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A condition on a Banach function space X is given under which the Coifman–Fefferman and Fefferman–Stein inequalities on X are equivalent.

We introduce a basis of the Orlik–Solomon algebra labeled by chambers, the so called chamber basis. We consider structure constants of the Orlik–Solomon algebra with respect to the chamber basis and prove that these structure constants recover D. Cohen’s minimal complex from the Aomoto complex.

Numerical modeling of wave propagation in heterogeneous media is important in many applications. Due to their complex nature, direct numerical simulations on the fine grid are prohibitively expensive. It is therefore important to develop efficient and accurate methods that allow the use of coarse grids. In this paper, we present a multiscale finite element method for wave propagation on a coarse grid. The proposed method is based on the generalized multiscale finite element method (GMsFEM) (see [Y. Efendiev, J. Galvis, and T. Hou, <i>J. Comput. Phys.</i>, 251 (2012), pp. 116--135]). To construct multiscale basis functions, we start with two snapshot spaces in each coarse-grid block, where one represents the degrees of freedom on the boundary and the other represents the degrees of freedom in the interior. We use local spectral problems to identify important modes in each snapshot space. These local spectral