New local \(T1\) theorems on non-homogeneous spaces (Q6589792)

The famous \(T1\) theorem was given by \textit{G. David} and \textit{J.-L. Journé} [Ann. Math. (2) 120, 371--397 (1984; Zbl 0567.47025)], which provides the \(L^2\) boundedness for singular integral operators of nonconvolution type. The name ``\(T1\)'' stems from its key conditions \(T1\in \mathrm{BMO}\) and \(T^\ast 1\in\mathrm{BMO}\).\N\NSome researchers have tried to find some alternative conditions for \(T\), especially by using the characteristic functions of cubes to replace the function \(1\), since there may exist better properties when \(T\) and \(T^\ast\) act on functions with compact supports instead of which support on the whole space. As the main achievement, \textit{M. Christ} [Colloq. Math. 60/61, No. 2, 601--628 (1990; Zbl 0758.42009)] obtained the local \(T1\) theorem in 1990. On the other hand, in recent decades, many results in harmonic analysis, including the \(T1\) theorem, have been extended from \(\mathbb{R}^n\) to general metric measure spaces, such as the space with non-doubling measures. Most of the conclusions on this topic are obtained from the randomization methods, see [\textit{F. Nazarov} et al., Int. Math. Res. Not. 1997, No. 15, 703--726 (1997; Zbl 0889.42013); Duke Math. J. 113, No. 2, 259--312 (2002; Zbl 1055.47027)].\N\NThis paper presents another method for obtaining the local \(T1\) theorem on spaces with power growth measure. Specifically, by using some grids of \(n\)-dimensional cubes to ``test'' the operator \(T\), the author gives some equivalent characterizations for the \(L^p\) boundedness of singular integral, which are better than results obtained by randomization methods, see Theorem 4.1. Besides, as an application of Theorem 4.1, the author describes the measure \(\mu\), which lets the operator generated by Cauchy integral be compact on \(L^p(\mathbb{C},\mu)\), see Theorem 4.4.\N\NThis paper is well written and the proofs are correct. The main conclusions are certainly new, non-trivial, and interesting to harmonic analysis.