Marcinkiewicz-type multiplier theorem for multi-parameter paraproducts

초록

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 Lp<inf>1</inf>×⋯×Lp<inf>n</inf>→Lp estimates for all 1<p<inf>1</inf>,…,p<inf>n</inf><∞ 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. © 2025 Elsevier B.V., All rights reserved.

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