Abstract: In the talk we first review the highly anisotropic Hardy spaces introduce in [1]. We will then discuss a careful approximation argument that is needed when analyzing dual spaces of Hardy spaces [2]. As explained in [3], 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).

[1] S. Dekel, P. Petrushev and T. Weissblat, Hardy spaces on $R^n$ with
pointwise variable anisotropy, Fourier Analysis and Applications 17
(2011), 1066-1107.

[2] S. Dekel and T. Weissblat, On duals of anisotropic Hardy spaces,
submitted.

[3] M. Bownik, Boundedness of operators on Hardy spaces via atomic
decompositions, Proc. Amer. Math Soc. 133, 3535-3542 (2005).