Multi-Parameter Exponential Sums with Product Hilbert Kernels

We establish necessary and sufficient conditions for the uniform boundedness of the multi-parameter singular exponential sum $$\sum_{|t_1|\le N_1,\dots,|t_k|\le N_k} \frac{e^{2\pi i P(t_1,\dots,t_k)}}{t_1\cdots t_k},$$ 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 coefficients $c_{\mathfrak{m}}$ and $N_i$'s. To prove this, we develop a higher-dimensional refinement of the multi-parameter circle method introduced in \cite{B4,KS}, thereby providing a more flexible framework for treating general polynomial mappings. Under the sufficient condition, we further prove $\ell^p$-boundedness of the associated discrete multiple Hilbert transform.