标题： Carleson 积分在加倍度量测度空间上的性质 显示英文标题

标题： Carleson operators on doubling metric measure spaces

摘要： 双重度量测度空间为奇异积分算子提供了自然的框架。 相比之下，对最大调制奇异积分算子，即所谓的Carleson算子的研究，主要局限于欧几里得空间，并且调制函数如由代数方法定义的多项式。 我们提出了一种在双重度量测度空间上调制函数的一般公理化方法，并在定理1.1和定理1.2中证明了相应的Carleson算子的$L^p$界。 这推广了关于Carleson算子的经典和现代结果。 除了这里提供的证明外，我们的主要结果已使用语言Lean和库mathlib进行了计算机验证，如兄弟通讯arXiv:2405.06423中所记录的。 显示英文摘要

摘要： Doubling metric measure spaces provide a natural framework for singular integral operators. In contrast, the study of maximally modulated singular integral operators, the so-called Carleson operators, has largely been limited to Euclidean space with modulation functions such as polynomials defined by algebraic means. We present a general axiomatic approach to modulation functions on doubling metric measure spaces and prove $L^p$ bounds for the corresponding Carleson operators in Theorem 1.1 and Theorem 1.2. This generalizes classical and modern results on Carleson operators. In addition to the proofs presented here, our main results have been computer verified using the language Lean and the library mathlib, as documented in the sibling communication arXiv:2405.06423.