Logarithmic Mean Oscillation on the Polydisc, Multi-Parameter Paraproducts and Iterated Commutators

Abstract

We introduce another notion of bounded logarithmic mean oscillation in the \(N\)-torus and give an equivalent definition in terms of boundedness of multi-parameter paraproducts from the dyadic little \(\mathrm {BMO}\), \(\mathrm {bmo}^d(\mathbb {T}^N)\) to the dyadic product \(\mathrm {BMO}\) space, \(\mathrm {BMO}^d(\mathbb {T}^N)\). We also obtain a sufficient condition for the boundedness of the iterated commutators from the subspace of \(\mathrm {bmo}(\mathbb {R}^N)\) consisting of functions with support in \([0,1]^N\) to \(\mathrm {BMO}(\mathbb {R}^N)\).