Abstract
The main purpose of this paper is to establish the Hörmander–Mihlin type theorem for Fourier multipliers with optimal smoothness on k-parameter Hardy spaces for k ≥ 3 using the multi-parameter Littlewood–Paley theory. For the sake of convenience and simplicity, we only consider the case k = 3, and the method works for all the cases k ≥ 3:
where \(x = \left( {{x_1},{x_2},{x_3}} \right) \in {\mathbb{R}^{{n_1}}} \times {\mathbb{R}^{{n_2}}} \times {\mathbb{R}^{{n_3}}}\) and \(\xi = \left( {{\xi _1},{\xi _2},{\xi _3}} \right) \in {\mathbb{R}^{{n_1}}} \times {\mathbb{R}^{{n_2}}} \times {\mathbb{R}^{{n_3}}}\). One of our main results is the following: Assume that m(ξ) is a function on \({\mathbb{R}^{{n_1} + {n_2} + {n_3}}}\) satisfying \(\mathop {\sup }\limits_{j,k,l \in \mathbb{Z}} {\left\| {{m_{j,k,l}}} \right\|_{{W^{\left( {{s_1},{s_2},{s_3}} \right)}}}} < \infty \) with \(s_{i} > n_{i}(\frac{1}{p}-\frac{1}{2})\) for 1 ≤ i ≤ 3. Then T m is bounded from \({H^p}\left( {{\mathbb{R}^{{n_1}}} \times {\mathbb{R}^{{n_2}}} \times {\mathbb{R}^{{n_3}}}} \right)\) to \({H^p}\left( {{\mathbb{R}^{{n_1}}} \times {\mathbb{R}^{{n_2}}} \times {\mathbb{R}^{{n_3}}}} \right)\) for all 0 < p ≤ 1 and \(\left\| {{T_m}} \right\|{}_{{H^P} \to {H^P}} \lesssim \mathop {\sup }\limits_{j,k,l \in \mathbb{Z}} {\left\| {{m_{j,k,l}}} \right\|_{{W^{\left( {{s_1},{s_{2,}}{s_3}} \right)}}}}\). Moreover, the smoothness assumption on s i for 1 ≤ i ≤ 3 is optimal. Here we have used the notations m j,k,l (ξ) = m(2j ξ 1, 2k ξ 2, 2l ξ 3)Ψ(ξ 1)Ψ(ξ 2)Ψ(ξ 3) and Ψ(ξ i ) is a suitable cut-off function on \({\mathbb{R}^{{n_i}}}\) for 1 ≤ i ≤ 3, and \({W^{\left( {{s_1},{s_2},{s_3}} \right)}}\) is a three-parameter Sobolev space on \({\mathbb{R}^{{n_1}}} \times {\mathbb{R}^{{n_2}}} \times {\mathbb{R}^{{n_3}}}\).
Because the Fefferman criterion breaks down in three parameters or more, we consider the L p boundedness of the Littlewood–Paley square function of T m f to establish its boundedness on the multi-parameter Hardy spaces.
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Acknowledgements
After the completion of this paper, the author learned from Professor Jill Pipher that Theorem 1.9 has been established by Carbery and Seeger in their earlier paper [2] using a very different method from ours by considering vector-valued rectangle atoms to avoid using the R. Fefferman criterion which breaks down in three or more parameters. The author wishes to thank Professor Pipher very much for bringing to his attention the work [2].
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Supported by NNSF of China (Grant No. 11371056)
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Chen, J. Hörmander type theorem for Fourier multipliers with optimal smoothness on Hardy spaces of arbitrary number of parameters. Acta. Math. Sin.-English Ser. 33, 1083–1106 (2017). https://doi.org/10.1007/s10114-017-6526-3
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DOI: https://doi.org/10.1007/s10114-017-6526-3
Keywords
- Hörmander multiplier
- minimal smoothness condition
- Littlewood–Paley’s inequality
- multi-parameter Hardy H p spaces
- multi-parameter Sobolev spaces