Weighted norm inequalities for multilinear strongly singular Calderón–Zygmund operators on RD‐spaces

Let (X,d,μ)$(\mathcal {X}, d, \mu )$ be an RD‐space, namely, a space of homogeneous type in the sense of Coifman and Weiss with the Borel measure μ satisfying the reverse doubling condition on X$\mathcal {X}$. Based on this space, the authors define a multilinear strongly singular Calderón–Zygmund operator whose kernel does not need any size condition and has more singularities near the diagonal than that of a standard multilinear Calderón–Zygmund operator. For such an operator, we establish its boundedness on product of weighted Lebesgue spaces by means of the pointwise estimate for the sharp maximal function. In addition, the endpoint estimates of the type L∞(X)×⋯×L∞(X)→BMO(X)$L^{\infty }(\mathcal {X})\times \cdots \times L^{\infty }(\mathcal {X}) \rightarrow BMO(\mathcal {X})$ are also obtained. Moreover, we prove weighted boundedness results for multilinear commutators generated by multilinear strongly singular Calderón–Zygmund operators and BMO functions. These results contribute to the extension of multilinear strongly singular Calderón–Zygmund operator theory in the Euclidean case to the context of space of homogeneous type.