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Jongchon Kim

Publications and source records attributed to Jongchon Kim.

16 recordsLinked to original sources
arXiv2026

Weighted decoupling with lower-dimensional frequency localization

We prove weighted $L^2$ and refined $L^p$ decoupling estimates for functions whose Fourier transforms are supported in a small neighborhood of the unit sphere or the truncated paraboloid with an additional lower-dimensional frequency localization property. As a special case, we recover the fractal $L^2$ restriction estimate of Du and Zhang, with a sharper dependence on the density of the weight. We also derive weighted refined decoupling estimates related to the Falconer distance set problem, improving earlier results under the stronger assumption that the underlying weight is $α$-dimensional at every scale.

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arXiv2026

A note on small cap square function and decoupling estimates for the parabola

In this paper, we prove small cap square function and decoupling estimates for the parabola, where the small caps are essentially axis-parallel rectangles of dimensions $δ\times δ^β$ for $0\leq β\leq 1$. Our estimates complement the known results for $1 \leq β\leq 2$ and are sharp up to polylogarithmic factors.

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arXiv2025

Weighted wave envelope estimates for the parabola

In this paper, we extend the Córdoba-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 Schrödinger equation.

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arXiv2024

Nikodym sets and maximal functions associated with spheres

We study spherical analogues of Nikodym sets and related maximal functions. In particular, we prove sharp $L^p$-estimates for Nikodym maximal functions associated with spheres. As a corollary, any Nikodym set for spheres must have full Hausdorff dimension. In addition, we consider a class of maximal functions which contains the spherical maximal function as a special case. We show that $L^p$-estimates for these maximal functions can be deduced from local smoothing estimates for the wave equation relative to fractal measures.

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arXiv2024

Weighted decoupling estimates and the Bochner-Riesz means

We prove new weighted decoupling estimates. As an application, we give an improved sufficient condition for almost everywhere convergence of the Bochner-Riesz means of arbitrary $L^p$ functions for $1<p<2$ in dimensions 2 and 3.

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arXiv2020

Almost-orthogonality principles for certain directional maximal functions

We develop almost-orthogonality principles for maximal functions associated with averages over line segments and directional singular integrals. Using them, we obtain sharp $L^2$-bounds for these maximal functions when the underlying direction set is equidistributed in $\mathbb{S}^{n-1}$.

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arXiv2020

$L^2$ Bounds for a maximal directional Hilbert transform

Given any finite direction set $Ω$ of cardinality $N$ in Euclidean space, we consider the maximal directional Hilbert transform $H_Ω$ associated to this direction set. Our main result provides an essentially sharp uniform bound, depending only on $N$, for the $L^2$ operator norm of $H_Ω$ in dimensions 3 and higher. The main ingredients of the proof consist of polynomial partitioning tools from incidence geometry and an almost-orthogonality principle for $H_Ω$. The latter principle can also be used to analyze special direction sets $Ω$, and derive sharp $L^2$ estimates for the corresponding operator $H_Ω$ that are typically stronger than the uniform $L^2$ bound mentioned above. A number of such examples are discussed.

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arXiv2019

Riesz means of Fourier series and integrals: Strong summability at the critical index

We consider spherical Riesz means of multiple Fourier series and some generalizations. While almost everywhere convergence of Riesz means at the critical index $(d-1)/2$ may fail for functions in the Hardy space $h^1(\mathbb T^d)$, we prove sharp positive results for strong summability almost everywhere. For functions in $L^p(\mathbb T^d)$, $1<p<2$, we consider Riesz means at the critical index $d(1/p-1/2)-1/2$ and prove an almost sharp theorem on strong summability. The results follow via transference from corresponding results for Fourier integrals. We include an endpoint bound on maximal operators associated with generalized Riesz means on Hardy spaces $H^p(\mathbb R^d)$ for $0<p<1$.

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arXiv2018

On the averaged Green's function of an elliptic equation with random coefficients

We consider a divergence-form elliptic difference operator on the lattice $\mathbb{Z}^d$, with a coefficient matrix that is an i.i.d. perturbation of the identity matrix. Recently, Bourgain introduced novel techniques from harmonic analysis to prove the convergence of the Feshbach-Schur perturbation series related to the averaged Green's function of this model. Our main contribution is a refinement of Bourgain's approach which improves the key decay rate from $-2d+ε$ to $-3d+ε$. (The optimal decay rate is conjectured to be $-3d$.) As an application, we derive estimates on higher derivatives of the averaged Green's function which go beyond the second derivatives considered by Delmotte-Deuschel and related works.

math.AP↗
arXiv2017

Some remarks on Fourier restriction estimates

We provide $L^p \to L^q$ refinements on some Fourier restriction estimates obtained using polynomial partitioning. Let $S\subset \mathbb{R}^3$ be a compact $C^\infty$ surface with strictly positive second fundamental form. We derive sharp $L^p(S) \to L^q(\mathbb{R}^3)$ estimates for the associated Fourier extension operator for $q> 3.25$ and $q\geq 2p'$ from an estimate of Guth that was used to obtain $L^\infty(S) \to L^q(\mathbb{R}^3)$ bounds for $q>3.25$. We present a slightly weaker result when $S$ is the hyperbolic paraboloid in $\mathbb{R}^3$ based on the work of Cho and Lee. Finally, we give some refinements for the truncated paraboloid in higher dimensions.

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arXiv2016

A characterization of maximal operators associated with radial Fourier multipliers

We give a simple necessary and sufficient condition for maximal operators associated with radial Fourier multipliers to be bounded on $L^p_{rad}$ and $L^p$ for certain $p$ greater than $2$. The range of exponents obtained for the $L^p_{rad}$ characterization is optimal for the given condition. The $L^p$ characterization is derived from an inequality of Heo, Nazarov, and Seeger regarding a characterization of radial Fourier multipliers.

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arXiv2016

Endpoint bounds for a class of spectral multipliers on compact manifolds

It is well known that the Stein-Tomas $L^2$ Fourier restriction theorem can be used to derive sharp $L^p$ bounds for radial Fourier multipliers such as the Bochner-Riesz means. In a similar manner, $L^p \to L^2$ estimates for spectral projection operators have been utilized in order to obtain sharp $L^p$ bounds for spectral multipliers of self-adjoint elliptic pseudo-differential operators on compact manifolds. In this paper, we refine an endpoint result for spectral multipliers due to Seeger, providing endpoint bounds in terms of Besov spaces. Our proof is based on the ideas from the recent work by Heo, Nazarov and Seeger, and Lee, Rogers and Seeger on radial Fourier multipliers.

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arXiv2016

Endpoint bounds for quasiradial Fourier multipliers

We consider quasiradial Fourier multipliers, i.e. multipliers of the form $m(a(ξ))$ for a class of distance functions $a$. We give a necessary and sufficient condition for the multiplier transformations to be bounded on $L^p$ for a certain range of $p$. In addition, when $m$ is compactly supported in $(0,\infty)$, we give a similar result for associated maximal operators.

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arXiv2015

Endpoint bounds of square functions associated with Hankel multipliers

We prove endpoint bounds for the square function associated with radial Fourier multipliers acting on $L^{p}$ radial functions. This is a consequence of endpoint bounds for a corresponding square function for Hankel multipliers. We obtain a sharp Marcinkiewicz-type multiplier theorem for multivariate Hankel multipliers and $L^p$ bounds of maximal operators generated by Hankel multipliers as corollaries. The proof is built on techniques developed by Garrigós and Seeger for characterizations of Hankel multipliers.

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arXiv2014

On discrete fractional integral operators and related Diophantine equations

We study discrete versions of fractional integral operators along curves and surfaces. $l^p \to l^q$ estimates are obtained from upper bounds of the number of solutions of associated Diophantine systems. In particular, this relates the discrete fractional integral along the curve $γ(m) = (m,m^2,...,m^k)$ to Vinogradov's mean value theorem. Sharp $l^p \to l^q$ estimates of the discrete fractional integral along the hyperbolic paraboloid in $\mathbf{Z}^3$ are also obtained except for endpoints.

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