AI 中文总结
研究多参数奇异指数和一致有界的充要条件，通过改进多参数圆法，给出在系数和参数上一致的界，在充分条件下还证明了相关离散多重希尔伯特变换的\(\ell^p\)有界性。
AI 中文摘要
我们建立了多参数奇异指数和\(\sum_{|t_1|\le N_1,\dots,|t_k|\le N_k} \frac{e^{2\pi i P(t_1,\dots,t_k)}}{t_1\cdots t_k}\)一致有界的充要条件，其中\(P:\mathbb{Z}^k\to\mathbb{R}\)是形如\(P(t)=\sum_{\mathfrak{m}\in \Lambda} c_{\mathfrak{m}}\, t^{\mathfrak{m}}\)的实系数多项式。所得界在系数\(c_{\mathfrak{m}}\)和\(N_i\)上是一致的。为此，我们对文献中引入的多参数圆法进行了高维改进，为处理一般多项式映射提供了更灵活的框架。在充分条件下，我们进一步证明了相关离散多重希尔伯特变换的\(\ell^p\)有界性。
英文摘要
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.
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