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Hyerim Ko

Publications and source records attributed to Hyerim Ko.

14 recordsLinked to original sources
arXiv2025

Weighted wave envelope estimates for the parabola

In this paper, we extend the C\'ordoba-Fefferman square function estimate for the parabola to a weighted setting. Our weighted square function estimate is derived from a weighted wave envelope estimate for the parabola. The bounds are formulated in terms of families of multiscale tubes together with weight parameters that quantify the distribution of the weight. As an application, we obtain some weighted L^p-estimates for a class of Fourier multiplier operators and for solutions to free Schr\"odinger equation.

math.CA↗
arXiv2025

Maximal estimates for orthonormal systems of wave equations

This paper investigates maximal estimates of the wave operators for orthonormal families of initial data. We extend the classical maximal estimates for the wave operator by making partial progress on maximal estimates for orthonormal systems in low dimensions. Our novel approach is based on a geometric analysis of the kernel of wave operators within the framework of Schatten $2$ estimates. In particular, we exploit Wolff's geometric lemma on the intersection patterns of thickened spheres.

math.AP↗
arXiv2025

Maximal estimates for orthonormal systems of wave equations with sharp regularity

We study maximal estimates for the wave equation with orthonormal initial data. In dimension $d=3$, we establish optimal results with the sharp regularity exponent up to the endpoint. In higher dimensions $d \ge 4$ and also in $d=2$, we obtain sharp bounds for the Schatten exponent (summability index) $\beta\in [2, \infty]$ when $d\ge4$, and $\beta\in[1, 2]$ when $d=2$, improving upon the previous estimates due to Kinoshita--Ko--Shiraki. Our approach is based on a novel analysis of a key integral arising in the case $\beta=2$, which allows us to refine existing techniques and achieve the optimal estimates.

math.AP↗
arXiv2025

Local smoothing and maximal estimates for average over surfaces of codimension 2 in $\mathbb R^4$

In this paper, we obtain local smoothing estimates for the averages over nondegenerate surfaces of codimension $2$ in $\mathbb R^4$. We make use of multilinear restriction estimates and decoupling inequalities for a hypersurface in $\mathbb R^5$, a conical extension of a two-dimensional nondegenerate surface along two flat directions. We also establish sharp $L^p$--$L^q$ estimates for maximal averages over nondegenerate surfaces of half the ambient dimension in $\mathbb R^{2n}$ for even $n \ge 2$.

math.CA↗
arXiv2022

Remarks on dimension of unions of curves

We study an analogue of Marstrand's circle packing problem for curves in higher dimensions. We consider collections of curves which are generated by translation and dilation of a curve $γ$ in $\mathbb R^d$, i.e., $ x + t γ$, $(x,t) \in \mathbb R^d \times (0,\infty)$. For a Borel set $F \subset \mathbb R^d\times (0,\infty)$, we show the unions of curves $\bigcup_{(x,t) \in F} ( x+tγ)$ has Hausdorff dimension at least $α+1$ whenever $F$ has Hausdorff dimension bigger than $α$, $α\in (0, d-1)$. We also obtain results for unions of curves generated by multi-parameter dilation of $γ$. One of the main ingredients is a local smoothing type estimate (for averages over curves) relative to fractal measures.

math.CA↗
arXiv2022

Circular average relative to fractal measures

We prove new $L^p$- $L^q$ estimates for averages over dilates of the circle with respect to $α$-dimensional fractal measure, which unify different types of maximal estimates for the circular average. Our results are consequences of $L^p$- $L^q$ smoothing estimates for the wave operator relative to fractal measures. We also discuss similar results concerning the spherical averages.

math.CA↗
arXiv2022

Pointwise convergence of sequential Schrödinger means

We study pointwise convergence of the fractional Schrödinger means along sequences $t_n$ which converge to zero. Our main result is that bounds on the maximal function $\sup_{n} |e^{it_n(-Δ)^{α/2}} f| $ can be deduced from those on $\sup_{0<t\le 1} |e^{it(-Δ)^{α/2}} f|$ when $\{t_n\}$ is contained in the Lorentz space $\ell^{r,\infty}$. Consequently, our results provide seemingly optimal results in higher dimensions, which extend the recent work of Dimou-Seeger, and Li-Wang-Yan to higher dimensions. Our approach based on a localization argument also works for other dispersive equations and provides alternative proofs of previous results on sequential convergence.

math.CA↗
arXiv2022

Sharp smoothing properties of averages over curves

We prove sharp smoothing properties of the averaging operator defined by convolution with a measure on a smooth nondegenerate curve $γ$ in $\mathbb R^d$, $d\ge 3$. Despite the simple geometric structure of such curves, the sharp smoothing estimates have remained largely unknown except for those in low dimensions. Devising a novel inductive strategy, we obtain the optimal $L^p$ Sobolev regularity estimates, which settle the conjecture raised by Beltran-Guo-Hickman-Seeger. Besides, we show the sharp local smoothing estimates for every $d$. As a result, we establish, for the first time, nontrivial $L^p$ boundedness of the maximal average over dilations of $γ$ for $d\ge 4$.

math.CA↗
arXiv2021

Pointwise convergence of the fractional Schrödinger equation in $\mathbb R^2$

We investigate the pointwise convergence of the solution to the fractional Schrödinger equation in $\mathbb R^2$. By establishing $H^s(\mathbb R^2)-L^3(\mathbb R^2)$ estimates for the associated maximal operator provided that $s>1/3$, we improve the previous result obtained by Miao, Yang, and Zheng. Our estimates extend the refined Strichartz estimates obtained by Du, Guth, and Li to a general class of elliptic functions.

math.AP↗
arXiv2021

Sharp Sobolev regularity of restricted X-ray transforms

We study $L^p$-Sobolev regularity estimate for the restricted X-ray transforms generated by nondegenerate curves. Making use of the inductive strategy in the recent work by the authors, we establish the sharp $L^p$-regularity estimates for the restricted X-ray transforms in $\mathbb R^{d+1}$, $d\ge 3$. This extends the result due to Pramanik and Seeger in $\mathbb R^3$ to every dimension.

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↗
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↗
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↗