1 Introduction↩︎

1.1 Restriction of eigenfunctions to fractal measures↩︎

Let \((M,g)\) be a compact, boundaryless, two-dimensional Riemannian manifold with metric \(g\). Let \(e_\lambda\) be an eigenfunction of \(-\Delta_g\) corresponding to the eigenvalue \(\lambda^2\), satisfying \[-\Delta_g e_\lambda = \lambda^2 e_\lambda.\] We assume that \(e_\lambda\) is normalized so that \(\|e_\lambda\|_{L^2(M)} = 1\).

There are several ways to quantify the concentration of eigenfunctions. One approach is to study the size of their \(L^p\) norms for \(p > 2\). In [4], Sogge proved that \[\label{soggees} \|e_\lambda\|_{L^p(M)} \leq C\lambda^{\delta_2(p)} \|e_\lambda\|_{L^2(M)},\tag{1}\] where \[\delta_2(p) = \begin{cases} \tfrac{1}{2}\bigl(\tfrac{1}{2} - \tfrac{1}{p}\bigr), & 2 \leq p \leq 6, \\[6pt] \tfrac{1}{2} - \tfrac{2}{p}, & 6 \leq p \leq \infty. \end{cases}\] These bounds are sharp on the standard sphere \(S^2\) and reveal two distinct types of eigenfunction concentration.

The eigenvalues of \(\sqrt{-\Delta_g}\) on \(S^2\) are \(\sqrt{k^2 + k}\), each with multiplicity \(d_k = 2k + 1\). The corresponding eigenspace \(\mathcal{E}_k\) consists of spherical harmonics of degree \(k\), obtained by restricting homogeneous harmonic polynomials of degree \(k\) to the unit sphere. Let \(\Pi_k(x, y)\) denote the kernel of the spectral projection operator onto \(\mathcal{E}_k\). The \(k\)-th \(L^2\)-normalized zonal function centered at \(x_0 \in S^2\) is \[\mathcal{Z}_k(y) = \bigl(\Pi_k(x_0, x_0)\bigr)^{-1/2} \Pi_k(x_0, y).\] The function \(\mathcal{Z}_k\) is sharply peaked at the poles \(\pm x_0\), where it attains the value \(\sqrt{d_k / 4\pi}\). It follows that \(\|\mathcal{Z}_k\|_{L^p(S^2)} \sim k^{\delta_2(p)}\) for \(p \geq 6\). Hence, on the standard sphere, Sogge’s bounds are optimal in this range of \(p\).

A different type of concentration is exhibited by the highest-weight spherical harmonic \(\mathcal{H}_k\), whose mass is concentrated along the equator \[\gamma_0 = \{(x_1, x_2, 0) : x_1^2 + x_2^2 = 1\}.\] The function \(\mathcal{H}_k\) is the restriction of the harmonic polynomial \(k^{1/4}(x_1 + i x_2)^k\) to \(S^2 = \{x : |x| = 1\}\). In this normalization, \(\mathcal{H}_k\) has \(L^2\) norm comparable to one, and its \(L^p\) norms satisfy \(\|\mathcal{H}_k\|_{L^p(S^2)} \sim k^{\delta_2(p)}\) for \(2 \leq p \leq 6\).

Another way to study the concentration of eigenfunctions is to examine their restriction to submanifolds. In [1], Burq–Gérard–Tzvetkov investigated the restriction of \(e_\lambda\) to submanifolds of compact Riemannian manifolds. In particular, for a smooth curve segment on a Riemannian surface, they proved the following result.

Burq–Gérard–Tzvetkov further showed that the exponent in 3 is sharp on \(S^2\) when \(\gamma\) is a geodesic. Although this bound cannot be improved on \(S^2\), several sharper estimates are known for manifolds with nonpositive curvature (see, for example, [5], [6]).

In this paper, we study the restriction of eigenfunctions to Borel subsets of \(M\), following the framework introduced by Eswarathasan and Pramanik [2]. To formulate results in this setting, we use the notion of \(\alpha\)-dimensional measures.

Eswarathasan and Pramanik [2] obtained restriction estimates for eigenfunctions on arbitrary Borel sets in all dimensions. In two dimensions, for any \(\alpha\)-dimensional probability measure \(\mu\) on \(M\), their results yield the following.

We note that the original statement in [2] is stronger than the streamlined formulation in Theorem 2, since the factor \(\lambda^\varepsilon\) can be removed for all \(p\in[2,\infty)\) except at the endpoint \(p=4\alpha\), where it can be replaced by a logarithmic loss. As our focus is on the exponents and thresholds in \(p\) and \(\alpha\), we do not track the precise loss here.

For \(p \ge 4\alpha\), these bounds interpolate between Sogge’s \(L^p(M)\) bounds and the curve restriction estimates of Burq–Gérard–Tzvetkov. In this range, \[\sigma(2,p)=\tfrac{1}{2}-\tfrac{2}{p}=\delta_2(p) \quad\text{and}\quad \sigma(1,p)=\tfrac{1}{2}-\tfrac{1}{p}=\delta_1(p).\] For \(\alpha=1\) the critical exponent is \(p=4\), agreeing with 3 . For \(\alpha=2\), the critical value \(p=8\) does not match Sogge’s critical exponent \(p=6\). Furthermore, Theorem 2 is not sharp for every pair \((\alpha,p)\) with \(\alpha \in (1,2]\) and \(p\in[2,4\alpha)\).

These comparisons leave a gap when \(\alpha\in(1,2]\) and \(2\le p<4\alpha\). Our primary objective is to establish sharp bounds, up to a \(\lambda^\varepsilon\) loss, for all \(p \ge 2\) and \(\alpha \in (0,2]\). Our main result is as follows.

From Theorem 3 we obtain sharp exponents, up to the usual \(\lambda^\varepsilon\) losses, for all \(p \ge 2\) and \(\alpha \in (0,2]\). At full dimension \(\alpha=2\), Theorem 3 identifies the critical Lebesgue exponent \(p=2\alpha+2=6\), recovering Sogge’s classical estimate. For intermediate fractal dimensions \(\alpha \in (1,2]\) and \(2 \le p < 4\alpha\), our bounds strictly improve those of Eswarathasan–Pramanik [2]. Figure 1 illustrates this comparison and shows that the sharp range in Theorem 3 extends beyond that of Theorem 2 when \(\alpha>1\).

As a direct consequence of Theorem 3 and Frostman’s lemma, we also obtain \(L^p\) bounds for the restriction of eigenfunctions to any Borel subset \(E \subseteq M\) with Hausdorff dimension \(\dim_{\mathcal{H}}(E)=\alpha \in (0,2]\). Given \(\varepsilon>0\), Frostman’s lemma guarantees a \((\alpha-\varepsilon)\)-dimensional probability measure \(\mu^{(\varepsilon)}\) supported on \(E\). Theorem 3 then implies the following.

We remark that the formulation of our fractal \(L^p\) restriction estimates differs from Theorem 1.3 of [2], where the critical value of \(p\) also depends on the choice of \(\varepsilon>0\). This discrepancy appears because we do not emphasize the exact size of the \(\lambda^\varepsilon\) losses. These losses arise naturally from the multi-scale analysis used in our proof. In contrast, [2] gives a more precise description of the loss factor. Nonetheless, 7 is sharp, modulo the \(\lambda^\varepsilon\) losses, for all \(p\ge 2\) on the standard sphere for suitable fractal sets \(E\). Apart from the \(\lambda^\varepsilon\) losses, our results unify the two-dimensional bounds of Sogge [4], Burq–Gérard–Tzvetkov [1], and Eswarathasan–Pramanik [2].

Sogge’s \(L^p\) eigenfunction bounds [4] can be viewed as a variable-coefficient analogue of the classical Fourier restriction theorem of Tomas–Stein. Shayya [7] proved a weighted Fourier restriction estimate that sharpens Tomas–Stein. This work partially inspired our proof of Theorem 3, which may be seen as a variable-coefficient counterpart of Shayya’s result.

The proof of Theorem 2 in [2] uses a \(TT^*\) argument. In contrast, our proof of Theorem 3 is based on a multi-scale wave-packet analysis. In particular, we use the broad–narrow argument of Bourgain and Guth [8]. To the best of our knowledge, this is the first application of modern multi-scale wave-packet methods to eigenfunction estimates. This opens a path to applying further techniques, including decoupling and polynomial methods, to spectral problems.

1.2 Restriction of eigenfunctions to the neighborhood of a curve↩︎

We now study eigenfunction concentration near curves with nonvanishing geodesic curvature. As a starting point, recall that in [1], Burq–Gérard–Tzvetkov showed that if a curve segment \(\gamma\) has nonvanishing geodesic curvature, i.e., if \(\gamma\) is parametrized by arc length \(s\) and \[\langle \nabla_{\gamma'(s)}\gamma'(s), \nabla_{\gamma'(s)}\gamma'(s)\rangle \neq 0,\] then the bound 2 can be improved. More precisely:

Our second result concerns mass distribution in tubular neighborhoods of such \(\gamma\). We show that the average \(L^2\) density of \(e_\lambda\) in tubes around \(\gamma\) decreases in a predictable way as the neighborhood size increases.

The proof of Theorem 5 uses a straightforward wave-packet analysis together with a geometric observation about how the tubes around \(\gamma\) intersect relevant wave-packets.

In comparison, as a direct consequence of 8 , we have \[\label{eq:032to321473} \|e_\lambda\|_{L^2_{\mathrm{avg}}(T_{\lambda^{-1+s}}(\gamma))} \le C\, \lambda^{\frac{1}{6}} \, \|e_\lambda\|_{L^2(M)},\tag{10}\] for every \(0 \le s \le \tfrac{1}{2}\). In particular, for \(0 \le s \le \tfrac{1}{3}\), 10 is sharp for the same reason that 9 is sharp when \(s=\tfrac{1}{3}\). Our estimate 9 improves this bound for \(\tfrac{1}{3} < s \le \tfrac{1}{2}\). In addition, our result identifies the critical scale for concentration near a curve with nonvanishing geodesic curvature as \(\lambda^{-2/3}\), which matches the scale at which a Dirichlet eigenfunction can concentrate near the boundary of a domain [9].

Theorem 5 is closely related to the Mizohata–Takeuchi conjecture. We show that a variable-coefficient analogue of this conjecture yields the same bound as 9 , which explains why our result is natural.

Finally, we remark that, as is standard in the study of eigenfunction Lebesgue norms, our bounds apply not only to the eigenfunctions \(e_\lambda\) but also to a smoothed spectral projector on \(M\).

1.3 Organization↩︎

We organize the paper as follows. Section 2 reduces the problem to oscillatory integral operators with Carleson–Sjölin phases, introduces a wave-packet decomposition, and reduces the proof of Theorem 3 to two key statements, Propositions 6 and 7. Section 3 proves Proposition 6, establishing weighted \(L^2\) estimates via a variable-coefficient bilinear restriction estimate and the Bourgain–Guth method. Section 4 proves Proposition 7 and extends the results to \(L^q\) for a range of exponents via interpolation and properties of fractal measures. Section 5 proves Theorem 5 and explores its connection to the Mizohata–Takeuchi conjecture. Finally, Section 6 discusses sharpness, demonstrating optimality through explicit examples and known behavior of spherical harmonics and eigenfunctions.

1.4 Notation↩︎

For \(r>0\), \(B(x, r)\) denotes a ball centered at \(x\) with radius \(r\). In local coordinates we do not distinguish between geodesic and Euclidean balls at the scales we consider. We abbreviate \(B_r\) when the center is not of particular interest. We use \(w_{B_r}\) to denote a Schwartz weight function that is adapted to \(B_r\) so that \(w_{B_r}\geq 1\) on \(B_r\) and decays rapidly away from it.

For non-negative quantities \(A\) and \(B\), we shall write \(A\lesssim B\), if there is an absolute constant \(C>0\) such that \(A\le CB\). We shall write \(A\approx B\), if \(\frac{1}{C} B\le A\le CB\) for some absolute constant \(C>1\).

We shall use \({\rm RapDec}(R)\) to denote a term that is rapidly decaying in \(R>1\), that is, for any \(N\in\mathbb{N}\), there exists a constant \(C_N\) such that \(|{\rm RapDec} (R)|\le C_N R^{-N}\).

1.5 Acknowledgements.↩︎

This project was supported by the National Key R&D Program of China: No. 2022YFA1007200, 2022YFA1005700, and 2024YFA1015400. C. Gao was supported by Natural Science Foundation of China grant: No. 12301121. C. Miao was partially supported by Natural Science Foundation of China grant: No. 12371095. Y. Xi was partially supported by Natural Science Foundation of China under Grant No. 12571107 and 12171424 and the Zhejiang Provincial Natural Science Foundation of China under Grant No. LR25A010001. The first author would like to express gratitude to Jianhui Li and Shaozhen Xu for their insightful discussions. The authors thank the anonymous referees for their careful reading and insightful comments, which have significantly improved the presentation of the paper.

2 Preliminaries↩︎

2.1 Reduction to oscillatory integral operators with Carleson–Sjölin phases↩︎

We begin by applying a standard reduction due to Sogge [10], specifically utilizing the variant presented in [1].

By employing a smooth partition of unity, we may assume that the support of \(\alpha(x, y, \lambda)\) is contained within \(B(x_0, \delta_0/10) \times B(y_0, \delta_0/10)\) for some points \(x_0, y_0 \in M\) satisfying \(\delta_0/2 < d_g(x_0, y_0) < \delta_0\). We work in geodesic normal coordinates centered at \(x_0=(0,0)\), so that the coordinates of \(y_0\) are \((0, t_0)\) for some \(t_0 \in (\delta_0/2, \delta_0)\).

Next, we express \(y\) in polar coordinates around \((0, 0)\) as \(y = (t \cos \xi, t \sin \xi)\), where \(t = |y|\). It is well-known (see, e.g., Chapter 5 in [10]) that for any fixed \(t \in (\delta_0/2, \delta_0)\), the phase function \[\phi(x, \xi) := -d_g\big(x, (t \cos \xi, t \sin \xi)\big)\] satisfies the Carleson–Sjölin conditions: \[\label{eq:CS-condition} \operatorname{rank} \begin{pmatrix} \partial_\xi \partial_{x} \phi(x, \xi) \\ \partial_\xi^2 \partial_{x} \phi(x, \xi) \end{pmatrix} = 2.\tag{12}\]

From this point forward, we focus on an oscillatory integral operator associated with a Carleson–Sjölin phase function \(\phi(x, \xi)\). Indeed, all of our main results can be rephrased to provide corresponding results for any oscillatory integral operator with a Carleson–Sjölin phase. Given \(\lambda > 1\), we define the rescaled phase function \(\phi^\lambda(x, \xi) := \lambda \phi(x/\lambda, \xi),\) and consider the oscillatory integral operator \[\mathcal{T}^\lambda f(x) := \int e^{i \phi^\lambda(x, \xi)} a^\lambda(x, \xi) f(\xi) \, d\xi.\] Here \(a^\lambda(x,\xi)\) is a smooth function whose \(x\)-support is contained in \(B(0,\lambda)\). In our application, it is given by \[a^\lambda(x,\xi) := \alpha\big(x/\lambda,\,(t\cos\xi, t\sin\xi),\,\lambda\big).\] Any \(L^2 \to L^p\) bound for \(\mathcal{T}^\lambda\) thus implies a corresponding \(L^2 \to L^p\) bound for the operator \(\chi_\lambda\), after integrating in \(t\).

Let \(H: \mathbb{R}^2 \to [0, 1]\) be a measurable function. Define \(A_\alpha(H)\) by \[\label{fractal} A_\alpha(H) := \inf \Big\{ C \ge 0 : \int_{B(x_0,r)} H(x)\, dx \le C r^\alpha \;\text{for all } x_0 \in \mathbb{R}^2 \text{ and } r \ge 1 \Big\}.\tag{13}\] We say that \(H\) is an \(\alpha\)-dimensional weight if \(A_\alpha(H) < \infty\). The associated weighted \(L^p\) norm is defined by \[\|f\|_{L^p(B(x_0,r);\, H \, dx)}: = \left( \int_{B(x_0,r)} |f(x)|^p H(x) \, dx \right)^{1/p}.\]

Given an \(\alpha\)-dimensional weight \(H\), we will show in Section 2.4 that the proof of Theorem 3 reduces to establishing estimates of the following form: \[\|\mathcal{T}^\lambda f\|_{L^p(B(0, \lambda);\,H dx)} \leq C_\varepsilon \lambda^{\beta(\alpha,p)+\varepsilon} \|f\|_{L^2}, \quad 2 \leq p \leq \infty,\] for some constant \(C_\varepsilon > 0\) independent of \(\lambda\) and suitable exponents \(\beta(\alpha,p) \geq 0\).

By interpolating between Propositions 6 and 7, we obtain

It is worth noting that the exponent \(\delta(\alpha, p)\) in equation 5 and \(\beta(\alpha, p)\) in equation ?? are related by the expression \[\delta(\alpha, p) = \beta(\alpha, p) + \frac{1}{2} - \frac{\alpha}{p}.\] Roughly speaking, the term \(-\frac{\alpha}{p}\) arises from the change of variable \(x \rightarrow \frac{x}{\lambda}\). This will be made precise in Lemma 6.

2.2 Wave-packet decomposition↩︎

We perform a wave-packet decomposition for \(\mathcal{T}^\lambda f\). The wave-packet viewpoint for oscillatory and Fourier integral operators goes back to Smith [11], [12] and to the curvelet-based analysis of Candès–Demanet [13]. These decompositions simplify the analysis of wavefronts and compositions, and clarify the operator’s classical action.

Without loss of generality, assume that the support of \(f\) is contained in the open interval \(I_0=[0,1]\). In this section we fix a parameter \(R \in [1,\lambda]\) and work in the ball \(B(0,R)\). First, we cover \(I_0\) by \(R^\frac{1}{2}\) open subintervals \(\theta\) of size \(R^{-\frac{1}{2}}\) with bounded overlap. Let \(\{\psi_\theta\}\) be a smooth partition of unity subordinate to this cover and write \[f=\sum_\theta f_\theta, \quad f_\theta:=f \psi_\theta.\] Next, we further decompose \(f_\theta\) in the physical side. Cover \(\mathbb{R}\) by a collection of finitely overlapping open intervals \(\{I_v\}_v\) with center \(v\) and length \(R^{\frac{1}{2}}\). Let \(\eta_v\) be a smooth partition of unity subordinate to this cover, we further decompose \(f\) into \[f=\sum_{\theta,v}f_{\theta,v},\quad f_{\theta,v}:=((f\psi_\theta)^{\wedge}\eta_v)^{\vee}.\] By Plancherel’s theorem and the fact that \(\{\psi_\theta\}\) and \(\{\eta_v\}\) form partitions of unity in frequency and in space, respectively, we obtain the following almost-orthogonality estimate.

Let \(\xi_\theta\) denote the center point of the interval \(\theta\). Since \(\phi\) is a Carleson–Sjölin phase, the associated nondegeneracy implies that the map \(x_1 \mapsto \partial_\xi \phi(x_1,x_2,\xi_\theta)\) is locally invertible. By the implicit function theorem, for each \(v \in [0,\lambda]\) there exists a smooth function \(\gamma_\theta(v/\lambda,x_2)\) such that \[\label{gamma95theta} \partial_{\xi}\phi(\gamma_\theta(v/\lambda,x_2),x_2,\xi_\theta) = v/\lambda .\tag{14}\] We denote the corresponding curve by \[\Gamma_{\xi_\theta,v}:=\{(\gamma_\theta(v/\lambda,x_2),x_2):\, |x_2|\leq 1\}.\] We define \(\gamma_\theta^\lambda(v,x_2):=\lambda \gamma_\theta(v/\lambda,x_2/\lambda)\) and the rescaled curve \(\Gamma_{\xi_\theta,v}^\lambda\) as \[\label{Gamma}\Gamma_{\xi_\theta,v}^\lambda:=\{(\gamma_\theta^\lambda(v,x_2),x_2):\, |x_2|\leq R\}.\tag{15}\] Given \(0<\varepsilon\ll 1\), we consider a curved tube \(T_{\theta,v}\) given by \[T_{\theta,v}:= \{(x_1,x_2): |x_1-\gamma_{\theta}^\lambda(v,x_2)|\leq R^{\frac{1}{2}+\varepsilon}, |x_2|\leq R\}.\] The next lemma shows that for each \(\theta,v\), \(\mathcal{T}^\lambda f_{\theta,v}\) is essentially supported on the tube \(T_{\theta,v}\).

2.3 Locally constant property↩︎

Roughly speaking, the locally constant property says that if the Fourier transform of a function \(f\) is supported in a ball of radius \(r\), then \(f\) can be treated as a constant in any ball of radius \(1/r\). The fact is extensively used in the research of restriction theory; one may refer to [3], [8], [14] for details. Let us formulate the locally constant property in our settings.

As a direct consequence of Lemma 4, we have

2.4 Proposition 8 implies Theorem 3↩︎

Recall that, we say a compactly supported probability measure \(\mu\) is \(\alpha\)-dimensional if there exists a constant \(C>0\), such that \[\mu(B(x,r))\leq C r^\alpha, \quad \forall x\in \mathbb{R}^2, r>0.\] The following lemma allows us to bound \(\|\mathcal{T}^\lambda f\|_{L^p(M;\, d\mu)}\) by \(\|\mathcal{T}^\lambda f\|_{L^p(B(0,\lambda);H dx)}\), where \(H\) is a suitable \(\alpha\)-dimensional weight function.

2.5 Variable-coefficient bilinear restriction estimate↩︎

Finally, we record a bilinear estimate that will be used to control the contribution of broad points in the next section. Let \(\tau_1,\tau_2\) be two disjoint intervals contained in \([0,1]\). Assume \(\operatorname{supp}\,f_{\tau_i} \subset \tau_i, i=1,2\), we say the transversality condition holds if there exists a constant \(\nu>0\) such that \[|{\rm det}(\partial_\xi\partial_x\phi(x,\xi_1),\partial_\xi\partial_x\phi(x,\xi_2))|\geq \nu>0,\quad \text{for all }\xi_i \in \tau_i,i=1,2.\]

3 Proof of Proposition 6↩︎

In this section, we prove Proposition 6. The key input is a weighted estimate for the oscillatory integral operator \(\mathcal{T}^\lambda\). We follow the Bourgain–Guth argument: we partition \(B(0,R)\) according to the local configuration of wave-packets at a point. A point is broad if at least two intersecting wave-packets with significant contributions have well-separated directions; otherwise, if the directions of all wave-packets with significant contributions concentrate near a single point of \(S^1\), it is narrow. We control the broad contribution via a bilinear estimate and treat the narrow contribution using an induction-on-scales argument.

3.1 Summary of notation used in the multiscale analysis↩︎

In this subsection we record the definitions and notation for the families of geometric objects—at various scales—used in the proof. This summary is intended to guide the reader through the argument.

First, fix the parameters \(R,\lambda,\varepsilon\) with \(1 \le R \le \lambda\) and \(0<\varepsilon\ll 1\). Set \(K = R^{\kappa}\) with \(\kappa = \varepsilon^{2}\), \(R_{1} = R/K^{2}\), and \(K_{1} = R_{1}^{\kappa}\). Let \(\rho\) range over \([1, R^{1/2}]\). See Table 1.

We work in a ball \(B(x_0,R)\subset B(0,\lambda)\). Without loss of generality, we may assume that \(x_0=0\) and write \(B_R=B(0,R)\). We use \(\mathcal{Q}=\{Q\}\) to denote a finitely overlapping collection of squares of side length \(K^2\).

By 14 , for each pair \((a,b)\in[0,1]\times[0,R]\) there exists a function \(\gamma_{a,b}=\gamma_{a,b}(x_2)\) such that \[\partial_\xi \phi\big(\gamma_{a,b}(x_2),x_2,a\big)=\frac{b}{\lambda}.\] We denote its rescaling by \(\gamma^\lambda_{a,b}({}\cdot{}):=\lambda\,\gamma_{a,b}({}\cdot/\lambda)\) and set \[\Gamma_{a,b}^\lambda:=\big\{ \big(\gamma_{a,b}^\lambda(x_2),\,x_2\big) \colon |x_2|\le R \big\}.\] This is consistent with our definition of \(\Gamma_{\xi_\theta,v}^\lambda\) as given in 15 . We shall refer to \(a\) as the direction of the curve.

For every \(1\le \rho \le R^{1/2}\) and \((a,b)\in[0,1]\times[0,R]\), let \(I_{a,b}\) denote the \(K^2\rho^{1-4\kappa}\)-neighborhood of \(\Gamma_{a,b}^\lambda\). We decompose \(I_{a,b}\) into a finitely overlapping collection \(\mathcal{I}_{a,b}\) of curved sub-tubes \(I\), each of length \(K^2\rho^{2-4\kappa}\) and the same width \(K^2\rho^{1-4\kappa}\); see Figure 2 for an illustration.

In this way we obtain the collection \[\mathcal{I} := \bigl\{\, I \colon I \in \mathcal{I}_{a,b}\text{ is a length } K^2\rho^{2-4\kappa} \text{ sub-tube of } I_{a,b}, \;\text{for some } (a,b)\in[0,1]\times[0,R] \,\bigr\}.\] Similarly, we define the collection of \(R/\rho\)-neighborhoods of \(\Gamma_{a,b}^\lambda\) by \[\mathcal{O} := \bigl\{\, O \colon O \text{ is the R/\rho-neighborhood of } \Gamma_{a,b}^\lambda,\;\text{for some }(a,b)\in[0,1]\times[0,R] \,\bigr\}.\] For simplicity, we shall continue to write \(O\) for an element of \(\mathcal{O}\).

For later use, we also introduce the notation \(Q'\), \(I'\), \(O'\), and \(T'_{\tau,v}\) for curved tubes at several scales produced by parabolic rescaling. Recall that \(R_1=R/K^2\) and \(K_1=R_1^{\kappa}\).

We use \(Q'_{a,b}\) to denote the \(K\,K_1^2\)–neighborhood of \(\Gamma_{a,b}^\lambda\). Each tube \(Q'_{a,b}\) is further decomposed along its longitudinal direction into a finitely overlapping collection \(\{Q'\}\) of shorter tubes of length \(K^2K_1^2\). We denote the collection of all such \(Q'\) over all possible \((a,b)\in[0,1]\times[0,R]\) by \(\mathcal{Q}'\).

Similarly, let \(I'_{a,b}\) denote the \(K\,K_1^2\rho^{\,1-4\kappa}\)–neighborhood of \(\Gamma_{a,b}^\lambda\). We partition \(I'_{a,b}\) into a finitely overlapping family of shorter tubes of length \(K^2K_1^2\rho^{\,2-4\kappa}\), and denote the collection of all such \(I'\) over all possible \((a,b)\in[0,1]\times[0,R]\) by \(\mathcal{I}'\). For convenience, we write \(I'\) for an element of \(\mathcal{I}'\).

Next, let \(\mathcal{O}'\) denote the collection of \(R/(K\rho)\)-neighborhoods of the curve segments \(\Gamma_{a,b}^\lambda\). For convenience, we write \(O'\) for an element of \(\mathcal{O}'\).

Finally, let \(\tau\subset[0,1]\) be an interval of length \(K^{-1}\) centered at \(\xi_\tau\). We choose a collection of parameters \(\{v\}\subset[0,R]\) that are \(R/K\)–separated, and denote by \(T_{\tau,v}'\) the \(R/K\)–neighborhood of the curve \(\Gamma_{\xi_\tau,v}^\lambda\). We write \(\mathfrak{T}_\tau'\) for the collection of all such tubes.

3.2 The case \(1\le \alpha \le 2\)↩︎

We now prove Proposition 6 in the range \(1\le \alpha \le 2\) via the Bourgain–Guth argument.

Our proof of ?? follows the approach of Zorin-Kranich [16]. The estimate ?? can be viewed as a variable-coefficient analogue of Theorem 1.6 in [3]. In particular, when \(\alpha=1\), Du–Zhang’s Theorem 1.6 yields sharp results on Carleson’s pointwise convergence for the Schrödinger operator in \(\mathbb{R}^{1+1}\). Theorem 1.6 in [3] also covers higher dimensions. We plan to study the corresponding variable-coefficient extensions in a sequel.

3.2.1 Reductions↩︎

First, decompose \(B(0,R)\) into a collection \(\mathcal{Q}\) of squares of side length \(K^2\) with bounded overlaps, and let \(k\) range over dyadic numbers with \(k\ge -C\log_2 K\). Define \[\mathcal{Q}_k := \{ Q \in \mathcal{Q} : \textstyle\int_Q H(x)\,dx \sim 2^{-k} \}.\] We may assume that \(R^{-100} \le 2^{-k} \le R\), since the contribution from the range \(2^{-k} < R^{-100}\) is negligible. By a dyadic pigeonholing argument, we may further assume that there exists a dyadic number \(k_0\) such that \[\label{eq:weighted-L2} \|\mathcal{T}^\lambda f\|_{L^2(B(0,R); H\,dx)}^2 \le C_\varepsilon R^\varepsilon \sum_{Q \in \mathcal{Q}_{k_0}} \|\mathcal{T}^\lambda f\|_{L^2(Q;\,Hdx)}^2.\tag{18}\] Set \[\mu_Q = \begin{cases} 1, & Q \in \mathcal{Q}_{k_0},\\[4pt] 0, & \text{otherwise}. \end{cases}\] Then the right-hand side of 18 is bounded by \[\label{eq:muL2} 2^{-k_0}\sum_{Q} \mu_Q^2 \|\mathcal{T}^\lambda f\|_{L^\infty(Q)}^2.\tag{19}\]

To incorporate the ball condition for \(H\) (see 13 ), namely \[\int_{B_r } H(x)\,dx \leq A_\alpha(H)\, r^{\alpha},\qquad K^2 \leq r\leq R,\] we require that \[\label{eq:mucondition} \sum_{Q\subset B_r} \mu_{Q} \leq A_\alpha(H)\,2^{k_0} r^{\alpha}, \qquad K^2 \le r \le R.\tag{20}\]

Define \[\label{eq:defW} \mathcal{W} := \sup_{1 \le \rho \le R^{1/2}} \rho^{-1/2} \sup_{O \in \mathcal{O}} \Bigg( \sum_{\substack{I \in \mathcal{I}\\ I \subset O}} \Big( \sum_{Q \subset I} \mu_Q^2 \Big)^2 \Bigg)^{1/4}.\tag{21}\] Here, the summation \(\sum_{I\in\mathcal{I},\, I\subset O}\) runs over the subcollection of tubes \(I\in\mathcal{I}\) that have the same direction as \(O\) and form a finitely overlapping cover of \(O\).

It is straightforward to show that \[\label{eq:Wbound} \mathcal{W} \le C_\varepsilon\,A_\alpha(H)^\frac{1}{2}\, 2^{\frac{k_0}{2}} R^{\tfrac{\alpha}{4}+\varepsilon}.\tag{22}\] Indeed, using that the tubes \(I\) have bounded overlap, we obtain \[\begin{align} \mathcal{W} &= \sup_{1 \le \rho \le R^{1/2}} \rho^{-1/2} \sup_{O \in \mathcal{O}} \Bigg( \sum_{\substack{I \in \mathcal{I}\\ I \subset O}} \Big( \sum_{Q \subset I} \mu_Q^2 \Big)^2 \Bigg)^{1/4} \\ &\le \sup_{1 \le \rho \le R^{1/2}} \rho^{-1/2} \sup_{O \in \mathcal{O}} \Big( \sum_{Q \subset O} \mu_Q^2 \Big)^{1/4} \sup_{I \in \mathcal{I}} \Big( \sum_{Q \subset I} \mu_Q^2 \Big)^{1/4} \\ &\le C_\varepsilon A_\alpha(H)^{1/2} R^{\varepsilon} \sup_{1 \le \rho \le R^{1/2}} \rho^{-1/2} \Big( 2^{k_0}\, \rho (R/\rho)^{\alpha} \Big)^{1/4} \Big( 2^{k_0}\, \rho \cdot \rho^{\alpha} \Big)^{1/4} \\ &\le C_\varepsilon A_\alpha(H)^{1/2} R^{\varepsilon} 2^{k_0/2} R^{\alpha/4}. \end{align}\] Here, in the second-to-last inequality, we applied the ball condition 20 twice, with radii \(r=R/\rho\) and \(r=\rho\), respectively. Consequently, to prove ?? , it suffices to establish the following proposition, which is the key technical estimate in the proof of our main theorem.

In fact, using 18 , 19 , 22 and ?? , we obtain \[\begin{align} \int_{B(0,R)} |\mathcal{T}^\lambda f|^2 H(x)\,dx &\le C_\varepsilon R^{3\varepsilon} 2^{-k_0} \sum_{Q} \mu_{Q}^2 \| \mathcal{T}^\lambda f \|_{L^\infty(Q)}^2 \\ &\le C_\varepsilon R^{3\varepsilon} A_\alpha(H)2^{-k_0} 2^{k_0} R^{\frac{\alpha}{2}} \|f\|_{L^2}^2. \end{align}\]

In general, let \(\mu_Q \ge 0\) be arbitrary for each \(Q \in \mathcal{Q}\). The definition of \(\mathcal{W}\) is unchanged. Define \(\mathcal{F}(\lambda, R)\) to be the smallest positive number such that \[\Bigg( \sum_{Q} \mu_Q^2 \|\mathcal{T}^\lambda f\|_{L^\infty(Q)}^2 \Bigg)^{1/2} \le \mathcal{F}(\lambda, R)\, \mathcal{W}\, \|f\|_{L^2}.\] To prove ?? , it suffices to show for each \(\varepsilon>0\), there exists \(\bar{C}_\varepsilon>0\) such that \[\label{inred} \mathcal{F}(\lambda,R)\leq \bar{C}_\varepsilon R^\varepsilon,\quad 1\leq R\leq \lambda.\tag{23}\]

Induction hypothesis. To prove 23 , we argue by induction on scales. First, it is straightforward to verify that, for \(N_\varepsilon\in\mathbb{N}\) to be chosen later, one can choose \(\bar C_\varepsilon\) sufficiently large so that \[\mathcal{F}(\lambda,R)\le \bar C_\varepsilon R^\varepsilon,\qquad \text{for all } 1\le R\le \min\{N_\varepsilon,\,\lambda\}.\] Now suppose \(R\ge N_\varepsilon\). As the induction hypothesis, assume that \[\label{assumption} \mathcal{F}(\bar\lambda,\bar R)\le \bar C_\varepsilon \bar R^\varepsilon,\tag{24}\] for all \(1\le \bar R\le R/2\) and \(\bar R\le \bar\lambda\le \lambda/2\). To close the induction, it suffices to establish \[\label{ingoal} \mathcal{F}(\lambda,R)\le \bar C_\varepsilon R^\varepsilon,\tag{25}\] under the assumptions \(R\ge N_\varepsilon\) and 24 . To achieve this, we invoke the following parabolic rescaling lemma, which links estimates across different scales.

3.2.2 Parabolic rescaling lemma↩︎

For technical reasons, to complete the induction argument in the narrow case, it is customary to prove estimates for a class of phase functions. To this end, we impose quantitative conditions on the phase function \(\phi\) and introduce the notion of reduced form.

Roughly speaking, a phase function \(\phi(x,\xi)\) satisfying the Carleson–Sjölin conditions can be regarded as a small perturbation of the translation-invariant case. More precisely, through a suitable change of variables, \(\phi\) can be written as \[\label{eq:09} \phi(x,\xi) = \langle x_1, \xi \rangle + x_2 h(\xi) + \mathcal{E}(x,\xi),\tag{26}\] where \(h\) and \(\mathcal{E}\) are smooth functions, \(h\) is quadratic in \(\xi\), and \(\mathcal{E}\) is quadratic in both \(x\) and \(\xi\).¹

As in the discussion of (3.7) in [17], to establish 23 , it suffices to consider the phase function in reduced form.

To derive the estimate for the narrow case, we require the following parabolic rescaling lemma, which connects estimates across different scales. Such parabolic rescaling lemmas are widely used in multi-scale wave-packet analysis; see, for example, [17]–[19] for similar variable-coefficient parabolic rescaling results. Here, we provide only an outline of the proof. For a detailed treatment, the reader is referred to the proof of inequality (3.7) in [17].

We cover the interval \([0,1]\) by \(K\) open subintervals \(\tau\) of length \(K^{-1}\) with bounded overlap. Let \(\{\tilde{\psi}_\tau\}\) be a smooth partition of unity subordinate to this cover. Correspondingly, define \[f_\tau = f\,\tilde{\psi}_\tau, \qquad \mathcal{T}^\lambda f = \sum_\tau \mathcal{T}^\lambda f_\tau.\] Recall that \(T'_{\tau,v}\) are tubes in the collection \(\mathfrak{T}'_\tau\). We have the following lemma.

3.2.3 Bourgain–Guth argument↩︎

Now we are ready to prove 23 .

It is straightforward to verify that, if \(Q\) is narrow, then there exists some \(\tau\) such that \[\|\mathcal{T}^\lambda f_\tau\|_{L^\infty(Q)} \ge \frac{1}{10000}\,\|\mathcal{T}^\lambda f\|_{L^\infty(Q)}.\] In contrast, if \(Q\) is broad, then there exist two significant caps \(\tau_1,\tau_2\) such that the corresponding wave packets associated to \(\tau_1\) and \(\tau_2\) both pass through \(Q\), and the angle between them is \(\ge 1000 K^{-1}\). See Figure 3 for a comparison of narrow and broad squares \(Q\).

Correspondingly, we decompose \(B_R\) into squares \(Q\) of side length \(K^2\) with bounded overlap and define \[\mathbf{B} := \bigcup_{\substack{Q\;\text{broad}}} Q, \qquad \mathbf{N} := \bigcup_{\substack{Q\;\text{narrow}}} Q.\] Note that \[\Bigg( \sum_{Q \subset B_R} \mu_Q^2 \,\|\mathcal{T}^\lambda f\|_{L^\infty(Q)}^2 \Bigg)^{1/2} \le \Bigg( \sum_{Q \subset \mathbf{B}} \mu_Q^2 \,\|\mathcal{T}^\lambda f\|_{L^\infty(Q)}^2 \Bigg)^{1/2} + \Bigg( \sum_{Q \subset \mathbf{N}} \mu_Q^2 \,\|\mathcal{T}^\lambda f\|_{L^\infty(Q)}^2 \Bigg)^{1/2}.\] We apply the bilinear estimate 17 to the broad part, and an induction-on-scales argument to the narrow part.

Broad case. First, we bound the Broad term. Note that if \(Q\) is broad, then there exist two intervals \(\tau_1,\tau_2\in\mathcal{S}(Q)\) with \[{\rm dist}(\tau_1,\tau_2)\geq 1000 K^{-1}.\] Thus by the locally constant property, we have \[\begin{align} \label{locala} \|\mathcal{T}^\lambda f\|_{L^\infty(Q)}\lesssim K^{C} \sum_{{\rm dist}(\tau_1,\tau_2 )\geq 1000 K^{-1}}\big\||\mathcal{T}^\lambda f_{\tau_1} \mathcal{T}^\lambda f_{\tau_2}|^{\frac{1}{2}}\big\|_{L^\infty(Q)}. \end{align}\tag{31}\] Applying the bilinear estimate Theorem 9 with \(\nu=1000K^{-1}\) and \(\varepsilon_1=\varepsilon/2\), we obtain \[\begin{align} ( \sum_{Q\subset \mathbf{B}} \mu_Q^2 \|\mathcal{T}^\lambda f\|_{L^\infty(Q)}^2)^{1/2} &\lesssim K^C ( \sum_{Q\subset \mathbf{B}} \mu_Q^4 )^{1/4}(\sum_{Q\subset \mathbf{B}}\|\mathcal{T}^\lambda f\|_{L^\infty(Q)}^4)^{1/4}\\ &\lesssim K^{C} (\sum_{Q\subset \mathbf{B}} \mu_Q^4)^{1/4} \!\!\! \!\!\!\!\!\!\sum_{{\rm dist}(\tau_1,\tau_2)>1000K^{-1}}(\sum_{Q\subset \mathbf{B}}\||\mathcal{T}^\lambda f_{\tau_1}\mathcal{T}^\lambda f_{\tau_2}|^{1/2}\|_{L^\infty(Q)}^4)^{1/4}\\ &\lesssim K^C(\sum_{Q\subset \mathbf{B}}\mu_Q^4)^{1/4} \!\!\! \!\!\!\!\!\!\sum_{{\rm dist}(\tau_1,\tau_2)>1000K^{-1}} \||\mathcal{T}^\lambda f_{\tau_1}\mathcal{T}^\lambda f_{\tau_2}|^{1/2}\|_{L^4(B_R)}\\ &\le C_{\varepsilon} K^C R^{\varepsilon/2} (\sum_{Q\subset \mathbf{B}} \mu_Q^4)^{1/4}\|f\|_{L^2}. \end{align}\]

In the second-to-last inequality, we bound each \(L^\infty(Q)\)-norm by its \(L^4(Q)\)-norm using the locally constant property (Lemma 5), and then sum over \(Q\subset \mathbf{B}\). The resulting loss is absorbed into the \(K^{C}\) factor. Note that, choosing \(\rho=1\) in the definition of \(\mathcal{W}\), we obtain \[\Big(\sum_{Q\subset \mathbf{B}} \mu_Q^4\Big)^{1/4}\le \mathcal{W}.\] Therefore \[\label{eq32B} \Bigg( \sum_{Q \subset \mathbf{B}} \mu_Q^2 \|\mathcal{T}^\lambda f\|_{L^\infty(Q)}^2 \Bigg)^{1/2} \le C_\varepsilon K^{C} R^{\varepsilon/2} \, \mathcal{W} \, \|f\|_{L^2}.\tag{32}\]

Narrow case. Now we estimate the narrow term. Note that if a square \(Q\) is narrow, then there exists some \(\tau\) such that \[10000\,\|\mathcal{T}^\lambda f_\tau\|_{L^\infty(Q)} \ge \|\mathcal{T}^\lambda f\|_{L^\infty(Q)}.\] Thus, for every narrow square \(Q\), we have \[\|\mathcal{T}^\lambda f\|_{L^\infty(Q)} \lesssim \Big( \sum_\tau \|\mathcal{T}^\lambda f_\tau\|_{L^\infty(Q)}^2 \Big)^{1/2}.\] Therefore, \[\label{narrow} \Big( \sum_{Q\subset \mathbf{N}} \mu_Q^2 \|\mathcal{T}^\lambda f\|_{L^\infty(Q)}^2 \Big)^{1/2} \lesssim \Big( \sum_{\tau} \sum_{Q\subset \mathbf{N}} \mu_Q^2 \|\mathcal{T}^\lambda f_\tau\|_{L^\infty(Q)}^2 \Big)^{1/2}.\tag{33}\] We decompose \(B_R\) into a collection of finitely overlapping curved tubes \(T'_{\tau,v}\) such that \[B_R \subset \bigcup_{v} T'_{\tau,v}.\] It follows that \[\begin{align} \Big( \sum_{Q\subset \mathbf{N}} \mu_Q^2 \|\mathcal{T}^\lambda f\|_{L^\infty(Q)}^2 \Big)^{1/2} &\lesssim \Big( \sum_{\tau} \sum_{Q\subset B_R} \mu_Q^2 \|\mathcal{T}^\lambda f_\tau\|_{L^\infty(Q)}^2 \Big)^{1/2} \\ &\lesssim \Big( \sum_{\tau} \sum_{T'_{\tau,v}\subset B_R} \sum_{Q\subset T'_{\tau,v}} \mu_Q^2 \|\mathcal{T}^\lambda f_\tau\|_{L^\infty(Q)}^2 \Big)^{1/2}. \end{align}\] Noting that \[\begin{align} \sum_{Q\subset T'_{\tau,v}} \mu_Q^2 \|\mathcal{T}^\lambda f_\tau\|_{L^\infty(Q)}^2 &\lesssim \sum_{Q'\subset T'_{\tau,v}} \sum_{Q\subset Q'} \mu_Q^2 \|\mathcal{T}^\lambda f_\tau\|_{L^\infty(Q)}^2, \end{align}\] we define \[\begin{align} \label{muq} \mu_{Q'}^2 := \sum_{Q\subset Q'} \mu_Q^2, \end{align}\tag{34}\] and observe that \[\sum_{Q\subset Q'} \mu_Q^2 \|\mathcal{T}^\lambda f_\tau\|_{L^\infty(Q)}^2 \lesssim \mu_{Q'}^2 \|\mathcal{T}^\lambda f_\tau\|_{L^\infty(Q')}^2.\] Hence, \[\sum_{Q\subset T'_{\tau,v}} \mu_Q^2 \|\mathcal{T}^\lambda f_\tau\|_{L^\infty(Q)}^2 \lesssim \sum_{Q'\subset T'_{\tau,v}} \mu_{Q'}^2 \|\mathcal{T}^\lambda f_\tau\|_{L^\infty(Q')}^2 .\] Applying 29 to the right-hand side yields \[\Big( \sum_{Q\subset T'_{\tau,v}} \mu_{Q}^2 \|\mathcal{T}^\lambda f_\tau\|_{L^\infty(Q)}^2 \Big)^{1/2} \lesssim \mathcal{F}(\lambda/K^2, R/K^2)\, K^{-1/2}\, \mathcal{W}' \|f_{\tau,v}\|_{L^2}.\] We claim that \[K^{-1/2}\,\mathcal{W}' \lesssim \mathcal{W}.\] Indeed, by the definitions of \(\mathcal{W}'\) and 34 , we have \[\mathcal{W}' = \sup_{1\le \rho \le R_1^{1/2}} \rho^{-1/2} \sup_{O'\in \mathcal{O}'} \Bigg( \sum_{\substack{I' \in \mathcal{I}' \\ I' \subset O'}} \Big( \sum_{Q \subset I'} \mu_{Q}^2 \Big)^{2} \Bigg)^{1/4}, \qquad R_1 = R/K^2.\] Each \(O' \in \mathcal{O}'\) at scale \(\rho\) corresponds to some \(O \in \mathcal{O}\) at scale \(K\rho\), and similarly \(I' \subset O'\) corresponds to \(I \subset O\). Thus, the factor \(\rho^{-1/2}\) in \(\mathcal{W}'\) becomes \((K\rho)^{-1/2}\) when viewed in \(\mathcal{W}\), which produces the extra \(K^{-1/2}\) and confirms the claim.

Summing over \(\tau\) and \(T'_{\tau,v}\) and using almost orthogonality, we obtain \[\label{eq32N} \Bigg( \sum_{Q \subset \mathbf{N}} \mu_Q^2 \,\|\mathcal{T}^\lambda f\|_{L^\infty(Q)}^2 \Bigg)^{1/2} \lesssim \, \mathcal{F}(\lambda/K^2, R/K^2)\, \mathcal{W} \,\|f\|_{L^2}.\tag{35}\]

Closing the induction. By combining the contributions from 32 (broad) and 35 (narrow), we obtain the recursive bound \[\label{eq:recursiveF} \mathcal{F}(\lambda,R) \le C_\varepsilon K^{C} R^{\varepsilon/2} + C\, \mathcal{F}(\lambda/K^2, R/K^2).\tag{36}\] Here \(K=R^\kappa\) with \(\kappa=\varepsilon^2\), and \(C\) is an absolute constant. Applying the induction hypothesis 24 at the smaller scale, \[\mathcal{F}(\lambda/K^2, R/K^2) \le \bar C_\varepsilon (R/K^2)^\varepsilon,\] turns 36 into \[\mathcal{F}(\lambda,R) \le C_\varepsilon K^{C} R^{\varepsilon/2} + C \bar C_\varepsilon (R/K^2)^{\varepsilon}.\] Hence, \[\mathcal{F}(\lambda,R) \le \big( C_\varepsilon K^{C} R^{-\varepsilon/2} + C \bar C_\varepsilon K^{-2\varepsilon} \big) R^{\varepsilon}.\] Since \(K=R^{\kappa}\) with \(\kappa=\varepsilon^2\) and \(R\ge N_\varepsilon\), choosing \(N_\varepsilon\) sufficiently large yields \[C_\varepsilon K^{C} R^{-\varepsilon/2} + C \bar C_\varepsilon K^{-2\varepsilon} \le \bar C_\varepsilon.\] This closes the induction and completes the proof of Propositions 11 and 10.

3.3 The case \(0 < \alpha < \frac{1}{2}\)↩︎

In this subsection, we prove Proposition 6 in the range \(0 < \alpha < \frac{1}{2}\). Indeed, we show that for every \(\varepsilon>0\), there exists \(C_\varepsilon>0\) such that \[\label{eql2e} \|\mathcal{T}^\lambda f\|_{L^q(B(0,R);\,H \, dx)} \le C_\varepsilon A_\alpha(H)^\frac{1}{q} R^\varepsilon \|f\|_{L^2}, \quad q\geq 2 .\tag{37}\] For convenience, set \(d\omega := H(x) \, dx\). Since, by the definition of \(\mathcal{T}^\lambda\), \[\|\mathcal{T}^\lambda f\|_{L^\infty(d\omega)} \leq C \|f\|_{L^2},\] it follows that \[\label{eq:dis} \|\mathcal{T}^\lambda f\|_{L^q(d\omega)}^q = q \int_0^{C \|f\|_{L^2}} t^{q-1} \,\omega\{ x : |\mathcal{T}^\lambda f(x)| > t \} \, dt .\tag{38}\] Note that the set \(\{x : |\mathcal{T}^\lambda f(x)| > t\}\) is contained in \[\{ ({\rm Re}\, \mathcal{T}^\lambda f)_+ > \tfrac{t}{4} \} \cup \{ ({\rm Re}\, \mathcal{T}^\lambda f)_- > \tfrac{t}{4} \} \cup \{ ({\rm Im}\, \mathcal{T}^\lambda f)_+ > \tfrac{t}{4} \} \cup \{ ({\rm Im}\, \mathcal{T}^\lambda f)_- > \tfrac{t}{4} \} .\] We will consider the first set as an example, since the others can be handled in the same manner. Let \[G := \{ ({\rm Re}\, \mathcal{T}^\lambda f)_+ > \tfrac{t}{4} \} .\] Thus, \[\label{eq32G} \frac{t}{4} \, \omega(G) \leq \int_G ({\rm Re}\, \mathcal{T}^\lambda f) \, d\omega \leq \left| \int \mathcal{T}^\lambda f(x) \chi_G(x) \, d\omega(x) \right|,\tag{39}\] where \(\chi_G\) denotes the characteristic function of \(G\). It follows that \[\begin{gather} \label{eq:bound95integral} \left| \int \mathcal{T}^\lambda f(x) \chi_G(x) \, d\omega(x) \right| = \left| \int f(\xi) \int e^{i \phi^\lambda(x,\xi)} a^\lambda(x,\xi) \chi_G(x) \, d\omega(x) \, d\xi \right| \\ \leq \left\| \int e^{i \phi^\lambda(x,\xi)} a^\lambda(x,\xi) \chi_G(x) \, d\omega(x) \right\|_{L^2} \|f\|_{L^2}. \end{gather}\tag{40}\] It suffices to consider \[\label{eq:dual} \left\| \int e^{i \phi^\lambda(x,\xi)} a^\lambda(x,\xi) \chi_G(x) \, d\omega(x) \right\|_{L^2}^2 .\tag{41}\] Expanding 41 , we obtain \[\label{eq:expan} \int \int \int e^{i (\phi^\lambda(x,\xi) - \phi^\lambda(y,\xi))} a^\lambda(x,\xi) a^\lambda(y,\xi) \, d\xi \, \chi_G(x) \, d\omega(x) \, \chi_G(y) \, d\omega(y) .\tag{42}\] By a standard stationary phase argument (see, e.g., [10]), \[\label{kernel32bd} \left| \int e^{i (\phi^\lambda(x,\xi) - \phi^\lambda(y,\xi))} a^\lambda(x,\xi) a^\lambda(y,\xi) \, d\xi \right| \lesssim \frac{1}{(1 + |x - y|)^{1/2}} .\tag{43}\] Therefore, 42 is controlled by \[\int \int \frac{1}{(1 + |x - y|)^{1/2}} \chi_G(x) \, d\omega(x) \, \chi_G(y) \, d\omega(y) .\] We claim that, uniformly in \(y\), \[\label{claima} \int \frac{1}{(1 + |x - y|)^{1/2}} \chi_G(x) \, d\omega(x) \lesssim A_\alpha(H) .\tag{44}\] Combining 39 –43 with the claim 44 , we obtain \[t^2 \omega(G)^2 \lesssim A_\alpha(H) \|f\|_{L^2}^2 \omega(G),\] which implies \[\|\mathcal{T}^\lambda f\|_{L^q(d\omega)}^q \lesssim A_\alpha(H) \|f\|_{L^2}^2 \int_0^{C \|f\|_{L^2}} t^{q - 3} \, dt \lesssim A_\alpha(H) \|f\|_{L^2}^q ,\] for \(q>2\). The case \(q=2\) also follows since we allow an \(R^\varepsilon\) loss. It remains to prove our claim 44 .

By our assumption on \(H\), for each ball \(B(x,r)\) with \(r \geq 1\), \[\label{grow} \omega(B(x,r)) \leq A_\alpha(H)\, r^\alpha .\tag{45}\] Using 45 , we obtain \[\begin{align} \int \frac{1}{(1 + |x - y|)^{1/2}} \chi_G(x) \, d\omega(x) &\leq \int_{B(y,1)} \frac{1}{(1 + |x - y|)^{1/2}} \chi_G(x) \, d\omega(x) \\ &\quad + \sum_{k \in \mathbb{Z}_+} \int_{B(y,2^k) \setminus B(y,2^{k-1})} \frac{1}{(1 + |x - y|)^{1/2}} \chi_G(x) \, d\omega(x) . \end{align}\] The first term is bounded by a constant multiple of \(A_\alpha(H)\). For each \(k \in \mathbb{Z}_+\), \[\int_{B(y,2^k) \setminus B(y,2^{k-1})} \frac{1}{(1 + |x - y|)^{1/2}} \chi_G(x) \, d\omega(x) \lesssim 2^{-k/2} \,\omega(B(y,2^k)) \lesssim A_\alpha(H)\, 2^{k (\alpha - 1/2)} .\] Since \(0 < \alpha < \tfrac{1}{2}\), the series \(\sum_{k=1}^\infty 2^{k (\alpha - 1/2)}\) converges. Therefore, we obtain 44 and complete the proof of 37 in this range of \(\alpha\).

3.4 The case \(\frac{1}{2}\leq\alpha<1\)↩︎

Finally, we prove Proposition 6 for \(\frac{1}{2}\leq\alpha<1\). This case is a direct consequence of Proposition 7, via Hölder’s inequality. Indeed, \[\begin{align} \int_{B(0,R)} |\mathcal{T}^\lambda f(x)|^2 H(x)\,dx &\leq \Big(\int_{B(0,R)} |\mathcal{T}^\lambda f(x)|^{4\alpha} H(x)\,dx \Big)^{\frac{1}{2\alpha}} \Big(\int_{B(0,R)} H(x)\,dx\Big)^{1-\frac{1}{2\alpha}}\\ &\lesssim \big(R^\varepsilon \|f\|_{L^2}^{4\alpha}\big)^{\frac{1}{2\alpha}}\, R^{\alpha-\frac{1}{2}}\\ &\lesssim R^{\alpha-\frac{1}{2}+\varepsilon}\,\|f\|_{L^2}^2. \end{align}\]

4 Proof of Proposition 7↩︎

The proof is analogous to that of Theorem 2.1 in [7], except that our estimates also extend to the endpoint since we allow \(\lambda^\varepsilon\) losses. For any Lebesgue measurable function \(F:\mathbb{R}^2\to[0,\infty)\), define \[M_\alpha F:=\Bigg(\sup_H \frac{1}{A_\alpha(H)}\int F(x)^\alpha H(x)\,dx\Bigg)^{1/\alpha},\] where the supremum is taken over all nonzero weights \(H\) on \(\mathbb{R}^2\) of dimension \(\alpha\). The following Hölder-type inequality is a key ingredient in the proof of Proposition 7.

We define \(\mathfrak Q(\alpha,p)\) to be the infimum of all exponents \(q\) satisfying the following property: for every \(\varepsilon>0\), there exists \(C_\varepsilon>0\) such that \[\label{qpestimate} \int_{B(0,R)} |\mathcal{T}^\lambda f(x)|^q H(x)\,dx \le C_\varepsilon\,A_\alpha(H)\,R^{\varepsilon}\,\|f\|_{L^p}^q .\tag{46}\] Equivalently, \[\mathfrak Q(\alpha,p) := \inf\Big\{ q : \forall\,\varepsilon>0 ,\exists\,C_\varepsilon>0 \text{ such that \eqref{qpestimate} holds}\Big\}.\] We first establish a monotonicity property of \(\mathfrak Q(\alpha,p)\) with respect to \(\alpha\): for \(0<\beta<\alpha\le 2\), \[\label{rela} \frac{\mathfrak Q(\alpha,p)}{\alpha}\le \frac{\mathfrak Q(\beta,p)}{\beta}.\tag{47}\] Indeed, fix \(\varepsilon>0\) and suppose \(q>\mathfrak Q(\beta,p)\). By the definition of \(\mathfrak Q(\beta,p)\), there exists \(C_\varepsilon>0\) such that \[\int_{B(0,R)} |\mathcal{T}^\lambda f(x)|^q H(x)\,dx \le C_\varepsilon\,R^\varepsilon\,A_\beta(H)\,\|f\|_{L^p}^q ,\] for all \(f\in L^p\) and all weights \(H\) of dimension \(\beta\). Set \[F = \chi_{B(0,R)}\,|\mathcal{T}^\lambda f|^{q/\beta}.\] Then it follows that \[M_\beta(F)\le (C_\varepsilon R^\varepsilon \|f\|_{L^p}^q)^{1/\beta}.\] Applying Theorem 12, we obtain \[M_\alpha(F)\le (C_\varepsilon R^\varepsilon \|f\|_{L^p}^q)^{1/\beta}.\] Hence, \[\bigg(\frac{1}{A_\alpha(H)}\int_{B(0,R)} |\mathcal{T}^\lambda f(x)|^{\frac{\alpha q}{\beta}} H(x)\,dx\bigg)^{1/\alpha} \le (C_\varepsilon R^\varepsilon \|f\|_{L^p}^q)^{1/\beta},\] for all \(f\in L^p\) and all weights \(H\) of dimension \(\alpha\). By the definition of \(\mathfrak Q(\alpha,p)\), this implies \[\frac{\alpha q}{\beta}\ge \mathfrak Q(\alpha,p).\] Since this holds for all \(q>\mathfrak Q(\beta,p)\), we conclude that \[\frac{\mathfrak Q(\beta,p)}{\beta}\ge \frac{\mathfrak Q(\alpha,p)}{\alpha},\] which proves 47 .

Now we are ready to prove Proposition 7. The case \(0<\alpha\le \tfrac{1}{2}\) was handled in Proposition 6. See the proof of 37 . As a direct consequence of 47 , for \(\tfrac{1}{2}<\alpha\le 1\) and \(q>4\alpha\), we have \[\int_{B(0,R)} |\mathcal{T}^\lambda f(x)|^q H(x)\,dx \le C_\varepsilon\, R^\varepsilon\, A_\alpha(H)\, \|f\|_{L^2}^q .\] Indeed, by 37 we have \[\frac{\mathfrak Q(\beta,2)}{\beta}=\frac{2}{\beta},\qquad 0<\beta<\tfrac12.\] Thus, by 47 , \[\frac{\mathfrak Q(\alpha,2)}{\alpha}\le \inf_{\beta\in(0,\tfrac12)}\frac{2}{\beta}=4,\] and the desired bound follows.

Finally, we consider the case \(1<\alpha\le 2\). By Proposition 10, we have \[\int_{B(0,R)} |\mathcal{T}^\lambda f(x)|^2 H(x)\,dx \le C_\varepsilon\, R^\varepsilon\, A_\alpha(H)\, R^{\tfrac{\alpha}{2}} \|f\|_{L^2}^2.\] Define \[\bar H(x):=\|f\|_{L^2}^{-2}\,|\mathcal{T}^\lambda f(x)|^2\,H(x).\] Then \(\bar H\) is a weight of dimension \(\bar\alpha:=\tfrac{\alpha}{2}+\varepsilon\), and moreover \[A_{\bar\alpha}(\bar H)\lesssim A_\alpha(H).\] Since \(1<\alpha<2\), we have \(1/2<\bar\alpha<1\) provided \(\varepsilon>0\) is sufficiently small. Thus, by the result for the range \(1/2<\bar\alpha<1\), \[\mathfrak Q(\bar\alpha,2)\le 4\bar\alpha.\] Consequently, for all \(g\in L^2\) and all \(q'>4\bar\alpha=2\alpha+4\varepsilon\), \[\int_{B(0,R)} |\mathcal{T}^\lambda g(x)|^{q'} \bar H(x)\,dx \le C_\varepsilon\, R^\varepsilon\, A_{\bar\alpha}(\bar H)\, \|g\|_{L^2}^{q'}.\] Replacing \(\bar H\) by \(\|f\|_{L^2}^{-2}|\mathcal{T}^\lambda f|^2 H\), setting \(g=f\), and choosing \(\varepsilon>0\) sufficiently small, we obtain \[\int_{B(0,R)} |\mathcal{T}^\lambda f(x)|^q H(x)\,dx \le C_\varepsilon\, R^\varepsilon\, A_\alpha(H)\, \|f\|_{L^2}^q,\] for all \(q>2\alpha+2\). This completes the proof.

5 Proof of Theorem 5↩︎

In this section we prove Theorem 5 and explain its connection to the Mizohata–Takeuchi conjecture. As in the proof of Theorem 3, Theorem 5 reduces to a bound for \(\mathcal{T}^\lambda f\). Specifically, we estimate \(\mathcal{T}^\lambda f\) on suitably rescaled tubular neighborhoods of a curve \(\gamma\) tangent to \(\{\Gamma_{\xi_\theta,v}\}_{\theta,v}\) to order at most \(1\). See Definition 4 below. We denote by \(T_{\lambda^{s}}(\gamma^\lambda)\) the \(\lambda^{s}\)-neighborhood of the rescaled curve \(\gamma^\lambda(\cdot) := \lambda \gamma(\cdot/\lambda)\). Theorem 5 is a consequence of the following proposition.

Our argument for proving Proposition 13 combines the wave-packet decomposition from the previous sections with a simple geometric observation. We begin with a geometric lemma. In \(\mathbb{R}^2\), the \(\delta\)-neighborhoods of a line and a parabola cannot overlap too much. We generalize this observation to the Riemannian setting.

5.1 A geometric lemma↩︎

Let \(\gamma\subset \mathbb{R}^2\) be a curve segment parametrized by arc length with nonvanishing curvature, that is, \[|\gamma''(s)|\geq c,\quad s\in (0,\varepsilon_0),\] for some fixed \(0<\varepsilon_0\ll 1\) and constant \(c>0\).

Let \(0<\delta\ll 1\), and write \(\gamma_\delta\) for the \(\delta\)-neighborhood of \(\gamma\). Similarly, for a line \(\ell\), write \(\ell_\delta\) for the \(\delta\)-neighborhood of \(\ell\). Whenever \(\gamma_\delta\) intersects \(\ell_\delta\), one has \[\Big|\ell_\delta\cap \gamma_\delta\Big|\le C\,\delta^{3/2},\] with equality in the extremal case when \(\ell\) is tangent to \(\gamma\). This fact extends to Riemannian manifolds, with the line \(\ell\) replaced by a geodesic and \(\gamma\) a curve with nonvanishing geodesic curvature.

To prove Lemma 8 we first record some basic geometric lemmas.

5.2 Proof of Proposition 13 and its connection to the Mizohata–Takeuchi conjecture↩︎

Before proving Proposition 13, we provide additional motivation by discussing the following conjecture of Mizohata–Takeuchi in the plane.

To relate the Mizohata–Takeuchi conjecture to Theorem 5, we propose the following variable-coefficient analogue.

To see the consequence of Conjecture 15 for our problem, we introduce the following notion.

After rescaling, we write \(T_{\lambda^{s}}(\gamma^\lambda)\) for the \(\lambda^s\)-neighborhood of \(\gamma^\lambda\). Assuming Conjecture 15, we obtain \[\int_{T_{\lambda^{s}}(\gamma^\lambda)} |\mathcal{T}^\lambda f(x)|^2\,dx \le C_\varepsilon \lambda^\varepsilon \sup_{\theta,v}\big|\Gamma_{\xi_\theta,v}^\lambda \cap T_{\lambda^{s}}(\gamma^\lambda)\big|\,\|f\|_{L^2}^2,\] where \(|\cdot|\) denotes arc length.

Since \(\gamma\) is tangent to the family \(\{\Gamma_{\xi_\theta,v}\}_{\theta,v}\) to order at most \(k\), we have uniformly in \((\theta,v)\) that \[\big|\Gamma_{\xi_\theta,v}^\lambda \cap T_{\lambda^{s}}(\gamma^\lambda)\big| \le \lambda^{\frac{k+s}{k+1}}.\] Thus, for \(0\le s\le \tfrac12\), \[\label{eq:mt1} \int_{T_{\lambda^{s}}(\gamma^\lambda)} |\mathcal{T}^\lambda f(x)|^2\,dx \le C_\varepsilon\,\lambda^{\frac{k+s}{k+1}+\varepsilon}\,\|f\|_{L^2}^2.\tag{53}\] In particular, if \(k=1\), then \[\label{eq:mt2} \int_{T_{\lambda^{s}}(\gamma^\lambda)} |\mathcal{T}^\lambda f(x)|^2\,dx \le C_\varepsilon\,\lambda^{\frac{1+s}{2}+\varepsilon}\,\|f\|_{L^2}^2.\tag{54}\]

We now prove that for \(0 \le s \le \tfrac12\) the estimate predicted by the variable-coefficient Mizohata–Takeuchi conjecture holds. We will show in the final section that, when \(k=1\), it is sharp for \(\tfrac13 \le s \le \tfrac12\).

Proposition 13 corresponds to the case \(k=1\) in the above proposition, and Theorem 5 is a direct consequence of Proposition 13, 11 , and Lemma 8. Moreover, Proposition 16 may be viewed as a variable-coefficient generalization of Corollary 3.4 in [21], which corresponds to the case \(k=0\).

For \(s<\tfrac{1}{3}\) and \(k=1\), the Mizohata–Takeuchi conjecture yields a nonsharp estimate in 53 . Indeed, if we localize \(\mathcal{T}^\lambda f\) to a ball of radius \(\lambda^{2s}\) and perform the wave-packet decomposition at that scale, then each wave packet \(T_{\theta,v}\) can intersect \(T_{\lambda^{s}}(\gamma^\lambda)\) in a segment of length \(\lambda^{\frac{1+s}{2}}\). To obtain an improvement from the Mizohata–Takeuchi conjecture, the intersection length \(\lambda^{(1+s)/2}\) must be dominated by the packet scale \(\lambda^{2s}\), i.e. \[\frac{1+s}{2}<2s,\] which is equivalent to \(s>\tfrac{1}{3}\).

For \(0<s<\tfrac{1}{3}\), the sharp bound instead follows directly from the theorem of Pan and Sogge [22], which quantifies the optimal gain for oscillatory integral operators with at most fold singularities. Similarly, this is exactly how Hu [23] proved his version of Theorem 4.

6 Sharpness of main results↩︎

6.1 Sharpness of Theorem 3↩︎

The sharpness of Theorem 3 for large \(p\) on \(S^2\) follows from Theorem 1.5 in [2]. In this regime, the exponent \(\delta(\alpha,p)=\tfrac12-\tfrac{\alpha}{p}\) captures the behavior of zonal spherical harmonics concentrating at a typical point in the support of an \(\alpha\)-dimensional measure, which shows that Theorem 3 is optimal there.

It remains to establish sharpness for the smaller ranges, namely \(\alpha\in(\tfrac12,1]\), \(p\in[2,4\alpha]\), and \(\alpha\in(1,2]\), \(p\in[2,2\alpha+2]\). In both cases, the bounds are saturated by highest-weight spherical harmonics.

Recall that the highest-weight spherical harmonic \(\mathcal{H}_k\) is the restriction to \(S^2=\{x\in\mathbb{R}^3:|x|=1\}\) of the harmonic polynomial \(k^{1/4}(x_1+i x_2)^k\). A direct computation shows that \(\mathcal{H}_k\) has \(L^2\) mass essentially concentrated in a \(k^{-1/2}\)-tubular neighborhood of the equator \(\gamma_0\), where \(|\mathcal{H}_k|\approx k^{1/4}\).

In particular, when \(\alpha\in(\tfrac12,1]\) and \(p\in[2,4\alpha]\), we have \(\delta(\alpha,p)=\tfrac14\). If \(\mu\) is any \(\alpha\)-dimensional probability measure supported on \(\gamma_0\), then \[\|\mathcal{H}_k\|_{L^p(M;\,d\mu)}\approx k^{1/4},\qquad p\ge2,\] so the bound in Theorem 3 is sharp up to a \(k^\varepsilon\) loss in this range.

The situation is a bit more involved in the case \(\alpha \in (1,2]\), \(p \in [2, 2\alpha+2]\), for which \(\delta(\alpha, p) = \frac{1}{4} - \frac{\alpha - 1}{2p}\). We shall prove the following.

6.2 Sharpness of Theorem 5↩︎

We use a construction due to Tacy [26] to prove the sharpness of Theorem 5. We first recall:

Now let \(\gamma\) have nonvanishing geodesic curvature and consider its \(\lambda^{-1+s}\)–tubular neighborhood with \(\tfrac13\le s\le \tfrac12\). The largest “rectangular’’ region of the form \(\lambda^{-1+2\beta}\times\lambda^{-1+\beta}\) that fits inside this tube has dimensions \(\lambda^{-1+\frac{1+s}{2}}\times \lambda^{-1+\frac{1+s}{4}}\) (see Figure 5). Hence, \[\lambda^{1-s}\int_{T_{\lambda^{-1+s}}(\gamma)} |e_\lambda|^2\,dx \gtrsim \lambda^{1-s}\cdot \lambda^{\,1-\frac{1+s}{4}}\cdot \lambda^{-1+\frac{1+s}{2}}\cdot \lambda^{-1+\frac{1+s}{4}} = \lambda^{\frac{1-s}{2}}.\] Therefore, \[\|e_\lambda\|_{L^2_{\mathrm{avg}}(T_{\lambda^{-1+s}}(\gamma))} \gtrsim \lambda^{\frac{1-s}{4}},\] which proves the sharpness of Theorem 5 for \(\tfrac13\le s\le \tfrac12\). When \(s=\tfrac13\), the lower bound is \(\lambda^{1/6}\), matching the restriction estimate 8 for \(\gamma\). Consequently, 10 is also sharp for \(0\le s\le \tfrac13\).

References↩︎

---

1. Here, “quadratic” means: for \((\mathbf{a},\mathbf{b})\in\mathbb{N}^2\times\mathbb{N}\),

   • \(\partial_\xi^{\mathbf{b}} h(0)=0\) and \(\partial_x^{\mathbf{a}}\partial_\xi^{\mathbf{b}}\mathcal{E}(x,0)=0\) for \(\mathbf{b}\le 1\);

   • \(\partial_x^{\mathbf{a}}\partial_\xi^{\mathbf{b}}\mathcal{E}(0,\xi)=0\) for \(|\mathbf{a}|\le 1\).

   ↩︎