We consider a class of multiparameter singular Radon integral operators on the Heisenberg group ${\mathbb H}^1$ where the underlying variety is the graph of a polynomial. A remarkable difference with the euclidean case, where Heisenberg convolution is replaced by euclidean convolution, is that the operators on the Heisenberg group are always $L^2$ bounded. This is not the case in the euclidean setting where $L^2$ boundedness depends on the polynomial defining the underlying surface. Here we uncover some new, interesting phenomena. For example, although the Heisenberg group operators are always $L^2$ bounded, the bounds are {\it not} uniform in the coefficients of polynomials with fixed degree. When we ask for which polynoimals uniform $L^2$ bounds hold, we arrive at the {\it same} class where uniform bounds hold in the euclidean case.

中文翻译：

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海森堡群上的多参数奇异积分：均匀估计

我们考虑海森堡群 ${\mathbb H}^1$ 上的一类多参数奇异 Radon 积分算子，其中潜在的多样性是多项式的图。海森堡卷积被欧氏卷积取代的欧几里得情况的一个显着区别是海森堡群上的算子总是$L^2$有界。在欧几里得设置中情况并非如此，其中 $L^2$ 有界取决于定义底层表面的多项式。在这里，我们发现了一些新的、有趣的现象。例如，尽管 Heisenberg 群算子总是 $L^2$ 有界，但在固定次数多项式的系数中，边界是{\it not}一致的。当我们询问哪些多项式一致 $L^2$ 边界成立时，我们到达了 {\it same} 类，其中一致边界在欧几里得情况下成立。