The dual of the martingale Hardy space \({\mathcal H}_{\mathbf{\Phi}}\) with general Young function \(\Phi\) (Q1826209)

It has been known for a while that for \(1\leq p\leq 2\) the dual of the Hardy space \({\mathcal H}_ p\) is the \({\mathcal K}_ q\) space, where \(p^{- 1}+q^{-1}=1\) (esp. \(q=\infty\) and \({\mathcal K}_{\infty}=BMO\) if \(p=1)\) [see \textit{A. M. Garsia}, ``Martingale inequalities. Seminar notes on recent progress.'' (1973; Zbl 0284.60046)]. \textit{N. L. Bassily} and \textit{J. Mogyoródi} [Ann. Univ. Sci. Budap. Rolando Eötvös, Sect. Math. 27(1985), 205-214, 215-227 (1984; Zbl 0581.60035, Zbl 0581.60036, respectively)] introduced the notion of the Hardy space \({\mathcal H}_{\Phi}\) with a general Young function \(\Phi\). In the present paper it is shown that the dual of the Hardy space \({\mathcal H}_{\Phi}\) is the space \({\mathcal K}_{\Psi}\), where (\(\Phi\),\(\Psi)\) is a pair of conjugate Young functions, both having finite power.