论文标题
在Bellow-Furstenberg问题的多参数变体中
On a multi-parameter variant of the Bellow-Furstenberg problem
论文作者
论文摘要
我们证明了在$ l^p $，$ p \ in（1，\ infty）$上几乎无处不在的融合，对于某些多参数多项式ergodic平均值，我们通过确定相应的多参数最大和振动不平等。尤其是我们的结果，对Bellow-Furstenberg问题的多参数变体给出了肯定的答案。本文也是对多参数振荡半月型的首次系统处理，该振荡半月型可以有效处理具有算术特征的多参数收敛问题。我们主要结果的证明方法开发了多参数指数总和的估计值，并在向后的牛顿图的几何形状的上下文中介绍了所谓的多参数圆方法的新思想，这些几何形状由定义我们的精神平均的多项式形状所决定。
We prove convergence in norm and pointwise almost everywhere on $L^p$, $p\in (1,\infty)$, for certain multi-parameter polynomial ergodic averages by establishing the corresponding multi-parameter maximal and oscillation inequalities. Our result, in particular, gives an affirmative answer to a multi-parameter variant of the Bellow-Furstenberg problem. This paper is also the first systematic treatment of multi-parameter oscillation semi-norms which allows an efficient handling of multi-parameter pointwise convergence problems with arithmetic features. The methods of proof of our main result develop estimates for multi-parameter exponential sums, as well as introduce new ideas from the so-called multi-parameter circle method in the context of the geometry of backwards Newton diagrams that are dictated by the shape of the polynomials defining our ergodic averages.
