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Moreover, we introduce the `column square functions' $\\norm{x}_{T,c,\\alpha}=\\Bnorm{\\Big(\\sum_{k=1}^{+\\infty}k^{2\\alpha-1}|T^{k-1}(I-T)^{\\alpha}(x)|^2\\Big)^{1/2}}_{L^p(M)}$ and the `row square functions' $\\norm{x}_{T,r,\\alpha}=\\Bnorm{\\Big(\\sum_{k=1}^{+\\infty}k^{2\\alpha-1} |\\Big(T^{k-1}(I-T)^{\\alpha}(x)\\Big)^*|^2\\Big)^{1/2}}_{L^p(M)}$ for any $\\alpha>0$ and any $x\\in L^p(M)$. 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Moreover, we introduce the `column square functions' $\\norm{x}_{T,c,\\alpha}=\\Bnorm{\\Big(\\sum_{k=1}^{+\\infty}k^{2\\alpha-1}|T^{k-1}(I-T)^{\\alpha}(x)|^2\\Big)^{1/2}}_{L^p(M)}$ and the `row square functions' $\\norm{x}_{T,r,\\alpha}=\\Bnorm{\\Big(\\sum_{k=1}^{+\\infty}k^{2\\alpha-1} |\\Big(T^{k-1}(I-T)^{\\alpha}(x)\\Big)^*|^2\\Big)^{1/2}}_{L^p(M)}$ for any $\\alpha>0$ and any $x\\in L^p(M)$. 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