}$ satisfies the bound corresponding to (6.2).

To verify these claims, we will utilize a key estimate from [Fra17b], which provides essential bounds for these spherical harmonics.

Lemma 6.1 (Proposition 15 in [Fra17b]).

If we parameterize points in $\mathbb{S}^{2}$ by usual spherical coordinates $(\theta,\varphi)\in[0,\pi]\times[0,2\pi)$ , then

(1)

there exist $\eta_{1}>0,~{}K\geq 1,$ and $c>0$ such that for all $k\geq K$ , $\eta_{1}t_{k}/k\leq\theta\leq\frac{\pi}{2}$ and $0\leq\varphi\leq 2\pi$ ,

$\sum_{\alpha_{k}\in[t_{k},2t_{k}]\cap\mathbb{Z}}|Y^{k}_{\alpha_{k}}(\theta,\varphi)|^{2}\geq c\,\frac{t_{k}}{\sin\theta};$

(2)

there exist $\eta_{2}>0,K\geq 1$ , and $c>0$ such that for all $k\geq K$ , $0\leq\theta\leq\eta_{2}(t_{k}/k)^{1/2}$ and $0\leq\varphi\leq 2\pi$ ,

\sum_{\alpha_{k}\in[k-2t_{k},k-t_{k}]\cap\mathbb{Z}}|Y^{k}_{\alpha_{k}}(\pi/2-\theta,\varphi)|^{2}\geq c\,k^{1/2}t_{k}^{1/2}.

Now, we are prepared to proceed with the proof of our sharpness result.

Theorem 6.2.

There exist $c>0$ , $K\geq 1$ and curves $\gamma_{1}$ and $\gamma_{2}$ on $\mathbb{S}^{2}$ so that for all $k\geq K$ and $2\leq p\leq\infty$ one has

(6.3)		$\displaystyle\Big{\\|}\sum_{\alpha_{k}\in[t_{k},2t_{k}]\cap\mathbb{Z}}\|Y^{k}_{\alpha_{k}}(\theta,\varphi)\|^{2}\Big{\\|}_{L^{p/2}(\gamma_{1})}\geq Ck^{1-\frac{2}{p}}t^{\frac{2}{p}}_{k};$
(6.4)		$\displaystyle\Big{\\|}\sum_{\alpha_{k}\in[k-2t_{k},k-t_{k}]\cap\mathbb{Z}}\|Y^{k}_{\alpha_{k}}(\theta,\varphi)\|^{2}\Big{\\|}_{L^{p/2}(\gamma_{2})}\geq Ck^{\frac{1}{2}}t^{\frac{1}{2}}_{k}.$

In particular, Theorem 1.2 is essentially saturated by (6.3) for $4\leq p\leq\infty$ , and by $\eqref{s2}$ for $2\leq p\leq 4$ .

Proof.

We first establish inequality (6.3). Consider the curve $\gamma_{1}$ on $\mathbb{S}^{2}$ parameterized by $\gamma_{1}(\theta)=(\sin\theta,~{}0,~{}\cos\theta)$ , with $\theta\in[0,\pi/2].$ Applying item (1) of Lemma 6.1, we obtain

	$\displaystyle\Big{\\|}\sum_{\alpha_{k}\in[t_{k},2t_{k}]\cap\mathbb{Z}}\|Y^{k}_{\alpha_{k}}(\theta,\varphi)\|^{2}\Big{\\|}^{p/2}_{L^{p/2}(\gamma_{1})}$	$\displaystyle\geq\int^{\pi/2}_{\eta_{1}t_{k}/k}\Big{(}\sum_{\alpha_{k}\in[t_{k},2t_{k}]\cap\mathbb{Z}}\|Y^{k}_{\alpha_{k}}(\theta,0)\|^{2}\Big{)}^{p/2}\,d\theta$
		$\displaystyle\gtrsim\int^{\pi/2}_{\eta_{1}t_{k}/k}\Big{(}\frac{t_{k}}{\sin\theta}\Big{)}^{\frac{p}{2}}\,d\theta\gtrsim t^{p/2}_{k}\int^{2\eta_{1}t_{k}/k}_{\eta_{1}t_{k}/k}{\theta}^{-\frac{p}{2}}\,d\theta$
		$\displaystyle\gtrsim k^{\frac{p}{2}-1}t_{k}.$

Next, to prove (6.4), consider the curve $\gamma_{2}$ as the equator, that is, $\theta=\pi/2$ . Applying item (2) of Lemma 6.1, we conclude that

	$\displaystyle\Big{\\|}\sum_{\alpha_{k}\in[k-2t_{k},k-t_{k}]\cap\mathbb{Z}}\|Y^{k}_{\alpha_{k}}(\theta,\varphi)\|^{2}\Big{\\|}^{p/2}_{L^{p/2}(\gamma_{2})}$	$\displaystyle=\int^{2\pi}_{0}\Big{(}\sum_{\alpha_{k}\in[k-2t_{k},k-t_{k}]\cap\mathbb{Z}}\|Y^{k}_{\alpha_{k}}(\pi/2,\varphi)\|^{2}\Big{)}^{p/2}d\varphi$
		$\displaystyle\geq\int^{2\pi}_{0}\big{(}Ck^{\frac{1}{2}}t_{k}^{\frac{1}{2}}\big{)}^{p/2}\,d\varphi$
		$\displaystyle\gtrsim(k^{1/2}t_{k}^{\frac{1}{2}})^{p/2}.$

∎

7. Sharpness in hgher dimensions

Finally, we address the sharpness in higher dimensions. Suppose $d\geq 3$ and $\dim\Sigma=k$ . We work on the standard $d$ -sphere. Let

\lambda^{2}=l(l+d-1),\quad l\in\mathbb{N},

and fix an orthonormal basis $\{Y_{i}\}_{i=1}^{\dim E_{\lambda}}$ of the eigenspace $E_{\lambda}$ . Define

Z_{\lambda}(p,x)=\sum_{i=1}^{\dim E_{\lambda}}Y_{i}(p)\,Y_{i}(x).

By symmetry, $Z_{\lambda}(p,p)=C(\lambda)\sim\lambda^{d-1}$ . For any $p\in\mathbb{S}^{d}$ , the normalized zonal eigenfunction concentrated at $p$ is

Z_{\lambda}^{p}(x)=\frac{Z_{\lambda}(p,x)}{\sqrt{Z_{\lambda}(p,p)}}.

Proposition 1.

For each $p\in\mathbb{S}^{d}$ ,

Z_{\lambda}^{p}(x)=\begin{cases}O\bigl{(}\lambda^{\frac{d-1}{2}}\bigr{)},&\mathrm{dist}(x,p)\leq\lambda^{-1},\\ O\bigl{(}\mathrm{dist}(x,p)^{-\frac{d-1}{2}}\bigr{)},&\mathrm{dist}(x,p)\geq\lambda^{-1}.\end{cases}

Proposition 2.

For any $p_{1},p_{2}\in\mathbb{S}^{d}$ ,

\bigl{\langle}Z_{\lambda}^{p_{1}},Z_{\lambda}^{p_{2}}\bigr{\rangle}=\frac{Z_{\lambda}(p_{1},p_{2})}{C(\lambda)}.

The proofs of these propositions can be found in [DX13, Lemma 1.2.5].

Let $0<\beta<1$ be a parameter to be chosen later. Fix a sufficiently large constant $C>0$ , and select a $C\lambda^{-1+\beta}$ –separated set $\{p_{j}\}_{j\in J}\subset\Sigma$ so that for all $i\neq j$ ,

C\,\lambda^{-1+\beta}\leq\mathrm{dist}(p_{i},p_{j})\leq\frac{\pi}{2}.

In this configuration,

|J|=O\bigl{(}C^{-1}\,\lambda^{\,k(1-\beta)}\bigr{)}.

Moreover, by the preceding propositions, for $i\neq j$ we have

\bigl{|}\langle Z_{\lambda}^{p_{i}},\,Z_{\lambda}^{p_{j}}\rangle\bigr{|}\lesssim\lambda^{-\frac{d-1}{2}\,\beta}.

We will need the following lemma, which follows from Gerschgorin’s Circle Theorem. For completeness, we include a proof here.

Lemma 7.1.

Let $H$ be a Hilbert space, and let $\{e_{i}\}_{i=1}^{n}\subset H$ be unit vectors satisfying

|\langle e_{i},e_{j}\rangle|<\frac{\delta}{n-1},\quad i\neq j,

for some $0<\delta<\tfrac{1}{2}$ . Then there exists an $n\times n$ matrix $A=[A_{ij}]$ with

|A_{ij}-\delta_{ij}|\leq\delta\quad(1\leq i,j\leq n),

such that the vectors

f_{i}=\sum_{j=1}^{n}A_{ij}e_{j},\quad i=1,\dots,n,

form an orthonormal system in $H$ .

Proof.

Let $G=[\langle e_{i},e_{j}\rangle]_{i,j=1}^{n}$ be the Gram matrix. By hypothesis,

G_{ii}=1,\qquad|G_{ij}|<\frac{\delta}{n-1}\quad(i\neq j),

with $0<\delta<\tfrac{1}{2}$ . Write $G=I+E$ with

E_{ij}=\begin{cases}\langle e_{i},e_{j}\rangle,&i\neq j,\\ 0,&i=j.\end{cases}

Then for each $i$ , the row–sum norm of the off–diagonal part satisfies

\|E\|_{\infty}=\max_{i}\sum_{j\neq i}|E_{ij}|<(n-1)\cdot\frac{\delta}{n-1}=\delta<1.

By Gerschgorin’s theorem, each eigenvalue $\lambda_{i}$ of $G=I+E$ lies in $\{\,z:|z-1|\leq\delta\}$ , so $G$ is positive‐definite and invertible. Since the spectral norm $\|E\|_{2}\leq\|E\|_{\infty}<1$ , the Neumann series for $(I+E)^{-1/2}$ converges in operator norm. Hence we may set

A=G^{-1/2}=(I+E)^{-1/2}.

Define

f_{i}=\sum_{j=1}^{n}A_{ij}\,e_{j}.

Then the Gram matrix of $\{f_{i}\}$ is

\langle f_{i},f_{j}\rangle=(A\,G\,A^{*})_{ij}=[G^{-1/2}\,G\,G^{-1/2}]_{ij}=\delta_{ij},

so $\{f_{i}\}$ is orthonormal.

Finally, we have for all $i,j$

\max_{i,j}|a_{ij}-\delta_{ij}|\leq\max_{1\leq i\leq n}{|\sqrt{\lambda_{i}}-1|}<\delta.

This completes the proof. ∎

Let us return to the example. For simplicity, let $Z_{j}=Z_{\lambda}^{p_{j}}$ . Then there exists a matrix $A=[a_{ij}]$ such that

f_{i}=\sum_{j\in J}a_{ij}\,Z_{j},\qquad\bigl{|}a_{ij}-\delta_{ij}\bigr{|}\lesssim|J|\,\lambda^{-\frac{d-1}{2}\beta},

and the family $\{f_{j}\}_{j\in J}$ is orthonormal.

Our goal is to choose $\beta$ so that $\lvert f_{i}(x)\rvert\approx\lambda^{\frac{d-1}{2}}$ whenever $x$ lies in the $\lambda^{-1}$ -neighborhood of $p_{i}$ . To ensure this, note that for such $x$ ,

\Bigl{|}\sum_{j\neq i}a_{ij}\,Z_{j}(x)\Bigr{|}\leq|J|\lambda^{\frac{d-1}{2}(1-\beta)}\lambda^{-\frac{d-1}{2}\beta}|J|\ll\lambda^{\frac{d-1}{2}}.

This holds provided

|J|\lambda^{-\frac{d-1}{2}\beta}=O\bigl{(}C^{-1}\,\lambda^{(1-\beta)k-\frac{d-1}{2}\beta}\bigr{)}\ll 1,

which is achieved by taking

(7.1)

\frac{2k}{2k+d-1}\leq\beta<1,

and $C$ sufficiently large. For the system $\{f_{j}\}$ constructed above, we then have

|f_{j}(x)|\geq\lambda^{\frac{d-1}{2}}\quad\text{on }B_{p_{j}}(\lambda^{-1}),

and $\mathrm{dist}(p_{i},p_{j})\gg\lambda^{-1}$ for $i\neq j$ . Hence, for $p\geq 2$ and $\dim\Sigma\leq d-1$ ,

	$\displaystyle\Bigl{\\|}\sum_{j\in J}t_{j}\,\|f_{j}\|^{2}\Bigr{\\|}_{L^{p/2}(\Sigma)}$	$\displaystyle\geq\Bigl{(}\sum_{j\in J}\int_{B_{p_{j}}(\lambda^{-1})\cap\Sigma}t_{j}^{\frac{p}{2}}\,\|f_{j}\|^{p}\Bigr{)}^{\frac{2}{p}}$
		$\displaystyle\geq\Bigl{(}\sum_{j\in J}t_{j}^{\frac{p}{2}}\,\lambda^{\frac{(d-1)p}{2}}\,\lambda^{-k}\Bigr{)}^{\frac{2}{p}}$
		$\displaystyle\approx\lambda^{\,d-1-\frac{2k}{p}}\,\bigl{\\|}t_{j}\bigr{\\|}_{\ell^{\frac{p}{2}}(J)}.$

This demonstrates the sharpness of Theorem 1.1, and Theorem 1.3 for the case $4\leq p\leq\infty$ .

In the case $k=d-1$ and $d\geq 3$ , one can likewise employ highest‐weight spherical harmonics to construct an orthonormal family $\{f_{j}\}_{j\in J}$ , with $|J|\sim\lambda^{(d-2)(1-\epsilon)}$ for any $0<\epsilon<1$ , each concentrating along the geodesic through a chosen point. Consequently, for $p=2$ the exponent $\alpha=1$ is achieved, confirming its sharpness.

Remark 7.2.

Finally, we point out that producing a sharpness example for Theorem 1.1 requires a nontrivial upper bound on $|J|$ —equivalently, a positive lower bound on $\beta$ (see, e.g., (7.1)). By contrast, Theorem 1.2 is sharp for all values of $|J|$ .

Indeed, take a codimension-2 submanifold $\Sigma\subset S^{d}$ and let $Q=E_{\lambda}$ , so that

|J|=\dim E_{\lambda}\sim\lambda^{d-1}.

The sharp pointwise Weyl law on the sphere gives

\sum_{\lambda_{j}\in I_{\lambda}}\bigl{|}e_{\lambda_{j}}(x)\bigr{|}^{2}\sim\lambda^{d-1},\quad x\in S^{d},

and hence

\Bigl{\|}\sum_{\lambda_{j}\in I_{\lambda}}|e_{\lambda_{j}}|^{2}\Bigr{\|}_{L^{p/2}(\Sigma)}\sim\lambda^{d-1}.

On the other hand, Theorem 1.1 yields the upper bound

\Bigl{\|}\sum_{\lambda_{j}\in I_{\lambda}}|e_{\lambda_{j}}|^{2}\Bigr{\|}_{L^{p/2}(\Sigma)}\lesssim\lambda^{{d-1}-\frac{2d-4}{p}+\frac{2(d-1)}{p}}=\lambda^{{d-1}+\frac{2}{p}}.

It follows that the exponent in Theorem 1.1 strictly exceeds the true exponent $d-1$ , so the bound cannot be sharp in this case unless $p=\infty$ .

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(4.4)	$\displaystyle\Big{\\|}WT_{\lambda}T^{*}_{\lambda}\overline{W}\Big{\\|}_{\mathfrak{S}^{1}(L^{2}(\Sigma))}$	$\displaystyle=\\|WT_{\lambda}\\|^{2}_{\mathfrak{S}^{2}(L^{2}(M)\to L^{2}(\Sigma))}$
		$\displaystyle=\int_{\Sigma}\int_{M}\|W(x)\|^{2}\|T_{\lambda}(x,y)\|^{2}\,dy\,dx$
		$\displaystyle\leq C\\|W\\|^{2}_{L^{2}(\Sigma)},$

	$\displaystyle\eqref{gg26}$	$\displaystyle\leq\\|W\\|^{2}_{L^{4}(\Sigma)}\cdot\Big{\\|}\int^{1}_{0}(1+\lambda\|s-t\|)^{-1}\|W(x(t))\|^{2}dt\Big{\\|}_{L^{2}(0,1)}$
		$\displaystyle\leq\\|W\\|_{L^{4}(\Sigma)}^{4}\cdot\int^{1}_{0}(1+\lambda t)^{-1}dt$
		$\displaystyle\lesssim\lambda^{-1}\log\lambda\\|W\\|_{L^{4}(\Sigma)}^{4}.$

	$\displaystyle\Big{\\|}WT_{\lambda}T_{\lambda}^{*}\overline{W}\Big{\\|}_{\mathfrak{S}^{\alpha^{\prime}}}$	$\displaystyle\leq\Big{(}\int^{1}_{0}\Big{(}\int^{1}_{0}\big{\|}W(x(s))\mathcal{K}(s,t)\overline{W(x(t))}\big{\|}^{\alpha}\,ds\Big{)}^{\alpha^{\prime}/\alpha}\,dt\Big{)}^{1/\alpha^{\prime}}$
		$\displaystyle\leq\\|W\\|_{L^{2\alpha^{\prime}}(\Sigma)}\Big{(}\int^{1}_{0}\Big{(}\int^{1}_{0}\|W(x(s))\|^{\alpha}(1+\lambda\|s-t\|)^{-\alpha/2}\,ds\Big{)}^{2\alpha^{\prime}/\alpha}\,dt\Big{)}^{1/2\alpha^{\prime}}$
		$\displaystyle\leq\\|W\\|^{2}_{L^{2\alpha^{\prime}}(\Sigma)}\Big{(}\int^{1}_{0}(1+\lambda\|s\|)^{-\frac{\alpha}{2}}\,ds\Big{)}^{1/\alpha}$
		$\displaystyle\lesssim\lambda^{-1/2}\\|W\\|^{2}_{L^{2\alpha^{\prime}}(\Sigma)}.$

	$\displaystyle\Big{\\|}WT_{\lambda}T^{\star}_{\lambda}\overline{W}\Big{\\|}_{\mathfrak{S}^{2}(L^{2}(\Sigma))}$	$\displaystyle=\Big{(}\int^{1}_{0}\int^{1}_{0}\|W(x(s))\|^{2}\|\mathcal{K}(s,t)\|^{2}\|\overline{W(x(t))}\|^{2}\,ds\,dt\Big{)}^{1/2}$
		$\displaystyle\leq\\|W\\|_{L^{2p_{0}/(p_{0}-2)}(\Sigma)}\Big{\\|}\int^{1}_{0}\|\mathcal{K}(s,t)\|^{2}\|W(x(t))\|^{2}\,dt\Big{\\|}^{1/2}_{L^{p_{0}/2}(0,1)}$
		$\displaystyle\leq\\|(1+\lambda\|s\|)^{-1}\\|^{1/2}_{L^{p_{0}/4}(0,1)}\\|W\\|^{2}_{L^{2p_{0}/(p_{0}-2)}(\Sigma)}$
		$\displaystyle\leq C_{p_{0}}\lambda^{-2/p_{0}}\\|W\\|_{L^{2p_{0}/(p_{0}-2)}(\Sigma)}^{2}.$

(5.9)		$\displaystyle\Big{\\|}W(\mathcal{T}_{\lambda}\mathcal{T}^{*}_{\lambda})_{j}\overline{W}\Big{\\|}_{\mathfrak{S}^{\infty}(L^{2}(\Sigma))}$	$\displaystyle\leq\\|W\\|^{2}_{L^{\infty}(\Sigma)}\\|(\mathcal{T}_{\lambda}\mathcal{T}^{*}_{\lambda})_{j}\\|_{\mathfrak{S}^{\infty}(L^{2}(\Sigma))}$
(5.9)			$\displaystyle=\\|W\\|^{2}_{L^{\infty}(\Sigma)}\\|(\mathcal{T}_{\lambda}\mathcal{T}^{*}_{\lambda})_{j}\\|_{L^{2}(\Sigma)\to L^{2}(\Sigma)}.$