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We consider a Cauchy problem of the Boltzmann equation without angular cutoff near the global Maxwellian. When a spatial domain is the torus, it is proved that the problem has a unique solution for small data in the Wiener space, the collection of functions whose Fourier coefficients are absolutely summable. Due to its Banach-algebra property, we do not need the Sobolev embedding in this space. Following this result, we consider the problem on the whole space. In this case, the control of the $L^1$ norm on the Fourier side is not sufficient due to low-frequency terms. Therefore, inspired by Kawashima-Nishibata-Nishikawa's work on the viscous conservation laws, we employ the $L^p$ norm estimates with respect to the frequency to control such parts. This $L^1 \cap L^p$ strategy will close a priori estimates when combined with a time-weighted energy method. This work is based on a joint work with Renjun Duan (Chinese University of Hong Kong) and Yoshihiro Ueda (Kobe University).

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