We consider the problem of the boundedness of maximal operators on \(\mathrm {BMO}_{}^{}\) on shapes in \({\mathbb {R}}^n\). We prove that for bases of shapes with an engulfing property, the corresponding maximal function is bounded from \(\mathrm {BMO}_{}^{}\) to \(\mathrm {BLO}_{}^{}\), generalising a known result of Bennett for the basis of cubes. When the basis of shapes does not possess an engulfing property but exhibits a product structure with respect to lower-dimensional shapes coming from bases that do possess an engulfing property, we show that the corresponding maximal function is bounded from \(\mathrm {BMO}_{}^{}\) to a space we define and call rectangular \(\mathrm {BLO}_{}^{}\).

我们考虑\（{\ mathbb {R}} ^ n \）上的形状上\（\ mathrm {BMO} _ {} ^ {} \）上最大算子的有界性问题。我们证明，对于具有吞没属性的形状基础，相应的最大函数从\（\ mathrm {BMO} _ {} ^ {} \）到\（\ mathrm {BLO} _ {} ^ {} \），以立方为基础概括了Bennett的已知结果。当形状的基础不具有吞噬性，但相对于来自具有吞噬性的碱基的低维形状表现出乘积结构时，我们表明相应的最大函数受\（\ mathrm {BMO} _ {} ^ {} \）到我们定义的空间，并将其称为矩形\（\ mathrm {BLO} _ {} ^ {} \）。