In the talk we first review the highly anisotropic Hardy spaces introduced in [1]. We will discuss a careful approximation argument that is needed when analyzing dual spaces of Hardy spaces [2]. This approach is essential, since a linear functional, uniformly bounded on all atoms, is not automatically bounded on spaces that have atomic representations [3].

[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).