$L^2\times L^2\times L^2\to L^{2/3}$ boundedness for trilinear multiplier operator

TL;DR

This paper establishes boundedness results for a trilinear multiplier operator acting from three L^2 spaces into L^{2/3}, based on the multiplier's L^q integrability and smoothness conditions, without requiring decay.

Contribution

It provides new boundedness criteria for trilinear multipliers with minimal smoothness assumptions and no decay, extending previous results in multilinear harmonic analysis.

Findings

Boundedness of T_m from L^2×L^2×L^2 to L^{2/3} established.

Boundedness depends on the L^q norm of the multiplier m.

The operator norm is controlled by orm{m}_{L^q}^{q/3}.

Abstract

This paper discusses the boundedness of the trilinear multiplier operator $T_{m}(f_1,f_2,f_3)$, when the multiplier satisfies a certain degree of smoothness but with no decaying condition and is $L^{q}$-integrable with an admissible range of $q$. The boundedness is stated in the terms of $\|m\|_{L^{q}}$. In particular, \begin{equation*}\|T_{m}\|_{L^2\times L^2\times L^2\to L^{2/3}}\lesssim\|m\|_{L^{q}}^{q/3}.\end{equation*}