We study the maximal estimates for the bilinear spherical average and the bilinear Bochner-Riesz operator. Firstly, we obtain $L^p\times L^q \to L^r$ estimates for the bilinear spherical maximal function on the optimal range. Thus, we settle the problem which was previously considered by Geba, Greenleaf, Iosevich, Palsson and Sawyer, later Barrionevo, Grafakos, D. He, Honzik and Oliveira, and recently Heo, Hong and Yang. Secondly, we consider $L^p\times L^q \to L^r$ estimates for the maximal bilinear Bochner-Riesz operators and improve the previous known results. For the purpose we draw a connection between the maximal estimates and the square function estimates for the classical Bochner-Riesz operators.

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双线性球面平均值和双线性 Bochner-Riesz 算子的最大估计值

我们研究了双线性球面平均值和双线性 Bochner-Riesz 算子的最大估计。首先，我们获得最佳范围内双线性球面极大函数的 $L^p\times L^q \to L^r$ 估计。因此，我们解决了先前由 Geba、Greenleaf、Iosevich、Palsson 和 Sawyer，后来的 Barrionevo、Grafakos、D. He、Honzik 和 Oliveira，以及最近的 Heo、Hong 和 Yang 考虑的问题。其次，我们考虑最大双线性 Bochner-Riesz 算子的 $L^p\times L^q \to L^r$ 估计并改进先前已知的结果。为此，我们在经典 Bochner-Riesz 算子的最大估计和平方函数估计之间建立了联系。