We introduce and investigate the weak weighted martingale
Hardy spaces Λp,∞s(ω)\documentclass[12pt]{minimal}
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\begin{document}$$\Lambda_{p,\infty}^s(\omega)$$\end{document}, where 0<p<∞\documentclass[12pt]{minimal}
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\begin{document}$$0<p<\infty$$\end{document}, ω\documentclass[12pt]{minimal}
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\begin{document}$$\omega$$\end{document} is a weight and s\documentclass[12pt]{minimal}
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\begin{document}$$s$$\end{document} is the conditional
square function. This new family of spaces provides a framework which unifies
various kinds of weak martingale Hardy spaces, including weak martingale
Orlicz–Hardy spaces, weak martingale Karamata–Hardy spaces, weak martingale
Orlicz–Karamata–Hardy spaces, and so on. We establish the atomic decompositions
for Λp,∞s(ω)\documentclass[12pt]{minimal}
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\begin{document}$$\Lambda_{p,\infty}^s(\omega)$$\end{document}, and then apply the atomic decompositions to deduce some new
martingale inequalities and duality theorems. We discuss similar results for the
Hardy spaces Λp,∞∗(ω)\documentclass[12pt]{minimal}
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\begin{document}$$\Lambda_{p,\infty}^*(\omega)$$\end{document}, Λp,∞S(ω)\documentclass[12pt]{minimal}
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\begin{document}$$\Lambda_{p,\infty}^S(\omega)$$\end{document}, Pp,∞(ω)\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal P_{p,\infty}(\omega)$$\end{document} and Qp,∞(ω)\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal Q_{p,\infty}(\omega)$$\end{document} as well. The results obtained
here generalize the corresponding known results in various weak martingale
Hardy spaces.
