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Sanghyuk Lee

Publications and source records attributed to Sanghyuk Lee.

At least 73 records · Page 4Linked to original sources
arXiv2019

Restriction estimates to complex hypersurfaces

The restriction problem is better understood for hypersurfaces and recent progresses have been made by bilinear and multilinear approaches and most recently polynomial partitioning method which is combined with those estimates. However, for surfaces with codimension bigger than 1, bilinear and multilinear generalization of restriction estimates are more involved and effectiveness of these multilinear estimates is not so well understood yet. Regarding the restriction problem for the surfaces with codimensions bigger than 1, the current state of the art is still at the level of $TT^*$ method which is known to be useful for obtaining $L^q$--$L^2$ restriction estimates. In this paper, we consider a special type of codimension 2 surfaces which are given by graphs of complex analytic functions and attempt to make progress beyond the $L^2$ restriction estimates.

math.CA↗
arXiv2019

Inhomogeneous Strichartz estimates in some critical cases

Strong-type inhomogeneous Strichartz estimates are shown to be false for the wave equation outside the so-called acceptable region. On a critical line where the acceptability condition marginally fails, we prove substitute estimates with a weak-type norm in the temporal variable. We achieve this by establishing such weak-type inhomogeneous Strichartz estimates in an abstract setting. The application to the wave equation rests on a slightly stronger form of the standard dispersive estimate in terms of certain Besov spaces.

math.CA↗
arXiv2018

XJTLUIndoorLoc: A New Fingerprinting Database for Indoor Localization and Trajectory Estimation Based on Wi-Fi RSS and Geomagnetic Field

In this paper, we present a new location fingerprinting database comprised of Wi-Fi received signal strength (RSS) and geomagnetic field intensity measured with multiple devices at a multi-floor building in Xi'an Jiatong-Liverpool University, Suzhou, China. We also provide preliminary results of localization and trajectory estimation based on convolutional neural network (CNN) and long short-term memory (LSTM) network with this database. For localization, we map RSS data for a reference point to an image-like, two-dimensional array and then apply CNN which is popular in image and video analysis and recognition. For trajectory estimation, we use a modified random way point model to efficiently generate continuous step traces imitating human walking and train a stacked two-layer LSTM network with the generated data to remember the changing pattern of geomagnetic field intensity against (x,y) coordinates. Experimental results demonstrate the usefulness of our new database and the feasibility of the CNN and LSTM-based localization and trajectory estimation with the database.

cs.LG↗
arXiv2018

Stability of the Brascamp-Lieb constant and applications

We prove that the best constant in the general Brascamp-Lieb inequality is a locally bounded function of the underlying linear transformations. As applications we deduce certain very general Fourier restriction, Kakeya-type, and nonlinear variants of the Brascamp-Lieb inequality which have arisen recently in harmonic analysis.

math.CA↗
arXiv2018

Square function estimates for the Bochner-Riesz means

We consider the square function (known as Stein's square function) estimate associated with the Bochner-Riesz means. The previously known range of sharp estimate is improved. Our results are based on vector valued extensions of Bennett-Carbery-Tao's multilinear (adjoint) restriction estimate and adaption of induction argument due to Bourgain-Guth. Unlike the previous work by Bourgain-Guth on $L^p$ boundedness of Bochner-Riesz means in which oscillatory operators associated to the kernel had been studied, we take more direct approach by working on the Fourier transform side. This enables us to obtain the correct order of smoothing which is essential for obtaining the sharp estimate for the square function.

math.CA↗
arXiv2018

Bounds on the maximal Bochner-Riesz means for elliptic operators

We investigate $L^p$ boundedness of the maximal Bochner-Riesz means for self-adjoint operators of elliptic type. Assuming the finite speed of propagation for the associated wave operator, from the restriction type estimates we establish the sharp $L^p$ boundedness of the maximal Bochner-Riesz means for the elliptic operators. As applications, we obtain the sharp $L^p$ maximal bounds for the Schrödinger operators on asymptotically conic manifolds, the harmonic oscillator and its perturbations or elliptic operators on compact manifolds.

math.AP↗
arXiv2018

Carleman estimates and boundedness of associated multiplier operators

Let $P(D)$ be the Laplacian $Δ,$ or the wave operator $\square$. The following type of Carleman estimate is known to be true on a certain range of $p,q$: \[ \|e^{v\cdot x}u\|_{L^q(\mathbb{R}^d)} \le C\|e^{v\cdot x}P(D)u\|_{L^p(\mathbb{R}^d)} \] with $C$ independent of $v\in \mathbb{R}^d$. The estimates are consequences of the uniform Sobolev type estimates for second order differential operators due to Kenig-Ruiz-Sogge \cite{KRS} and Jeong-Kwon-Lee \cite{JKL}. The range of $p,q$ for which the uniform Sobolev type estimates hold was completely characterized for the second order differential operators with nondegenerate principal part. But the optimal range of $p,q$ for which the Carleman estimate holds has not been clarified before. When $P(D)=Δ$, $\square$, or the heat operator, we obtain a complete characterization of the admissible $p,q$ for the aforementioned type of Carleman estimate. For this purpose we investigate $L^p$-$L^q$ boundedness of related multiplier operators. As applications, we also obtain some unique continuation results.

math.AP↗
arXiv2018

Weighted fractional chain rule and nonlinear wave equations with minimal regularity

We consider the local well-posedness for 3-D quadratic semi-linear wave equations with radial data: $\Box u = a |\partial_t u|^2+b|\nabla_x u|^2$, $u(0,x)=u_0(x)\in H^{s}_{\mathrm{rad}}$, $\partial_t u(0,x)=u_1(x)\in H^{s-1}_{\mathrm{rad}}$. It has been known that the problem is well-posed for $s\ge 2$ and ill-posed for $s<3/2$. In this paper, we prove unconditional well-posedness up to the scaling invariant regularity, that is to say, for $s>3/2$ and thus fill the gap which was left open for many years. For the purpose, we also obtain a weighted fractional chain rule, which is of independent interest. Our method here also works for a class of nonlinear wave equations with general power type nonlinearities which contain the space-time derivatives of the unknown functions. In particular, we prove the Glassey conjecture in the radial case, with minimal regularity assumption.

math.AP↗
arXiv2018

Sharp $L^p$-$L^q$ estimates for the spherical harmonic projection

We consider $L^p$-$L^q$ estimates for the spherical harmonic projection operators and obtain sharp bounds on a certain range of $p$, $q$. As an application, we provide a proof of off-diagonal Carleman estimates for the Laplacian, which extends the earlier results due to Jerison and Kenig \cite{JK}, and Stein \cite{St-append}.

math.CA↗
arXiv2017

A Scalable Deep Neural Network Architecture for Multi-Building and Multi-Floor Indoor Localization Based on Wi-Fi Fingerprinting

One of the key technologies for future large-scale location-aware services covering a complex of multi-story buildings --- e.g., a big shopping mall and a university campus --- is a scalable indoor localization technique. In this paper, we report the current status of our investigation on the use of deep neural networks (DNNs) for scalable building/floor classification and floor-level position estimation based on Wi-Fi fingerprinting. Exploiting the hierarchical nature of the building/floor estimation and floor-level coordinates estimation of a location, we propose a new DNN architecture consisting of a stacked autoencoder for the reduction of feature space dimension and a feed-forward classifier for multi-label classification of building/floor/location, on which the multi-building and multi-floor indoor localization system based on Wi-Fi fingerprinting is built. Experimental results for the performance of building/floor estimation and floor-level coordinates estimation of a given location demonstrate the feasibility of the proposed DNN-based indoor localization system, which can provide near state-of-the-art performance using a single DNN, for the implementation with lower complexity and energy consumption at mobile devices.

cs.NI↗
arXiv2017

Improved bound for the bilinear Bochner-Riesz operator

We study $L^p\times L^q\to L^r$ bounds for the bilinear Bochner-Riesz operator $\mathcal{B}^α$, $α>0$ in $\mathbb{R}^d,$ $d\ge2$, which is defined by \[ {\mathcal B}^α(f,g)=\iint_{\mathbb{R}^d\times\mathbb{R}^d} e^{2πi x\cdot(ξ+η)} (1-|ξ|^2-|η|^2 )^α_+ ~ \widehat{f}(ξ)\,\widehat{g}(η)\,dξdη.\] We make use of a decomposition which relates the estimates for $\mathcal{B}^α$ to those of the square function estimates for the classical Bochner-Riesz operators. In consequence, we significantly improve the previously known bounds.

math.CA↗
arXiv2017

Bilinear restriction estimates for surfaces of codimension bigger than one

In connection with the restriction problem in $\mathbb R^n$ for hypersurfaces including the sphere and paraboloid, the bilinear (adjoint) restriction estimates have been extensively studied. However, not much is known about such estimates for surfaces with codimension (and dimension) larger than one. In this paper we show sharp bilinear $L^2 \times L^2 \to L^q$ restriction estimates for general surfaces of higher codimension. In some special cases, we can apply these results to obtain the corresponding linear estimates.

math.CA↗
arXiv2017

Large-Scale Location-Aware Services in Access: Hierarchical Building/Floor Classification and Location Estimation using Wi-Fi Fingerprinting Based on Deep Neural Networks

One of key technologies for future large-scale location-aware services in access is a scalable indoor localization technique. In this paper, we report preliminary results from our investigation on the use of deep neural networks (DNNs) for hierarchical building/floor classification and floor-level location estimation based on Wi-Fi fingerprinting, which we carried out as part of a feasibility study project on Xi'an Jiaotong-Liverpool University (XJTLU) Campus Information and Visitor Service System. To take into account the hierarchical nature of the building/floor classification problem, we propose a new DNN architecture based on a stacked autoencoder for the reduction of feature space dimension and a feed-forward classifier for multi-label classification with argmax functions to convert multi-label classification results into multi-class classification ones. We also describe the demonstration of a prototype DNN-based indoor localization system for floor-level location estimation using real received signal strength (RSS) data collected at one of the buildings on the XJTLU campus. The preliminary results for both building/floor classification and floor-level location estimation clearly show the strengths of DNN-based approaches, which can provide near state-of-the-art performance with less parameter tuning and higher scalability.

cs.NI↗
arXiv2017

Smoothing estimates for the kinetic transport equation at the critical regularity

We prove smoothing estimates for velocity averages of the kinetic transport equation in hyperbolic Sobolev spaces at the critical regularity, leading to a complete characterisation of the allowable regularity exponents. Such estimates will be deduced from some mixed-norm estimates for the cone multiplier operator at a certain critical index. Our argument is not particular to the geometry of the cone and we illustrate this by establishing analogous estimates for the paraboloid.

math.AP↗
arXiv2017

On the Strichartz estimates for orthonormal systems of initial data with regularity

The classical Strichartz estimates for the free Schrödinger propagator have recently been substantially generalised to estimates of the form \[ \bigg\|\sum_jλ_j|e^{itΔ}f_j|^2\bigg\|_{L^p_tL^q_x}\lesssim\|λ\|_{\ell^α} \] for orthonormal systems $(f_j)_j$ of initial data in $L^2$, firstly in work of Frank--Lewin--Lieb--Seiringer and later by Frank--Sabin. The primary objective is identifying the largest possible $α$ as a function of $p$ and $q$, and in contrast to the classical case, for such estimates the critical case turns out to be $(p,q) = (\frac{d+1}{d},\frac{d+1}{d-1})$. We consider the case of orthonormal systems $(f_j)_j$ in the homogeneous Sobolev spaces $\dot{H}^s$ for $s \in (0,\frac{d}{2})$ and we establish the sharp value of $α$ as a function of $p$, $q$ and $s$, except possibly an endpoint in certain cases, at which we establish some weak-type estimates. Furthermore, at the critical case $(p,q) = (\frac{d+1}{d-2s},\frac{d(d+1)}{(d-1)(d-2s)})$ for general $s$, we show the veracity of the desired estimates when $α= p$ if we consider frequency localised estimates, and the failure of the (non-localised) estimates when $α= p$; this exhibits the difficulty of upgrading from frequency localised estimates in this context, again in contrast to the classical setting.

math.FA↗
arXiv2017

Estimates for the kinetic transport equation in hyperbolic Sobolev spaces

We establish smoothing estimates in the framework of hyperbolic Sobolev spaces for the velocity averaging operator $ρ$ of the solution of the kinetic transport equation. If the velocity domain is either the unit sphere or the unit ball, then, for any exponents $q$ and $r$, we find a characterisation of the exponents $β_+$ and $β_-$, except possibly for an endpoint case, for which $D_+^{β_+}D_-^{β_-} ρ$ is bounded from space-velocity $L^2_{x,v}$ to space-time $L^q_tL^r_x$. Here, $D_+$ and $D_-$ are the classical and hyperbolic derivative operators, respectively. In fact, we shall provide an argument which unifies these velocity domains and the velocity averaging estimates in either case are shown to be equivalent to mixed-norm bounds on the cone multiplier operator acting on $L^2$. We develop our ideas further in several ways, including estimates for initial data lying in certain Besov spaces, for which a key tool in the proof is the sharp $\ell^p$ decoupling theorem recently established by Bourgain and Demeter. We also show that the level of permissible smoothness increases significantly if we restrict attention to initial data which are radially symmetric in the spatial variable.

math.AP↗
arXiv2016

Fractal Strichartz estimate for the wave equation

We consider Strichartz estimates for the wave equation with respect to general measures which satisfy certain growth condition. In $\mathbb R^{3+1}$ we obtain the sharp estimate and in higher dimensions improve the previous results.

math.AP↗