The Littlewood–Paley decomposition for periodic functions and applications to the Boussinesq equations

The Littlewood–Paley decomposition for functions defined on the whole space [Formula: see text] and related Besov space techniques have become indispensable tools in the study of many partial differential equations (PDEs) with [Formula: see text] as the spatial domain. This paper intends to develop parallel tools for the periodic domain [Formula: see text]. Taking advantage of the boundedness and convergence theory on the square-cutoff Fourier partial sum, we define the Littlewood–Paley decomposition for periodic functions via the square cutoff. We remark that the Littlewood–Paley projections defined via the circular cutoff in [Formula: see text] with [Formula: see text] in the literature do not behave well on the Lebesgue space [Formula: see text] except for [Formula: see text]. We develop a complete set of tools associated with this decomposition, which would be very useful in the study of PDEs defined on [Formula: see text]. As an application of the tools developed here, we study the periodic weak solutions of the [Formula: see text]-dimensional Boussinesq equations with the fractional dissipation [Formula: see text] and without thermal diffusion. We obtain two main results. The first assesses the global existence of [Formula: see text]-weak solutions for any [Formula: see text] and the existence and uniqueness of the [Formula: see text]-weak solutions when [Formula: see text] for [Formula: see text]. The second establishes the zero thermal diffusion limit with an explicit convergence rate.