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Yakun Xi

Publications and source records attributed to Yakun Xi.

33 records · Page 2Linked to original sources
arXiv2023

A two term Kuznecov sum formula

The Kuznecov sum formula, proved by Zelditch in the Riemannian setting, is an asymptotic sum formula $$N(λ) := \sum_{λ_j \leq λ} \left| \int_H e_j \, dV_H \right|^2 = C_{H,M} λ^{\operatorname{codim} H} + O(λ^{\operatorname{codim} H - 1})$$ where $e_j$ constitute a Hilbert basis of Laplace-Beltrami eigenfunctions on a Riemannian manifold $M$ with $Δ_g e_j = -λ_j^2 e_j$, and $H$ is an embedded submanifold. We show for some suitable definition of `$\sim$', $$ N(λ) \sim C_{H,M} λ^{\operatorname{codim} H} + Q(λ) λ^{\operatorname{codim} H - 1} + o(λ^{\operatorname{codim} H - 1}) $$ where $Q$ is a bounded oscillating term and is expressed in terms of the geodesics which depart and arrive $H$ in the normal directions. In work by Canzani, Galkowski, and Toth, they establish (as a corollary to a stronger result involving defect measures) that if the set of recurrent directions of geodesics normal to $H$ has measure zero, then we obtain improved bounds on the individual terms in the sum -- the period integrals. We are able to give a dynamical condition such that $Q$ is uniformly continuous and `$\sim$' can be replaced with `$=$'. This implies improved bounds on period integrals, and this condition is weaker than the recurrent directions having measure zero. Moreover, our result implies improved bounds for period integrals if there is no $L^1$ measure on $SN^*H$ that is invariant under the first return map. This generalizes a theorem of Sogge and Zelditch and of Galkowski.

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arXiv2023

Fourier coefficients of restrictions of eigenfunctions

Let $\{e_j\}$ be an orthonormal basis of Laplace eigenfunctions of a compact Riemannian manifold $(M,g)$. Let $H \subset M$ be a submanifold and let $\{ψ_k\}$ be an orthonormal basis of Laplace eigenfunctions of $H$ with the induced metric. We obtain joint asymptotics for the Fourier coefficients \[ \langle γ_H e_j, ψ_k \rangle_{L^2(H)} = \int_H e_j \overline ψ_k \, dV_H, \] of restrictions $γ_H e_j$ of $e_j$ to $H$. In particular, we obtain asymptotics for the sums of the norm-squares of the Fourier coefficients over the joint spectrum $\{(μ_k, λ_j)\}_{j,k - 0}^{\infty}$ of the (square roots of the) Laplacian $Δ_M$ on $M$ and the Laplacian $Δ_H$ on $H$ in a family of suitably `thick' regions in $\mathbb R^2$. Thick regions include (1) the truncated cone $μ_k/λ_j \in [a,b] \subset (0,1)$ and $λ_j \leq λ$, and (2) the slowly thickening strip $|μ_k - cλ_j| \leq w(λ)$ and $λ_j \leq λ$, where $w(λ)$ is monotonic and $1 \ll w(λ) \lesssim λ^{1 - 1/n}$. Key tools for obtaining these asymptotics include the composition calculus of Fourier integral operators and a new multidimensional Tauberian theorem.

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arXiv2023

Microlocal decoupling inequalities and the distance problem on Riemannian manifolds

We study the generalization of the Falconer distance problem to the Riemannian setting. In particular, we extend the result of Guth-Iosevich-Ou-Wang for the distance set in the plane to general Riemannian surfaces. Key new ingredients include a family of refined microlocal decoupling inequalities, which are related to the work of Beltran-Hickman-Sogge on Wolff-type inequalities, and an analog of Orponen's radial projection lemma which has proved quite useful in recent work on distance sets.

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arXiv2023

Geodesic bi-angles and Fourier coefficients of restrictions of eigenfunctions

This article concerns joint asymptotics of Fourier coefficients of restrictions of Laplace eigenfunctions $ϕ_j$ of a compact Riemannian manifold to a submanifold $H \subset M$. We fix a number $c \in (0,1)$ and study the asymptotics of the thin sums, $$ N^{c} _{ε, H }(λ): = \sum_{j, λ_j \leq λ} \sum_{k: |μ_k - c λ_j | < ε} \left| \int_{H} ϕ_j \overline{ψ_k}dV_H \right|^2 $$ where $\{λ_j\}$ are the eigenvalues of $\sqrt{-Δ}_M,$ and $\{(μ_k, ψ_k)\}$ are the eigenvalues, resp. eigenfunctions, of $\sqrt{-Δ}_H$. The inner sums represent the `jumps' of $ N^{c} _{ε, H }(λ)$ and reflect the geometry of geodesic c-bi-angles with one leg on $H$ and a second leg on $M$ with the same endpoints and compatible initial tangent vectors $ξ\in S^c_H M, π_H ξ\in B^* H$, where $π_H ξ$ is the orthogonal projection of $ξ$ to $H$. A c-bi-angle occurs when $\frac{|π_H ξ|}{|ξ|} = c$. Smoothed sums in $μ_k$ are also studied, and give sharp estimates on the jumps. The jumps themselves may jump as $ε$ varies, at certain values of $ε$ related to periodicities in the c-bi-angle geometry. Subspheres of spheres and certain subtori of tori illustrate these jumps. The results refine those of the previous article (arXiv:2011.11571) where the inner sums run over $k: | \frac{μ_k}{λ_j} - c| \leq ε$ and where geodesic bi-angles do not play a role.

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arXiv2019

$\ell^2$ Decoupling in $\mathbb R^2$ for curves with Vanishing Curvature

We expand the class of curves $(φ_1(t),φ_2(t)),\ t\in[0,1]$ for which the $\ell^2$ decoupling conjecture holds for $2\leq p\leq 6$. Our class of curves includes all real-analytic regular curves with isolated points of vanishing curvature and all curves of the form $(t,t^{1+ν})$ for $ν\in (0,\infty)$.

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arXiv2018

Improved Generalized Periods Estimates Over Curves on Riemannian Surfaces with Nonpositive Curvature

We show that on compact Riemann surfaces of nonpositive curvature, the generalized periods, i.e. the $ν$-th order Fourier coefficients of eigenfunctions $e_λ$ over a closed smooth curve $γ$ which satisfies a natural curvature condition, go to 0 at the rate of $O((\logλ)^{-1/2})$, if $0<|ν|/λ<1-δ$, for any fixed $0<δ<1$. Our result implies, for instance, the generalized periods over geodesic circles on any surfaces with nonpositive curvature would converge to zero at the rate of $O((\logλ)^{-1/2})$. A direct corollary of our results and the QER theorem of Toth and Zelditch is that for a geodesic circle $γ$ on a compact hyperbolic surface, the restriction $e_{λ_j}|_γ$ of an orthonormal basis $\{e_{λ_j}\}$ has a full density subsequence that goes to zero in weak-$L^2(γ)$. One key step of our proof is a microlocal decomposition of the measure over $γ$ into tangential and transversal parts.

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arXiv2018

Improved Generalized Periods estimates on Riemannian Surfaces with Nonpositive Curvature

We show that on compact Riemann surfaces of negative curvature, the generalized periods, i.e. the $ν$-th order Fourier coefficient of eigenfunctions $e_λ$ over a period geodesic $γ$ goes to 0 at the rate of $O((\logλ)^{-1/2})$, if $0<ν<c_0λ$, given any $0<c_0<1$. No such result is possible for the sphere $S^2$ or the flat torus $\mathbb T^2$. Combined with the quantum ergodic restriction result of Toth and Zelditch, our results imply that for a generic closed geodesic $γ$ on a compact hyperbolic surface, the restriction $e_{λ_j}|_γ$ of an orthonormal basis $\{e_{λ_j}\}$ has a full density subsequence that goes to zero weakly in $L^2(γ)$. Our proof consists of a further refinement of a recent paper by Sogge, Xi and Zhang on the geodesic period integrals ($ν=0$), which featured the Gauss-Bonnet Theorem as a key quantitative tool to avoid geodesic rectangles on the universal cover of $M$. In contrast, we shall employ the Gauss-Bonnet Theorem to quantitatively avoid geodesic parallelograms. The use of Gauss-Bonnet also enables us to weaken our curvature condition, by allowing the curvature to vanish at an averaged rate of finite type.

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arXiv2018

An Endpoint Version of Uniform Sobolev Inequalities

We prove an endpoint version of the uniform Sobolev inequalities in Kenig-Ruiz-Sogge [8]. It was known that strong type inequalities no longer hold at the endpoints; however, we show that restricted weak type inequalities hold there, which imply the earlier classical result by real interpolation. The key ingredient in our proof is a type of interpolation first introduced by Bourgain [2]. We also prove restricted weak type Stein-Tomas restriction inequalities on some parts of the boundary of a pentagon, which completely characterizes the range of exponents for which the inequalities hold.

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arXiv2018

Inner product of eigenfunctions over curves and generalized periods for compact Riemannian surfaces

We show that for a smooth closed curve $γ$ on a compact Riemannian surface without boundary, the inner product of two eigenfunctions $e_λ$ and $e_μ$ restricted to $γ$, $|\int e_λ\overline{e_μ}\,ds|$, is bounded by $\min\{λ^\frac12,μ^\frac12\}$. Furthermore, given $0<c<1$, if $0<μ<cλ$, we prove that $\int e_λ\overline{e_μ}\,ds=O(μ^\frac14)$, which is sharp on the sphere $S^2$. These bounds unify the period integral estimates and the $L^2$-restriction estimates in an explicit way. Using a similar argument, we also show that the $ν$-th order Fourier coefficient of $e_λ$ over $γ$ is uniformly bounded if $0<ν<cλ$, which generalizes a result of Reznikov for compact hyperbolic surfaces, and is sharp on both $S^2$ and the flat torus $\mathbb T^2$. Moreover, we show that the analogs of our results also hold in higher dimensions for the inner product of eigenfunctions over hypersurfaces.

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arXiv2017

On Instability of the Nikodym Maximal Function bounds over Riemannian Manifolds

We show that, for odd $d$, the $L^{\frac{d+2}2}$ bounds of Sogge and Xi for the Nikodym maximal function over manifolds of constant sectional curvature, are unstable with respect to metric perturbation, in the spirit of the work of Sogge and Minicozzi. A direct consequence is the instability of the bounds for the corresponding oscillatory integral operator. Furthermore, we extend our construction to show that the same phenomenon appears for any $d$-dimensional Riemannian manifold with a local totally geodesic submanifold of dimension $\lceil{\frac{d+1}2}\rceil$ if $d\ge 3$. In contrast, Sogge's $L^\frac73$ bound for the Nikodym maximal function on 3-dimensional variably curved manifolds is stable with respect to metric perturbation.

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arXiv2017

Geodesic period integrals of eigenfunctions on Riemannian surfaces and the Gauss-Bonnet Theorem

We use the Gauss-Bonnet theorem and the triangle comparison theorems of Rauch and Toponogov to show that on compact Riemann surfaces of negative curvature period integrals of eigenfunctions $e_λ$ over geodesics go to zero at the rate of $O((\logλ)^{-1/2})$ if $λ$ are their frequencies. As discussed in \cite{CSPer}, no such result is possible in the constant curvature case if the curvature is $\ge0$. Notwithstanding, we also show that these bounds for period integrals are valid provided that integrals of the curvature over all geodesic balls of radius $r\le 1$ are pinched from above by $-δr^N$ for some fixed $N$ and $δ>0$. This allows, for instance, the curvature to be nonpositive and to vanish of finite order at a finite number of isolated points. Naturally, the above results also hold for the appropriate type of quasi-modes.

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arXiv2016

Improved critical eigenfunction restriction estimates on Riemannian surfaces with nonpositive curvature

We show that one can obtain improved $L^4$ geodesic restriction estimates for eigenfunctions on compact Riemannian surfaces with nonpositive curvature. We achieve this by adapting Sogge's strategy in proving improved critical $L^p$ estimates. We first combine the improved $L^2$ restriction estimate of Blair and Sogge and the classical improved $L^\infty$ estimate of Bérard to obtain an improved weak-type $L^4$ restriction estimate. We then upgrade this weak estimate to a strong one by using the improved Lorentz space estimate of Bak and Seeger. This estimate improves the $L^4$ restriction estimate of Burq, Gérard and Tzvetkov and Hu by a power of $(\log\logλ)^{-1}$. Moreover, in the case of compact hyperbolic surfaces, we obtain further improvements in terms of $(\logλ)^{-1}$ by applying the ideas from recent works of Chen, Sogge and Blair, Sogge. We are able to compute various constants that appeared in the work of Chen and Sogge explicitly, by proving detailed oscillatory integral estimates and lifting calculations to the universal cover $\mathbb H^2$.

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arXiv2015

On Kakeya-Nikodym type maximal inequalities

We show that for any dimension $d\ge3$, one can obtain Wolff's $L^{(d+2)/2}$ bound on Kakeya-Nikodym maximal function in $\mathbb R^d$ for $d\ge3$ without the induction on scales argument. The key ingredient is to reduce to a 2-dimensional $L^2$ estimate with an auxiliary maximal function. We also prove that the same $L^{(d+2)/2}$ bound holds for Nikodym maximal function for any manifold $(M^d,g)$ with constant curvature, which generalizes Sogge's results for $d=3$ to any $d\ge3$. As in the 3-dimensional case, we can handle manifolds of constant curvature due to the fact that, in this case, two intersecting geodesics uniquely determine a 2-dimensional totally geodesic submanifold, which allows the use of the auxiliary maximal function.

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