Multilinear analysis on metric spaces

The multilinear Calderón–Zygmund theory is developed in the setting of RD-spaces which are spaces of homogeneous type equipped with measures satisfying a reverse doubling condition. The multiple-weight multilinear Calderón–Zygmund theory in this context is also developed in this work. The bilinear T1-theorems for Besov and Triebel–Lizorkin spaces in the full range of exponents are among the main results obtained. Multilinear vector-valued T1 type theorems on Lebesgue spaces, Besov spaces, and Triebel–Lizorkin spaces are also proved. Applications include the boundedness of paraproducts and bilinear multiplier operators on products of Besov and Triebel–Lizorkin spaces. Acknowledgements. Loukas Grafakos is supported by grant DMS 0900946 of the National Science Foundation of the USA. Liguang Liu is supported by the National Natural Science Foundation of China (grant No. 11101425). Diego Maldonado is supported by grant DMS 0901587 of the National Science Foundation of the USA. Dachun Yang (the corresponding author) is supported by the National Natural Science Foundation of China (grant nos. 11171027 & 11361020) and the Specialized Research Fund for the Doctoral Program of Higher Education of China (grant No. 20120003110003). All authors would like to thank the copy editor, Jerzy Trzeciak, for his valuable remarks which made this article more readable. 2010 Mathematics Subject Classification: Primary 42B20, 42B25, 42B35; Secondary 35S50, 42C15, 47G30, 30L99.