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Sanghyuk Lee

Publications and source records attributed to Sanghyuk Lee.

At least 55 records · Page 3Linked to original sources
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

$L^p-L^q$ estimates for the circular maximal operators on Heisenberg radial functions

$L^p$ boundedness of the circular maximal function $\mathcal M_{\mathbb{H}^1}$ on the Heisenberg group $\mathbb{H}^1$ has received considerable attentions. While the problem still remains open, $L^p$ boundedness of $\mathcal M_{\mathbb{H}^1}$ on Heisenberg radial functions was recently shown for $p>2$ by Beltran, Guo, Hickman, and Seeger [2]. In this paper we extend their result considering the local maximal operator $M_{\mathbb{H}^1}$ which is defined by taking supremum over $1<t<2$. We prove $L^p-L^q$ estimates for $M_{\mathbb{H}^1}$ on Heisenberg radial functions on the optimal range of $p,q$ modulo the borderline cases. Our argument also provides a simpler proof of the aforementioned result due to Beltran et al.

math.CA↗
arXiv2021

Resolvent estimates for the Lamé operator and failure of Carleman estimates

In this paper, we consider the Lamé operator $-Δ^\ast$ and study resolvent estimate, uniform Sobolev estimate, and Carleman estimate for $-Δ^\ast$. First, we obtain sharp $L^p$--$L^q$ resolvent estimates for $-Δ^\ast$ for admissible $p,q$. This extends the particular case $q=\frac p{p-1}$ due to Barceló et al. \cite{BFPRV} and Cossetti \cite{Co19}. Secondly, we show failure of uniform Sobolev estimate and Carleman estimate for $-Δ^\ast$. For the purpose we directly analyze the Fourier multiplier of the resolvent. This allows us to prove not only the upper bound but also the lower bound on the resolvent, so we get the sharp $L^p$--$L^q$ bounds for the resolvent of $-Δ^\ast$. Strikingly, the relevant uniform Sobolev and Carleman estimates turn out to be false for the Lamé operator $-Δ^\ast$ even though the uniform resolvent estimates for $-Δ^\ast$ are valid for certain range of $p, q$. This contrasts with the classical result regarding the Laplacian $Δ$ due to Kenig, Ruiz, and Sogge \cite{KRS87} in which the uniform resolvent estimate plays crucial role in proving the uniform Sobolev and Carleman estimates for $Δ$. We also describe locations of the $L^q$-eigenvalues of $-Δ^\ast+V$ with complex potential $V$ by making use of the sharp $L^p$--$L^q$ resolvent estimates for $-Δ^\ast$.

math.CA↗
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

Dimension of divergence set of the wave equation

We consider the Hausdorff dimension of the divergence set on which the pointwise convergence $\lim_{t\rightarrow 0} e^{it\sqrt{-Δ}} f(x) = f(x)$ fails when $f \in H^s(\mathbb R^d)$. We especially prove the conjecture raised by Barceló, Bennett, Carbery and Rogers \cite{BBCR} for $d=3$, and improve the previous results in higher dimensions $d\ge4$. We also show that a Strichartz type estimate for $f\to e^{it\sqrt{-Δ}} f$ with the measure $ dt\,dμ(x)$ is essentially equivalent to the estimate for the spherical average of $\widehat μ$ which has been extensively studied for the Falconer distance set problem. The equivalence provides shortcuts to the recent results due to B. Liu and K. Rogers.

math.AP↗
arXiv2020

A sharp regularity estimate for the Schrödinger propagator on the sphere

Let $Δ_{\mathbb S^n}$ denote the Laplace-Beltrami operator on the $n$-dimensional unit sphere $\mathbb S^n$. In this paper we show that $$ \| e^{it Δ_{\mathbb S^n}}f \|_{L^4([0, 2π) \times \mathbb S^n)} \leq C \| f\|_{W^{α, 4} (\mathbb S^n)} $$ holds provided that $n\geq 2$, $α> {(n-2)/4}.$ The range of $α$ is sharp up to the endpoint. As a consequence, we obtain space-time estimates for the Schrödinger propagator $e^{it Δ_{\mathbb S^n}}$ on the $L^p$ spaces for $2\leq p\leq \infty.$ We also prove that for zonal functions on ${\mathbb S}^n$, the Schrödinger maximal operator $\sup_{0\leq t<2π} |e^{itΔ_{\mathbb S^n}} f|$ is bounded from $W^{α, 2}(\mathbb S^n) $ to $L^{\frac{6n}{3n-2}}(\mathbb S^n)$ whenever $α>{1/3}$.

math.AP↗
arXiv2020

Almost everywhere convergence of Bochner-Riesz means for the Hermite operators

Let $H = -Δ+ |x|^2$ be the Hermite operator in ${\mathbb R}^n$. In this paper we study almost everywhere convergence of the Bochner-Riesz means associated with $H$ which is defined by $S_R^λ(H)f(x) = \sum\limits_{k=0}^{\infty} \big(1-{2k+n\over R^2}\big)_+^λ P_k f(x).$ Here $P_k f$ is the $k$-th Hermite spectral projection operator. For $2\le p<\infty$, we prove that $$ \lim\limits_{R\to \infty} S_R^λ(H) f=f \ \ \ \text{a.e.} $$ for all $f\in L^p(\mathbb R^n)$ provided that $λ> λ(p)/2$ and $λ(p)=\max\big\{ n\big({1/2}-{1/p}\big)-{1/ 2}, \, 0\big\}.$ Conversely, we also show the convergence generally fails if $λ< λ(p)/2$ in the sense that there is an $f\in L^p(\mathbb R^n)$ for $2n/(n-1)\le p$ such that the convergence fails. This is in surprising contrast with a.e. convergence of the classical Bochner-Riesz means for the Laplacian. For $n\geq 2$ and $p\ge 2$ our result tells that the critical summability index for a.e. convergence for $S_R^λ(H)$ is as small as only the \emph{half} of the critical index for a.e. convergence of the classical Bochner-Riesz means. When $n = 1$, we show a.e. convergence holds for $f\in L^p({\mathbb R})$ with $ p\geq 2$ whenever $λ>0$. Compared with the classical result due to Askey and Wainger who showed the optimal $L^p$ convergence for $S_R^λ(H)$ on ${\mathbb R}$ we only need smaller summability index for a.e. convergence.

math.CA↗
arXiv2020

Uniqueness in the Calderón problem and bilinear restriction estimates

Uniqueness in the Calderón problem in dimension bigger than two was usually studied under the assumption that conductivity has bounded gradient. For conductivities with unbounded gradients uniqueness results have not been known until recent years. The latest result due to Haberman basically relies on the optimal $L^2$ restriction estimate for hypersurface which is known as the Tomas-Stein restriction theorem. In the course of developments of the Fourier restriction problem bilinear and multilinear generalizations of the (adjoint) restriction estimates under suitable transversality condition between surfaces have played important roles. Since such advanced machineries usually provide strengthened estimates, it seems natural to attempt to utilize these estimates to improve the known results. In this paper, we make use of the sharp bilinear restriction estimates, which is due to Tao, and relax the regularity assumption on conductivity. We also consider the inverse problem for the Schrödinger operator with potentials contained in the Sobolev spaces of negative orders.

math.AP↗
arXiv2020

Strichartz estimates for orthonormal families of initial data and weighted oscillatory integral estimates

We establish new Strichartz estimates for orthonormal families of initial data in the case of the wave, Klein-Gordon and fractional Schrödinger equations. Our estimates extend those of Frank-Sabin in the case of the wave and Klein-Gordon equations, and generalize work of Frank-Lewin-Lieb-Seiringer and Frank-Sabin for the Schrödinger equation. Due to a certain technical barrier, except for the classical Schrödinger equation, the Strichartz estimates for orthonormal families of initial data have not previously been established up to the sharp summability exponents in the full range of admissible pairs. We obtain the optimal estimates in various notable cases and improve the previous results. The main novelty of this paper is the use of estimates for weighted oscillatory integrals which we combine with an approach due to Frank and Sabin. This strategy also leads us to proving new estimates for weighted oscillatory integrals with optimal decay exponents which we believe to be of wider independent interest. Applications to the theory of infinite systems of Hartree type, weighted velocity averaging lemmas for kinetic transport equations, and refined Strichartz estimates for data in Besov spaces are also provided.

math.AP↗
arXiv2019

Maximal estimates for the bilinear spherical averages and the bilinear Bochner-Riesz operators

We study the maximal estimates for the bilinear spherical average and the bilinear Bochner-Riesz operator. Firstly, we obtain $L^p\times L^q \to L^r$ estimates for the bilinear spherical maximal function on the optimal range. Thus, we settle the problem which was previously considered by Geba, Greenleaf, Iosevich, Palsson and Sawyer, later Barrionevo, Grafakos, D. He, Honzík and Oliveira, and recently Heo, Hong and Yang. Secondly, we consider $L^p\times L^q \to L^r$ estimates for the maximal bilinear Bochner-Riesz operators and improve the previous known results. For the purpose we draw a connection between the maximal estimates and the square function estimates for the classical Bochner-Riesz operators.

math.CA↗
arXiv2019

Remarks on estimates for the adjoint restriction operator to curves over the sphere

Recently, two of the authors obtained estimates for the adjoint restriction operator to finite type curves with respect to general measures. Strikingly, it turns out that some of such estimates are sharp, especially when the measures are given by surface measures under certain condition. A typical example is the surface measure on the sphere. We demonstrate sharpness of such estimates by constructing an example and, also, discuss related estimates over different type of surfaces.

math.CA↗
arXiv2019

Atomic Scheduling of Appliance Energy Consumption in Residential Smart Grid

The current formulation of the optimal scheduling of appliance energy consumption uses as optimization variables the vectors of appliances' scheduled energy consumption over equally-divided time slots of a day, which does not take into account the atomicity of appliances' operations (i.e., the unsplittable nature of appliances' operations and resulting energy consumption). In this paper, we provide a new formulation of atomic scheduling of energy consumption based on the optimal routing framework; the flow configurations of users over multiple paths between the common source and destination nodes of a ring network are used as optimization variables, which indicate the starting times of scheduled energy consumption, and optimal scheduling problems are now formulated in terms of the user flow configurations. Because the atomic optimal scheduling results in a Boolean-convex problem for a convex objective function, we propose a successive convex relaxation technique for efficient calculation of an approximate solution, where we iteratively drop fractional-valued elements and apply convex relaxation to the resulting problem until we find a feasible suboptimal solution. Numerical results for the cost and peak-to-average ratio minimization problems demonstrate that the successive convex relaxation technique can provide solutions close to, often identical to, global optimal solutions.

cs.NI↗
arXiv2019

Maximal estimates for the Schrödinger equation with orthonormal initial data

For the one-dimensional Schrödinger equation, we obtain sharp maximal-in-time and maximal-in-space estimates for systems of orthonormal initial data. The maximal-in-time estimates generalize a classical result of Kenig--Ponce--Vega and allow us obtain pointwise convergence results associated with systems of infinitely many fermions. The maximal-in-space estimates simultaneously address an endpoint problem raised by Frank--Sabin in their work on Strichartz estimates for orthonormal systems of data, and provide a path toward proving our maximal-in-time estimates.

math.AP↗
arXiv2019

Sharp resolvent estimates outside of the uniform boundedness range

In this paper we are concerned with resolvent estimates for the Laplacian $Δ$ in Euclidean spaces. Uniform resolvent estimates for $Δ$ were shown by Kenig, Ruiz and Sogge \cite{KRS} who established rather a complete description of the Lebesgue spaces allowing such estimates. However, the problem of obtaining sharp $L^p$--$L^q$ bounds depending on $z$ has not been considered in a general framework which admits all possible $p,q$. In this paper, we present a complete picture of sharp $L^p$--$L^q$ resolvent estimates, which may depend on $z$. We also obtain the sharp resolvent estimates for the fractional Laplacians and a new result for the Bochner--Riesz operators of negative index.

math.CA↗
arXiv2019

A Beaconless Asymmetric Energy-Efficient Time Synchronization Scheme for Resource-Constrained Multi-Hop Wireless Sensor Networks

The ever-increasing number of WSN deployments based on a large number of battery-powered, low-cost sensor nodes, which are limited in their computing and power resources, puts the focus of WSN time synchronization research on three major aspects, i.e., accuracy, energy consumption and computational complexity. In the literature, the latter two aspects have not received much attention compared to the accuracy of WSN time synchronization. Especially in multi-hop WSNs, intermediate gateway nodes are overloaded with tasks for not only relaying messages but also a variety of computations for their offspring nodes as well as themselves. Therefore, not only minimizing the energy consumption but also lowering the computational complexity while maintaining the synchronization accuracy is crucial to the design of time synchronization schemes for resource-constrained sensor nodes. In this paper, focusing on the three aspects of WSN time synchronization, we introduce a framework of reverse asymmetric time synchronization for resource-constrained multi-hop WSNs and propose a beaconless energy-efficient time synchronization scheme based on reverse one-way message dissemination. Experimental results with a WSN testbed based on TelosB motes running TinyOS demonstrate that the proposed scheme conserves up to 95% energy consumption compared to the flooding time synchronization protocol while achieving microsecond-level synchronization accuracy.

cs.NI↗
arXiv2019

On the Transformation of Latent Space in Autoencoders

Noting the importance of the latent variables in inference and learning, we propose a novel framework for autoencoders based on the homeomorphic transformation of latent variables, which could reduce the distance between vectors in the transformed space, while preserving the topological properties of the original space, and investigate the effect of the latent space transformation on learning generative models and denoising corrupted data. The experimental results demonstrate that our generative and denoising models based on the proposed framework can provide better performance than conventional variational and denoising autoencoders due to the transformation, where we evaluate the performance of generative and denoising models in terms of the Hausdorff distance between the sets of training and processed i.e., either generated or denoised images, which can objectively measure their differences, as well as through direct comparison of the visual characteristics of the processed images.

cs.LG↗
arXiv2019

Hierarchical Auxiliary Learning

Conventional application of convolutional neural networks (CNNs) for image classification and recognition is based on the assumption that all target classes are equal(i.e., no hierarchy) and exclusive of one another (i.e., no overlap). CNN-based image classifiers built on this assumption, therefore, cannot take into account an innate hierarchy among target classes (e.g., cats and dogs in animal image classification) or additional information that can be easily derived from the data (e.g.,numbers larger than five in the recognition of handwritten digits), thereby resulting in scalability issues when the number of target classes is large. Combining two related but slightly different ideas of hierarchical classification and logical learning by auxiliary inputs, we propose a new learning framework called hierarchical auxiliary learning, which not only address the scalability issues with a large number of classes but also could further reduce the classification/recognition errors with a reasonable number of classes. In the hierarchical auxiliary learning, target classes are semantically or non-semantically grouped into superclasses, which turns the original problem of mapping between an image and its target class into a new problem of mapping between a pair of an image and its superclass and the target class. To take the advantage of superclasses, we introduce an auxiliary block into a neural network, which generates auxiliary scores used as additional information for final classification/recognition; in this paper, we add the auxiliary block between the last residual block and the fully-connected output layer of the ResNet. Experimental results demonstrate that the proposed hierarchical auxiliary learning can reduce classification errors up to 0.56, 1.6 and 3.56 percent with MNIST, SVHN and CIFAR-10 datasets, respectively.

cs.LG↗