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Joonil Kim

Publications and source records attributed to Joonil Kim.

10 recordsLinked to original sources
arXiv2026

Multi-Parameter Exponential Sums with Product Hilbert Kernels

We establish necessary and sufficient conditions for the uniform boundedness of the multi-parameter exponential sums with product Hilbert kernels $$\sum_{1\le|t_1|\le N_1,\cdots,1\le|t_k|\le N_k} \frac{e^{2\pi i P(t_1,\dots,t_k)}}{t_1\cdots t_k},$$ where $P:\mathbb{Z}^k\to\mathbb{R}$ is a polynomial of the form $P(t)=\sum_{\mathfrak{m}\in \Lambda} c_{\mathfrak{m}}\, t^{\mathfrak{m}},$ with real coefficients. The resulting bound is uniform in both the coefficients $c_{\mathfrak m}$ and the truncation parameters $N_1,\ldots,N_k$. To this end, we develop a higher-dimensional version of the multi-parameter circle method. Under the sufficient condition, we further prove $\ell^p$-boundedness of the associated discrete multiple Hilbert transform.

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arXiv2026

Discrete analogues in harmonic analysis: Multi-parameter Radon averages

In this paper we study maximal and oscillation inequalities for multi-parameter discrete Radon averaging operators. We develop a robust variant of the multi-parameter circle method within the framework of Discrete Analogues in Harmonic Analysis. In particular, this gives quantitative estimates for these averages and their underlying Fourier multipliers which reveals an interesting major-arcs rigidity phenomenon. As a consequence, we completely resolve in the affirmative the multi-parameter Bellow--Furstenberg problem in pointwise ergodic theory.

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arXiv2025

Restricted weak type endpoint estimate for the spherical maximal operators on the Heisenberg group

Let $\mathbb{H}^n$ denote the Heisenberg group, identified with $\mathbb{R}^d \times \mathbb{R}$, where $d = 2n$ and $n \in \mathbb{N}$. We consider the spherical maximal operator $\mathcal{M}$ associated with the sphere $S^{d-1}$ embedded in the horizontal subspace $\mathbb{R}^d \times \{0\}$ of $\mathbb{H}^n$. It is known that $\mathcal{M}$ is bounded on $L^p(\mathbb{H}^n)$ if and only if $p \in (\tfrac{d}{d-1}, \infty]$. In this paper, we establish a restricted weak type $(p,p)$ estimate at the endpoint $p = \tfrac{d}{d-1}$ for $\mathcal{M}$, provided $d \ge 3$.

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arXiv2025

Lacunary elliptic maximal operator on the Heisenberg group

In this paper, we prove \( L^p \) boundedness results for lacunary elliptic maximal operators on the Heisenberg group. Furthermore, we extend these \( L^p \) estimates from skew-symmetric matrices, which naturally arise in Heisenberg group operations, to arbitrary matrices \( A \), investigating how the curvature induced by \( A \) governs the \( L^p \) boundedness of lacunary circular and elliptic maximal operators. Specifically, we provide necessary and sufficient conditions on \( A \) that determine whether these operators are bounded or unbounded on \( L^p \).

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arXiv2022

Sublevel Set Estimates over Global Domains

Since Varchenko's seminal paper, the asymptotics of oscillatory integrals and related problems have been elucidated through the Newton polyhedra associated with the phase $P$. The supports of those integrals are concentrated on sufficiently small neighborhoods. The aim of this paper is to investigate the estimates of sub-level-sets and oscillatory integrals whose supports are global domains $D$. A basic model of $D$ is $ \mathbb{R}^d$. For this purpose, we define the Newton polyhedra associated with $(P,D)$ and establish analogues of Varchenko's theorem in global domains $D$, under non-degeneracy conditions of $P$.

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arXiv2022

Discrete Double Hilbert Transforms Along Polynomial Surfaces

We obtain a necessary and sufficient condition on a polynomial $P(t_1,t_2)$ for the $\ell^{p}$ boundedness of the discrete double Hilbert transforms associated with $P(t)$ for $1 < p < \infty$. The proof is based on the multi-parameter circle method treating the cases of $|t_1|\not\approx |t_2|$ arising from $1/t_1$ and $1/t_2$.

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arXiv2020

Annulus Maximal Averages on Variable Hyperplanes

By giving a thin width of $0<δ\ll 1$ to both a unit circle and a unit line, we set an annulus and a tube on the Euclidean plane $\mathbb{R}^2$. Consider the maximal means $M_δ$ over dilations of the annulus, and $N_δ$ over rotations of the tube. It is known that their operator norms on $L^2(\mathbb{R}^2)$ are $O(|\log 1/δ|^{1/2})$. In this paper, we study the maximal averages $\mathcal{M}^A_δ$ and $\mathcal{N}^A_δ$ over those annuli and tubes now imbedded on the variable hyperplanes $(x,x_3)+\left\{\left(y, \langle A(x), y\rangle\right): y\in\mathbb{R}^2\right\}\subset \mathbb{R}^3$ where $A$ is a $2\times 2$ matrix. The model hyperplane is the horizontal plane of the Heisenberg group when $A$ is the skew--symmetric matrix denoted by $E$. It turns out that a rank of matrix $EA+(EA)^T$ or $A+A^T$ determines $\|\mathcal{M}^A_δ\|_{op} $ or $\|\mathcal{N}^A_δ\|_{op}$ respectively. In the higher dimension, the corresponding spherical maximal means is bounded in $L^p$ if $A$ has only complex eigenvalues.

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arXiv2019

Circular maximal functions on the Heisenberg group

We prove the $L^p$ boundedness of the circular maximal function on the Heisenberg group $\mathbb{H}^1$ for $2<p\le \infty$. The proof is based on the square sum estimate associated with the $2\times 2$ cone $|(\xi_1',\xi_2')|= |(\xi_3',\xi_4')| $ of the phase space arising from the vector fields $X_1,X_2,tX_3,\partial/\partial t$ on the Heisenberg group, rather than the $2\times 1$ cone $ |(\xi_1,\xi_2)|= |\xi_3|$ of the frequency space arising from $\partial/\partial x_1, \partial/\partial x_2, \partial/\partial t$ on the Euclidean space.

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