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Sanghyuk Lee

Publications and source records attributed to Sanghyuk Lee.

At least 91 records · Page 5Linked to original sources
arXiv2016

Uniform Sobolev inequalities for second order non-elliptic differential operators

We study uniform Sobolev inequalities for the second order differential operators $P(D)$ of non-elliptic type. For $d\ge3$ we prove that the Sobolev type estimate $\|u\|_{L^q(\mathbb{R}^d)}\le C \|P(D)u\|_{L^p(\mathbb{R}^d)}$ holds with $C$ independent of the first order and the constant terms of $P(D)$ if and only if $1/p-1/q=2/d$ and $\frac{2d(d-1)}{d^2+2d-4}<p<\frac{2(d-1)}d$. We also obtain restricted weak type endpoint estimates for the critical $(p,q)=(\frac{2(d-1)}{d},\frac{2d(d-1)}{(d-2)^2})$, $(\frac{2d(d-1)}{d^2+2d-4}, \frac{2(d-1)}{d-2})$. As a consequence, the result extends the class of functions for which the unique continuation for the inequality $|P(D)u|\le|Vu|$ holds.

math.AP↗
arXiv2016

Simulation Study of an Energy-Efficient Time Synchronization Scheme based on Source Clock Frequency Recovery in Asymmetric Wireless Sensor Networks

In this paper we report preliminary results of a simulation study on an energy-efficient time synchronization scheme based on source clock frequency recovery (SCFR) at sensor nodes in asymmetric wireless sensor networks (WSNs), where a head node --- serving as a gateway between wired and wireless networks --- is equipped with a powerful processor and supplied power from outlet, and sensor nodes --- connected only through wireless channels --- are limited in processing and battery-powered. In the SCFR-based WSN time synchronization scheme, we concentrate on battery-powered sensor nodes and reduce their energy consumption by minimizing the number of message transmissions from sensor nodes to the head node. Through simulation experiments we analyze the performance of the SCFR-based WSN time synchronization scheme, including the impact of SCFR on time synchronization based on two-way message exchange, and demonstrate the feasibility of the proposed time synchronization scheme.

cs.NI↗
arXiv2016

Energy-Efficient Time Synchronization Based on Asynchronous Source Clock Frequency Recovery and Reverse Two-Way Message Exchanges in Wireless Sensor Networks

We consider energy-efficient time synchronization in a wireless sensor network where a head node (i.e., a gateway between wired and wireless networks and a center of data fusion) is equipped with a powerful processor and supplied power from outlet, and sensor nodes (i.e., nodes measuring data and connected only through wireless channels) are limited in processing and battery-powered. It is this asymmetry that our study focuses on; unlike most existing schemes to save the power of all network nodes, we concentrate on battery-powered sensor nodes in minimizing energy consumption for time synchronization. We present a time synchronization scheme based on asynchronous source clock frequency recovery and reverse two-way message exchanges combined with measurement data report messages, where we minimize the number of message transmissions from sensor nodes, and carry out the performance analysis of the estimation of both measurement time and clock frequency with lower bounds for the latter. Simulation results verify that the proposed scheme outperforms the schemes based on conventional two-way message exchanges with and without clock frequency recovery in terms of the accuracy of measurement time estimation and the number of message transmissions and receptions at sensor nodes as an indirect measure of energy efficiency.

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arXiv2015

Fourier transform and regularity of characteristic functions

Let $E$ be a bounded domain in $\mathbb R^d$. We study regularity property of $χ_E$ and integrability of $\widehat {χ_E }$ when its boundary $\partial E$ satisfies some conditions. At the critical case these properties are generally known to fail. By making use of Lorentz and Lorentz-Sobolev spaces we obtain the endpoint cases of the previous known results. Our results are based on a refined version of Littlewood-Paley inequality, which makes it possible to exploit cancellation effectively.

math.AP↗
arXiv2015

On enery-efficient time synchronization based on source clock frequency recovery in wireless sensor networks

In this paper we study energy-efficient time synchronization schemes with focus on asymmetric wireless sensor networks, where a head node, which is connected to both wired & wireless networks, is equipped with a powerful processor and supplied power from outlet, and sensor nodes, which are connected only through wireless channels, are limited in processing and battery-powered. It is this asymmetry that we focus our study on; unlike existing schemes saving the power of all sensor nodes in the network (including the head node), we concentrate on battery-powered sensor nodes in minimizing energy consumption for synchronization. Specifically, we discuss a time synchronization scheme based on source clock frequency recovery, where we minimize the number of message transmissions from sensor nodes to the head node, and its extension to network-wide, multi-hop synchronization through gateway nodes.

cs.NI↗
arXiv2015

Restriction Estimates for space curves with respect to general measures

In this paper we consider adjoint restriction estimates for space curves with respect to general measures and obtain optimal estimates when the curves satisfy a finite type condition. The argument here is new in that it doesn't rely on the \emph{offspring curve} method, which has been extensively used in the previous works. Our work was inspired by the recent argument due to Bourgain and Guth which was used to deduce linear restriction estimates from multilinear estimates for hypersurfaces.

math.CA↗
arXiv2014

A Sobolev estimate for the adjoint restriction operator

In this note we consider the adjoint restriction estimate for hypersurface under additional regularity assumption. We obtain the optimal $H^s$-$L^q$ estimate and its mixed norm generalization. As applications we prove some weighted Strichartz estimates for the propagator $e^{it(-Δ)^{α/2}}φ$, $α>0$.

math.CA↗
arXiv2014

On radial Fourier multipliers and almost everywhere convergence

We study a.e. convergence on $L^p$, and Lorentz spaces $L^{p,q}$, $p>\tfrac{2d}{d-1}$, for variants of Riesz means at the critical index $d(\tfrac 12-\tfrac 1p)-\tfrac12$. We derive more general results for (quasi-)radial Fourier multipliers and associated maximal functions, acting on $L^2$ spaces with power weights, and their interpolation spaces. We also include a characterization of boundedness of such multiplier transformations on weighted $L^2$ spaces, and a sharp endpoint bound for Stein's square-function associated with the Riesz means.

math.CA↗
arXiv2014

Well-posedness and Ill-posedness for the cubic fractional Schrödinger equations

We study the low regularity well-posedness of the 1-dimensional cubic nonlinear fractional Schrödinger equations with Lévy indices $1 < α< 2$. We consider both non-periodic and periodic cases, and prove that the Cauchy problems are locally well-posed in $H^s$ for $s \geq \frac {2-α}4$. This is shown via a trilinear estimate in Bourgain's $X^{s,b}$ space. We also show that non-periodic equations are ill-posed in $H^s$ for $\frac {2 - 3α}{4(α+ 1)} < s < \frac {2-α}4$ in the sense that the flow map is not locally uniformly continuous.

math.AP↗
arXiv2014

Profile decompositions of fractional Schrödinger equations with angularly regular data

We study the fractional Schrödinger equations in $\mathbb R^{1+d}, d \geq 3$ of order ${d}/({d-1}) < \al < 2$. Under the angular regularity assumption we prove linear and nonlinear profile decompositions which extend the previous results \cite{chkl2} to data without radial assumption. As applications we show blowup phenomena of solutions to mass-critical fractional Hartree equations.

math.AP↗
arXiv2012

Strichartz Estimates in Spherical Coordinates

In this paper we study Strichartz estimates for dispersive equations which are defined by radially symmetric pseudo-differential operators, and of which initial data belongs to spaces of Sobolev type defined in spherical coordinates. We obtain the space time estimates on the best possible range including the endpoint cases.

math.AP↗
arXiv2012

Profile decompositions and Blowup phenomena of mass critical fractional Schrödinger equations

We study, under the radial symmetry assumption, the solutions to the fractional Schrödinger equations of critical nonlinearity in $\mathbb R^{1+d}, d \geq 2$, with Lévy index ${2d}/({2d-1}) < \al < 2$. We firstly prove the linear profile decomposition and then apply it to investigate the properties of the blowup solutions of the nonlinear equations with mass-critical Hartree type nonlineartity.

math.AP↗
arXiv2012

Square functions and maximal operators associated with radial Fourier multipliers

We begin with an overview on square functions for spherical and Bochner-Riesz means which were introduced by Eli Stein, and discuss their implications for radial multipliers and associated maximal functions. We then prove new endpoint estimates for these square functions, for the maximal Bochner-Riesz operator, and for more general classes of radial Fourier multipliers.

math.CA↗