Multilinear Spectral Multipliers on Lie Groups of Polynomial Growth
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Multilinear Spectral Multipliers on Lie Groups of Polynomial Growth Jingxuan Fang1 · Hongbo Li1 · Jiman Zhao2 Received: 3 June 2020 / Accepted: 25 September 2020 © Mathematica Josephina, Inc. 2020
Abstract In this paper, on Lie groups of polynomial growth, we make an estimation of the kernel function of multilinear spectral multipliers. Then as an application of this estimation, we prove the L p boundedness and weighted L p boundedness of such multilinear spectral multipliers. Keywords Multilinear spectral multipliers · Lie groups of polynomial growth · L p boundedness · Weighted L p boundedness Mathematics Subject Classification 42B15 · 43A80 · 47B40
1 Introduction Fourier multiplier is of great importance in harmonic analysis. Some classical operators, such as convolution operators, fractional differential operators, and pseu-
Jingxuan Fang: Supported by Chinese Postdoctoral Science Foundation Grant No. 2020M670489. Hongbo Li: supported by NSFC Grant No. 11671388, National Key Basic Research Program 2018YFA0704705. Jiman Zhao: the corresponding author, supported by National Natural Science Foundation of China (Grant Nos. 11471040 and 11761131002).
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Jiman Zhao [email protected] Jingxuan Fang [email protected] Hongbo Li [email protected]
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Academy of Mathematics and Systems Science, University of Chinese Academy of Sciences, Chinese Academy of Sciences, Beijing 100190, China
2
School of Mathematical Sciences, Key Laboratory of Mathematics and Complex Systems, Ministry of Education, Institution of Mathematics and Mathematical Education, Beijing Normal University, Beijing 100875, China
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J. Fang et al.
dodifferential operators with constant coefficients, have a close relationship with Fourier multipliers. The boundedness properties of Fourier multipliers are of great value in investigating the formal solution of linear PDEs with constant coefficients and the convergence of Fourier series. Marcinkiewicz [31] proved the original multiplier theorem in 1939. After that, Mihlin multiplier theorem and Hörmander multiplier theorem are obtained by the best known works Mihlin [36] and Hörmander [27], respectively. Fourier multipliers have attracted much attention. For example, Auscher [3], Bernicot and Kova [4], Besov [5], Grafakos et al. [23–25,37,44], Chen et al. [8], Christ et al. [12], Kolomoitsev [28], Lu et al. [30], Yabuta [49], Yang et al. [50], and Zhao et al. [51] study the boundedness of Fourier multipliers and multilinear Fourier multipliers. Mathematical spectral theory can be traced back to the 20th century with the classical works of Hilbert [26]. Spectral theory can be applied in studying many physics operators see e.g. [13]. Besides, spectral theory has been applied in many branches of modern mathematics, physics, and engineering science, especially in theory of partial differential equations, quantum mechanics, signal processing, and ergodic theory. Wendel [48] constructs the forerunner work of spectral multipliers. After that, the boundedness properties of the spectral integrals attract a lot o
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