论文标题
在二维中反面的调查状态
Counterexample to the off-testing condition in two dimensions
论文作者
论文摘要
储层计算是预测湍流的有力工具，其简单的架构具有处理大型系统的计算效率。然而，其实现通常需要完整的状态向量测量和系统非线性知识。我们使用非线性投影函数将系统测量扩展到高维空间，然后将其输入到储层中以获得预测。我们展示了这种储层计算网络在时空混沌系统上的应用，该系统模拟了湍流的若干特征。我们表明，使用径向基函数作为非线性投影器，即使只有部分观测并且不知道控制方程，也能稳健地捕捉复杂的系统非线性。最后，我们表明，当测量稀疏、不完整且带有噪声，甚至控制方程变得不准确时，我们的网络仍然可以产生相当准确的预测，从而为实际湍流系统的无模型预测铺平了道路。
In proving the local $T_b$ Theorem for two weights in one dimension [SaShUT] Sawyer, Shen and Uriarte-Tuero used a basic theorem of Hytönen [Hy] to deal with estimates for measures living in adjacent intervals. Hytönen's theorem states that the off-testing condition for the Hilbert transform is controlled by the Muckenhoupt's $A_2$ and $A^*_2$ conditions. So in attempting to extend the two weight $T_b$ theorem to higher dimensions, it is natural to ask if a higher dimensional analogue of Hytönen's theorem holds that permits analogous control of terms involving measures that live on adjacent cubes. In this paper we show that it is not the case even in the presence of the energy conditions used in one dimension [SaShUT]. Thus, in order to obtain a local $T_b$ theorem in higher dimensions, it will be necessary to find some substantially new arguments to control the notoriously difficult nearby form. More precisely, we show that Hytönen's off-testing condition for the two weight fractional integral and the Riesz transform inequalities is not controlled by Muckenhoupt's $A_2^α$ and $A_2^{α,*}$ conditions and energy conditions.
