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 $L^{p_1}\times \cdots \times L^{p_n}\to L^p$ estimates for all $1<p_1,\dots ,p_n<\infty$ and $0<p<\infty$, under the condition that $\frac{1}{p}=\frac{1}{p_1}+\cdots +\frac{1}{p_n}$. 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 $L^2$-average of the multiplier and its derivatives. This approach allows us to address a wider variety of multipliers compared to the Mihlin-type condition.