We prove a general sparse domination theorem in a space of homogeneous type, in which a vector-valued operator is controlled pointwise by a positive, local expression called a sparse operator. We use the structure of the operator to get sparse domination in which the usual \(\ell ^1\)-sum in the sparse operator is replaced by an \(\ell ^r\)-sum. This sparse domination theorem is applicable to various operators from both harmonic analysis and (S)PDE. Using our main theorem, we prove the \(A_2\)-theorem for vector-valued Calderón–Zygmund operators in a space of homogeneous type, from which we deduce an anisotropic, mixed-norm Mihlin multiplier theorem. Furthermore, we show quantitative weighted norm inequalities for the Rademacher maximal operator, for which Banach space geometry plays a major role.

我们证明了在齐次类型空间中的一般稀疏控制定理，其中向量值算子由称为稀疏算子的正局部表达式逐点控制。我们使用运算符的结构来获得稀疏控制，其中稀疏运算符中通常的\（\ ell ^ 1 \）- sum被\（\ ell ^ r \）- sum代替。这种稀疏的控制定理适用于谐波分析和（S）PDE的各种算子。使用主定理，我们证明\（A_2 \）齐型空间中的矢量值Calderón–Zygmund算子定理，从中我们得出各向异性的，混合范数的Mihlin乘子定理。此外，我们显示了Rademacher极大算子的量化加权范数不等式，对此Banach空间几何起主要作用。