Maximal functions, Hardy spaces and Fourier multiplier theorems on unbounded Vilenkin groups

We consider relations between various existing concepts of Hardy spaces on Vilenkin groups. The problem of an appropriate Fourier multiplier theorem for unbounded Vilenkin groups is revisited and the conditions made more precise under which such a theorem is established. Theorem 3.1 from Avdispahić and Memić (2010) [1] is rectified.