Abstract: In the talk we first review the highly anisotropic Hardy spaces introduce in . We will then discuss a careful approximation argument that is needed when analyzing dual spaces of Hardy spaces . As explained in , one cannot assume that a linear functional, uniformly bounded on all atoms, is automatically bounded on spaces that have atomic representations (e.g. Hardy spaces).
 S. Dekel, P. Petrushev and T. Weissblat, Hardy spaces on $R^n$ with
pointwise variable anisotropy, Fourier Analysis and Applications 17
 S. Dekel and T. Weissblat, On duals of anisotropic Hardy spaces,
 M. Bownik, Boundedness of operators on Hardy spaces via atomic
decompositions, Proc. Amer. Math Soc. 133, 3535-3542 (2005).