Trilinear Fourier multipliers on Hardy spaces

In this paper, we obtain the H1 × H2 × H3 → H boundedness for trilinear Fourier multiplier operators, which is a trilinear analogue of the multiplier theorem of Calderón and Torchinsky [4]. Our result improves the trilinear estimate in [22] by additionally assuming an appropriate vanishing moment condition, which is natural in the boundedness into the Hardy space H for 0 < p ≤ 1.

[1]  Bae Jun Park.  Fourier multiplier theorems for Triebel–Lizorkin spaces , 2017, Mathematische Zeitschrift.

[2]  A. Calderón,et al.  Parabolic maximal functions associated with a distribution, II , 1977 .

[3]  Loukas Grafakos,et al.  The Hörmander multiplier theorem, I: The linear case revisited , 2016, 1607.02620.

[4]  Y. Sawano,et al.  Conditions for boundedness into Hardy spaces , 2017, Mathematische Nachrichten.

[5]  Akihiko Miyachi,et al.  Minimal smoothness conditions for bilinear Fourier multipliers , 2013 .

[6]  A limit case of the Hörmander multiplier theorem , 1988 .

[7]  Akihito Uchiyama.  Characterization of $H^{p}(R^{n})$ in terms of generalized Littlewood-Paley g-functions , 1985 .

[8]  L. Grafakos,et al.  On Multilinear Fourier Multipliers of Limited Smoothness , 2013, Canadian Journal of Mathematics.

[9]  L. Hörmander,et al.  Estimates for translation invariant operators inLp spaces , 1960 .

[10]  Hanh Van Nguyen,et al.  Multilinear multipliers and singular integrals with smooth kernels on Hardy spaces , 2021 .

[11]  A. Calderón,et al.  An atomic decomposition of distributions in parabolic Hp spaces , 1977 .

[12]  Angel B. E. Gatto,et al.  An atomic decomposition of distributions in parabolic Hp spaces , 1980 .

[13]  Multilinear Fourier Multipliers with Minimal Sobolev Regularity, I , 2015, 1504.06915.

[14]  B. Jawerth,et al.  A discrete transform and decompositions of distribution spaces , 1990 .

[15]  Naohito Tomita.  A Hörmander type multiplier theorem for multilinear operators , 2010 .

[16]  Vladimir I. Clue.  Harmonic analysis , 2004, 2004 IEEE Electro/Information Technology Conference.

[17]  Yoshihiro Sawano,et al.  Multiplier conditions for Boundedness into Hardy spaces , 2017, Annales de l'Institut Fourier.

[18]  A. Seeger.  Estimates nearL1 for Fourier multipliers and maximal functions , 1989 .

[19]  R. Song,et al.  A Maximal Function Characterization of the Class H p , 2011 .

[20]  E. Stein,et al.  Hp spaces of several variables , 1972 .

[21]  The Hörmander multiplier theorem, II: The bilinear local $$L^2$$L2 case , 2016, 1607.02622.

[22]  Y. Meyer,et al.  Commutateurs d'intégrales singulières et opérateurs multilinéaires , 1978 .

[23]  Bae Jun Park.  Fourier Multipliers on a Vector-Valued Function Space , 2019, Constructive Approximation.

[24]  B. Jawerth,et al.  The φ-transform and applications to distribution spaces , 1988 .

[25]  Michael Frazier,et al.  Decomposition of Besov Spaces , 2009 .

[26]  Bae Jun Park,et al.  The Hörmander multiplier theorem for n-linear operators , 2021, Mathematische Annalen.

[27]  Weighted norm inequalities for multilinear Fourier multipliers , 2012 .

[28]  Bae Jun Park,et al.  Sharp Hardy space estimates for multipliers , 2019, 1912.01749.

[29]  A. Seeger,et al.  Embeddings for spaces of Lorentz–Sobolev type , 2018, Mathematische Annalen.