Singular integrals on Carleson measure spaces \(\text{CMO}^{p}\) on product spaces of homogeneous type (Q2839361)

It is well known that in the classical one-parameter case the dual space \(\text{BMO}\) of the Hardy space \(H^1\) can be characterized by Carleson measures. \textit{S.-Y. A. Chang} and \textit{R. Fefferman} [Ann. Math. (2) 112, 179--201 (1980; Zbl 0451.42014)] proved that the dual of the product \(H^1\) is characterized by the product Carleson measures. \textit{Y. Han, J. Li} and \textit{G. Lu} [Trans. Am. Math. Soc. 365, No. 1, 319-360 (2013; Zbl 1275.42035)] introduced the generalized Carleson measure space \(\text{CMO}^p\) and proved that the dual of \(H^p(M_1 \times M_2)\) is \(\text{CMO}^p (M_1 \times M_2)\), where \(M_1\) and \(M_2\) are spaces of homogeneous type in the sense of Coifman and Weiss. \textit{A. Nagel} and \textit{E. M. Stein} [Rev. Mat. Iberoam. 20, No. 2, 531--561 (2004; Zbl 1057.42016)] proved that non-isotropic smooth operators are bounded on \(L^p(M)\) where \(1<p<\infty\).NEWLINENEWLINEThe authors prove that non-isotropic smooth operators are bounded on \(\text{CMO}^p (M_1 \times M_2)\) where \(p_0 < p <1\). Here \(p_0\) depends on the homogeneous dimensions of the measures and the quasi-metrics on \(M_1\) and \(M_2\).