Marcinkiewicz-type multiplier theorem for multi-parameter paraproducts

In this paper, we establish Marcinkiewicz-type multiplier theorems for multi-linear and multi-parameter Fourier multiplier operators. For n-linear and multi-parameter Fourier multiplier operators, Muscalu et al. (2004,2006) obtained L^p₁×⋯×L^p_n→L^p estimates for all 1<p₁,…,p_n<∞ and 0<p<∞, under the condition that [Formula presented]=[Formula presented]+⋯+[Formula presented]. Their approach assumed that the multipliers and their derivatives satisfy specific pointwise estimates (the Mihlin-type condition). In contrast, we will consider multi-linear and multi-parameter operators whose multipliers exhibit limited smoothness, characterized by a function space rather than pointwise conditions (the Marcinkiewicz-type condition). The Marcinkiewicz-type multiplier condition addressed in this paper involves the L²-average of the multiplier and its derivatives. This approach allows us to address a wider variety of multipliers compared to the Mihlin-type condition.