arXiv subjects

Hoyoung Song

Publications and source records attributed to Hoyoung Song.

5 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 $$\sum_{1\le|t_1|\le N_1,\cdots,1\le|t_k|\le N_k} \frac{e^{2πi P(t_1,\dots,t_k)}}{t_1\cdots t_k},$$ where $P:\mathbb{Z}^k\to\mathbb{R}$ is a polynomial of the form $P(t)=\sum_{\mathfrak{m}\in Λ} c_{\mathfrak{m}}\, t^{\mathfrak{m}},$ 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.

math.CA↗

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.

math.CA↗

arXiv2025

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 $1/t_2$.

math.CA↗

arXiv2023

Weak type 1-1 bound of multi-parameter maximal function

We define the mulati-parameter maximal function $\mathcal{M}$ as $$ \mathcal{M} f(x)=\sup _{0<h_1,h_2,\cdots,h_n<1} \frac{1}{h_1h_2\cdots h_n}\left|\int_0^{h_1}\cdots \int_0^{h_n} f(x-P(t_1,\cdots,t_n)) \mathrm{d}t_1\cdots \mathrm{d} t_n\right| $$ where $P(t_1,t_2,\cdots,t_n)$ is a real-valued multi-parameter polynomial of real variables $t_1,t_2,\cdots,t_n$. Then, we prove that $\mathcal{M}$ is of weak-type 1-1 with a bound that depends only on the coefficients of $P(t_1,t_2,\cdots,t_n)$.

math.CA↗

arXiv2023

The Proof of restriction conjecture In $\mathbb{R}^{3}$

If S is a smooth compact surface in $\mathbb{R}^{3}$ with strictly positive second fundamental form, and $E_S$ is the corresponding extension operator, then we prove that for all $p > 3$, $\left\|E_S f\right\|_{L^p\left(\mathbb{R}^3\right)} \leq C(p, S)\|f\|_{L^{\infty}(S)}.$ The proof of restriction conjecture in $\mathbb{R}^{3}$ implies that Kakeya set conjecture is true when n=3.

math.CA↗