Multi-Parameter Exponential Sums with Product Hilbert Kernels

Abstract

We establish necessary and sufficient conditions for the uniform boundedness of the multi-parameter exponential sums with product Hilbert kernels $$\sum _{1\le |t_1|\le N_1,\cdots ,1\le |t_k|\le N_k}\frac{e^{2\ddot{I}€iP(t_1,\dots ,t_k)}}{t_1\cdots t_k},$$ where $P:{\mathbb{Z}}^k\to \mathbb{R}$ is a polynomial of the form $P(t)={\sum }_{\mathfrak{m}\in \hat{I}›}{c}_{\mathfrak{m}}{t}^{\mathfrak{m}},$ with real coefficients. The resulting bound is uniform in both the coefficients $c_{\mathfrak{m}}$ and the truncation parameters $N_1,\dots ,N_k$. 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.

Minor typos were corrected, with additional explanations added