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 × ⋯ × L p n → L p estimates for all 1 < p 1 , … , p n < ∞ and 0 < p < ∞ , under the condition that 1 p = 1 p 1 + ⋯ + 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.