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Jaehyeon Ryu

Publications and source records attributed to Jaehyeon Ryu.

16 recordsLinked to original sources
arXiv2026

The Endpoint Eigenfunction Bound for the Hermite Operator in Two Dimensions

In this paper, we establish the optimal \(L^2(\mathbb{R}^2)\to L^{10/3}(\mathbb{R}^2)\) endpoint estimate for the spectral projection operator associated with the Hermite operator on \(\mathbb{R}^2\). This completes a long-standing line of inquiry into sharp eigenfunction bounds for the Hermite operator, developed through the works of Thangavelu, Karadzhov, Koch--Tataru, and others. In higher dimensions \(d\ge 3\), the corresponding endpoint estimates \[ L^2(\mathbb{R}^d)\to L^{\frac{2(d+3)}{d+1}}(\mathbb{R}^d) \] were recently established by the present authors. Together with these earlier results, the present work fully resolves the problem of optimal \(L^2\to L^q\) eigenfunction bounds for Hermite spectral projections in all dimensions. Although our approach builds on our previous method, we overcome its limitations through a multiscale decomposition in space and time relative to the degeneracy set, combined with an asymmetric refinement on the input side and almost orthogonality.

math.CA↗
arXiv2026

On almost everywhere convergence of Bochner--Riesz means below the critical index

In this paper, we study the almost everywhere convergence problem for the Bochner--Riesz means $S_t^δf$ for $f\in L^p(\mathbb R^d)$ in the subcritical range \[ 0\le δ< δ(d,p):=d\Big(\frac12-\frac1p\Big)-\frac12, \qquad \frac{2d}{d-1} 0$, even as a tempered distribution. Nevertheless, the family $\{S_t^δf\}_{t>0}$ can still be interpreted as a distribution on $\mathbb R^d\times(0,\infty)$. We introduce an admissible class $\mathcal C_{p,δ}\subset L^p(\mathbb R^d)$ on which this distribution has sufficient regularity in the $t$ variable to formulate the almost everywhere convergence problem. We establish three results concerning this class. First, we show that $\mathcal C_{p,δ}\neq L^p(\mathbb R^d)$, and thus the almost everywhere convergence problem cannot be formulated for all $L^p$ functions below the critical index. Second, we show that this admissible class is nevertheless large in the sense that, for every $f\in L^p(\mathbb R^d)$, the pullback $f_V=f(V^{-1}\cdot)$ is admissible for Haar-a.e. volume-preserving upper triangular matrix $V$ with positive diagonal entries. Finally, we construct an $f\in \mathcal C_{p,δ}$ for which $S_t^δf$ fails to converge almost everywhere as $t\to\infty$. A key ingredient in our argument is a multiparameter variant of the Bochner--Riesz means.

math.CA↗
arXiv2026

Failure of stability of a maximal operator bound for perturbed Nevo-Thangavelu means

Let $G$ be a two-step nilpotent Lie group, identified via the exponential map with the Lie-algebra $\mathfrak g=\mathfrak g_1\oplus\mathfrak g_2$, where $[\mathfrak g,\mathfrak g]\subset \mathfrak g_2$. We consider maximal functions associated to spheres in a $d$-dimensional linear subspace $H$, dilated by the automorphic dilations. $L^p$ boundedness results for the case where $H=\mathfrak g_1$ are well understood. Here we consider the case of a tilted hyperplane $H\neq \mathfrak g_1$ which is not invariant under the automorphic dilations. In the case of Métivier groups it is known that the $L^p$-boundedness results are stable under a small linear tilt. We show that this is generally not the case for other two-step groups, and provide new necessary conditions for $L^p$ boundedness. We prove these results in a more general setting with tilted versions of submanifolds of $\mathfrak g_1$.

math.CA↗
arXiv2024

Spherical maximal functions on two step nilpotent Lie groups

Consider $\mathbb R^d\times \mathbb R^m$ with the group structure of a two-step nilpotent Lie group and natural parabolic dilations. The maximal function originally introduced by Nevo and Thangavelu in the setting of the Heisenberg group deals with noncommutative convolutions associated to measures on spheres or generalized spheres in $\mathbb R^d$. We drop the nondegeneracy condition in the known results on Métivier groups and prove the sharp $L^p$ boundedness result for all two step nilpotent Lie groups with $d\ge 3$.

math.CA↗
arXiv2024

Semiclassical $L^p$ quasimode restriction estimates in two dimensions

We establish the $L^p$ restriction estimates for quasimodes on a smooth curve in two dimensions. Our estimates are sharp for all smooth curves. As an application, we address $L^p$ eigenfunction restriction estimates for Laplace-Beltrami eigenfunctions on $2$-dimensional compact Riemannian manifolds without boundary and Hermite functions on $\mathbb R^2$. Our method involves a geometric analysis of the contact order between the curve and the bicharacteristic flow of the semiclassical pseudodifferential operator.

math.AP↗
arXiv2023

Endpoint eigenfunction bounds for the Hermite operator

We establish the optimal $L^p$, $p=2(d+3)/(d+1),$ eigenfunction bound for the Hermite operator $\mathcal H=-Δ+|x|^2$ on $\mathbb R^d$. Let $Π_λ$ denote the projection operator to the vector space spanned by the eigenfunctions of $\mathcal H$ with eigenvalue $λ$. The optimal $L^2$--$L^p$ bounds on $Π_λ$, $2\le p\le \infty$, have been known by the works of Karadzhov and Koch-Tataru except $p=2(d+3)/(d+1)$. For $d\ge 3$, we prove the optimal bound for the missing endpoint case. Our result is built on a new phenomenon: improvement of the bound due to asymmetric localization near the sphere $\sqrtλ\mathbb S^{d-1}$.

math.CA↗
arXiv2023

Bochner-Riesz mean for the twisted Laplacian in $\mathbb R^2$

We study the Bochner-Riesz problem for the twisted Laplacian $\mathcal L$ on $\mathbb R^2$. For $p\in [1, \infty]\setminus\{2\}$, it has been conjectured that the Bochner-Riesz means $S_λ^δ(\mathcal L) f$ of order $δ$ converges in $L^p$ for every $f\in L^p$ if and only if $δ> \max(0,|(p-2)/p|-1/2)$. We prove the conjecture by obtaining uniform $L^p$ bounds on $S_λ^δ(\mathcal L)$ up to the sharp summability indices.

math.CA↗
arXiv2023

Almost everywhere convergence of Bochner--Riesz means for the twisted Laplacian

Let $\mathcal L$ denote the twisted Laplacian in $\mathbb C^d$. We study almost everywhere convergence of the Bochner--Riesz mean $S^δ_{t}(\mathcal L) f$ of $f\in L^p(\mathbb C^d)$ as $t\to \infty$, which is an expansion of $f$ in the special Hermite functions. For $2\le p\le \infty$, we obtain the sharp range of the summability indices $δ$ for which the convergence of $S^δ_{t}(\mathcal L) f$ holds for all $f\in L^p(\mathbb C^d)$.

math.CA↗
arXiv2022

Bounds on the Hermite spectral projection operator

We study $L^p$-$L^q$ bounds on the spectral projection operator $Π_λ$ associated to the Hermite operator $H=|x|^2-Δ$ in $\mathbb R^d$. We are mainly concerned with a localized operator $χ_EΠ_λχ_E$ for a subset $E\subset\mathbb R^d$ and undertake the task of characterizing the sharp $L^p$--$L^q$ bounds. We obtain sharp bounds in extended ranges of $p,q$. First, we provide a complete characterization of the sharp $L^p$--$L^q$ bounds when $E$ is away from $\sqrtλ\mathbb S^{d-1}$. Secondly, we obtain the sharp bounds as the set $E$ gets close to $\sqrtλ\mathbb S^{d-1}$. Thirdly, we extend the range of $p,q$ for which the operator $Π_λ$ is uniformly bounded from $L^p(\mathbb R^d)$ to $L^q(\mathbb R^d)$.

math.CA↗
arXiv2022

Unique continuation for the heat operator with potentials in weak spaces

We prove strong unique continuation property for the differential inequality $|(\partial_t +Δ)u(x,t)|\le V(x,t)|u(x,t)|$ with $V$ contained in weak spaces. In particular, we establish the strong unique continuation property for $V\in L^\infty_t L^{d/2,\infty}_x$, which has been left open since the works of Escauriaza [6] and Escauriaza-Vega [8]. Our results are consequences of the Carleman estimates for the heat operator in the Lorentz spaces.

math.AP↗
arXiv2022

Effects of Crystalline Disorder on Interfacial and Magnetic Properties of Sputtered Topological Insulator/Ferromagnet Heterostructures

Thin films of Topological insulators (TIs) coupled with ferromagnets (FMs) are excellent candidates for energy-efficient spintronics devices. Here, the effect of crystalline structural disorder of TI on interfacial and magnetic properties of sputter-deposited TI/FM, Bi2Te3/Ni80Fe20, heterostructures is reported. Ni and a smaller amount of Fe from Py was found to diffuse across the interface and react with Bi2Te3. For highly crystalline c-axis oriented Bi2Te3 films, a giant enhancement in Gilbert damping is observed, accompanied by an effective out-of-plane magnetic anisotropy and enhanced damping-like spin-orbit torque (DL-SOT), possibly due to the topological surface states (TSS) of Bi2Te3. Furthermore, a spontaneous exchange bias is observed in hysteresis loop measurements at low temperatures. This is because of an antiferromagnetic topological interfacial layer formed by reaction of the diffused Ni with Bi2Te3 which couples with the FM, Ni80Fe20. For increasing disorder of Bi2Te3, a significant weakening of exchange interaction in the AFM interfacial layer is found. These experimental results Abstract length is one paragraph.

cond-mat.mtrl-sci↗
arXiv2021

Hermite spectral projection operator

We study $L^p$-$L^q$ estimate for the spectral projection operator $Π_λ$ associated to the Hermite operator $H=|x|^2-Δ$ in $\mathbb R^d$. Here $Π_λ$ denotes the projection to the subspace spanned by the Hermite functions which are the eigenfunctions of $H$ with eigenvalue $λ$. Such estimates were previously available only for $q=p'$, equivalently with $p=2$ or $q=2$ (by $TT^*$ argument) except for the estimates which are straightforward consequences of interpolation between those estimates. As shown in the works of Karadzhov, Thangavelu, and Koch and Tataru, the local and global estimates for $Π_λ$ are of different nature. Especially, $Π_λ$ exhibits complicated behaviors near the set $\sqrtλ\mathbb S^{d-1}$. Compared with the spectral projection operator associated to the Laplacian, $L^p$-$L^q$ estimate for $Π_λ$ is not so well understood up to now for general $p,q$. In this paper we consider $L^p$--$L^q$ estimate for $Π_λ$ in a general framework including the local and global estimates with $1\le p\le 2\le q\le \infty$ and undertake the work of characterizing the sharp bounds on $Π_λ$. We establish various new sharp estimates in extended ranges of $p,q$. First of all, we provide a complete characterization of the local estimate for $Π_λ$ which was first considered by Thangavelu. Secondly, for $d\ge5$, we prove the endpoint $L^2$--$L^{2(d+3)/(d+1)}$ estimate for $Π_λ$ which has been left open since the work of Koch and Tataru. Thirdly, we extend the range of $p,q$ for which the operator $Π_λ$ is uniformly bounded from $L^p$ to $L^q$.

math.CA↗
arXiv2021

Robust solutions for stochastic and distributionally robust chance-constrained binary knapsack problems

We consider chance-constrained binary knapsack problems, where the weights of items are independent random variables with the means and standard deviations known. The chance constraint can be reformulated as a second-order cone constraint under some assumptions for the probability distribution of the weights. The problem becomes a second-order cone-constrained binary knapsack problem, which is equivalent to a robust binary knapsack problem with an ellipsoidal uncertainty set. We demonstrate that optimal solutions to robust binary knapsack problems with inner and outer polyhedral approximations of the ellipsoidal uncertainty set can provide both upper and lower bounds on the optimal value of the second-order cone-constrained binary knapsack problem, and they can be obtained by solving ordinary binary knapsack problems repeatedly. Moreover, we prove that the solution providing the upper bound converges to the optimal solution to the second-order cone-constrained binary knapsack problem as the approximation of the uncertainty set becomes more accurate. Based on this, a pseudo-polynomial time algorithm is obtained under the assumption that the coefficients of the problem are integer-valued. We also propose an exact algorithm, which iteratively improves the accuracy of the approximation until an exact optimal solution is obtained. The exact algorithm also runs in pseudo-polynomial time. Computational results are presented to show the quality of the upper and lower bounds and efficiency of the exact algorithm compared to CPLEX and a previous study.

math.OC↗
arXiv2021

Bochner-Riesz means for the Hermite and special Hermite expansions

We consider the Bochner-Riesz means for the Hermite and special Hermite expansions and study their $L^p$ boundedness with the sharp summability index in a local setting. In two dimensions we establish the boundedness on the optimal range of $p$ and extend the previously known range in higher dimensions. Furthermore, we prove a new lower bound on the $L^p$ summability index for the Hermite Bochner-Riesz means in $\mathbb R^d$, $d\ge 2.$ This invalidates the conventional conjecture which was expected to be true.

math.CA↗
arXiv2021

A branch-and-price algorithm for the robust single-source capacitated facility location problem under demand uncertainty

We consider the robust single-source capacitated facility location problem with uncertainty in customer demands. A cardinality-constrained uncertainty set is assumed for the robust problem. To solve it efficiently, we propose an allocation-based formulation derived by Dantzig-Wolfe decomposition and a branch-and-price algorithm. The computational experiments show that our branch-and-price algorithm outperforms CPLEX in many cases, which solves the ordinary robust reformulation. We also examine the trade-off relationship between the empirical probability of infeasibility and the additional costs incurred and observe that the robustness of solutions can be improved significantly with small additional costs.

math.OC↗