Multi-Parameter Exponential Sums with Product Hilbert Kernels

2026/07/28 by Joonil Kim, Hoyoung Song
Mathematics · #math.CA

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Minor typos were corrected, with additional explanations added

arxiv created 2026/07/31 · arxiv updated 2026/08/04

Abstract

We establish necessary and sufficient conditions for the uniform boundedness of the multi-parameter exponential sums with product Hilbert kernels ∑_{1≤|t1|≤ N1,⋯,1≤|tk|≤ Nk} \frace^{2πi P(t₁,…,t_k)}t₁⋯ t_k, where P:ℤ^k→ℝ is a polynomial of the form P(t)=∑_\mathfrakm∈ Λ c_\mathfrakm t^\mathfrakm, with real coefficients. The resulting bound is uniform in both the coefficients c_{\mathfrak m} and the truncation parameters N₁,…,N_k. To this end, we develop a higher-dimensional version of the multi-parameter circle method. Under the sufficient condition, we further prove ℓ^p-boundedness of the associated discrete multiple Hilbert transform.

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