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Eunhee Jeong

Publications and source records attributed to Eunhee Jeong.

14 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↗
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

Carleman inequalities and unique continuation for the polyharmonic operators

We obtain a complete characterization of $L^p-L^q$ Carleman estimates with weight $e^{v\cdot x}$ for the polyharmonic operators. Our result extends the Carleman inequalities for the Laplacian due to Kenig--Ruiz--Sogge. Consequently, we obtain new unique continuation properties of higher order Schrödinger equations relaxing the integrability assumption on the solution spaces.

math.AP↗
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↗
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↗
arXiv2020

Fourier duality in the Brascamp-Lieb inequality

It was observed recently in work of Bez, Buschenhenke, Cowling, Flock and the first author, that the euclidean Brascamp-Lieb inequality satisfies a natural and useful Fourier duality property. The purpose of this paper is to establish an appropriate discrete analogue of this. Our main result identifies the Brascamp-Lieb constants on (finitely-generated) discrete abelian groups with Brascamp-Lieb constants on their (Pontryagin) duals. As will become apparent, the natural setting for this duality principle is that of locally compact abelian groups, and this raises basic questions about Brascamp-Lieb constants formulated in this generality.

math.CA↗
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↗
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↗
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↗
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↗