Let \((X,\|\cdot\|)\) be a Banach space, \(f= \{f_n, n\geq 0\}\) be an \(X\)-valued martingale defined on a stochastic basis \((\Omega,{\mathcal F},\mathbb{F}, P)\), \(f_0= 0\), \({\mathcal F}_0= \{\emptyset,\Omega\}\), \({\mathcal F}= \bigvee_{n\geq 0}{\mathcal F}_n\). For \(1< p< \infty\) and \(0<r\leq\infty\) denote the \(X\)-valued martingale Hardy spaces \[ \begin{aligned} H_r(X)&= \{f:\|f\|_{H_r(X)}:=\|f^*\|_r< \infty\},\\{_pH^S_r}(X)&= \{f:\|f\|_{{_pH^S_r(X)}}:=\|S^{(p)}(f)\|_r< \infty\}\\ \text{and}{_pH^\sigma_r}(X)&: \{f:\|f\|_{{_pH^\sigma_r(X)}}:= \|\sigma^{(p)}(f)\|_r< \infty\},\end{aligned} \] where \(f^*= \sup_{n\geq 1}\|f_n\|\), \(S^{(p)}(f)= (\sum^\infty_{k=1}\|f_k- f_{k-1}\|^p)^{1/p}\) and \(\sigma^{(p)}(f)= (\sum^\infty_{k=1} E(\|f_k- f_{k-1}\|^p\mid{\mathcal F}_{k-1}))^{1/p}\). Characterizing the duals of \({_pH^S_r}(X)\) and \({_pH^\sigma_r(X)}\), two new spaces of \(X\)-valued martingales are introduced for \(0< p\leq r\leq\infty\), \(p\neq \infty\): \[ \begin{split} _pK^S_r(X)=\left\{f:\exists\gamma\in L_r(R),\;\gamma\geq 0,\text{ such that}\right. \\ \left. E\left(\sum^\infty_{k=n}\|f_k- f_{k-1}\|^p\mid{\mathcal F}_n\right)\leq E(\gamma^p\mid{\mathcal F}_n),\;\forall n\geq 1,\;\|f\|_{{_pK^S_r(X)}}:= \inf_\gamma\|\gamma\|_r< \infty\right\}\end{split} \] and \[ \begin{split} _pK^\sigma_r(X)=\left\{f:\exists\gamma\in L_r(R),\;\gamma\geq 0,\text{ such that }\right. \\ E\left(\sum^\infty_{k=n+1} E(\|f_k- f_{k-1}\|^p\mid{\mathcal F}_{k- 1})\;\Bigl|\;{\mathcal F}_n\right)\leq\Bigr. \\ \leq \left. E(\gamma^p\mid {\mathcal F}_n), \forall n\geq 1, \|f\|_p K^\sigma_r(X):= \inf_\gamma\|\gamma\|_r< \infty\right\}.\end{split} \] The properties of these spaces, related to the \(p\)-uniform smoothness and \(q\)-uniform convexity of \(X\), are investigated. For a reflexive Banach space \(X\) and \(1\leq r\leq p<\infty\) it is proved that \[ ({_pH^S_r(X))}^*\approx {_qK^S_{r'}(X^*)}\quad\text{and}\quad ({_pH^\sigma_r(X))}^*\approx {_qK^\sigma_{r'}(X^*)}, \] where \(1/p+ 1/q= 1\) and \(1/r+ 1/r'= 1\).