The authors characterize weighted weak type and strong type inequalities for the maximal operator associated to the Cesàro-\(\alpha\) averages. Let \(\varphi_R(x)=\frac{1}{R^n}\varphi(\frac{x}{R})\), \(R>0\), be the dilated functions of a nonnegative integrable function \(\varphi\) defined on \(\mathbb{R}^n\) such that \(\int \varphi =1\). When \(\varphi(x)=\varphi^\alpha (x)=C_{n, \alpha}(1-|x|_{\infty})^{\alpha}\chi_{Q(0, 1)}(x)\), where \(x=(x_1, \cdots, x_n)\), \(|x|_{\infty}=\max_{1 \leq i \leq n}|x_i|\), \(-10} \frac{1}{|Q(x, R)|^{1+\alpha/n}}\int_{Q(x, R)}|f(y)|d(y, \partial Q(x, R))^{\alpha}dy. \] If \(\alpha =0\), \(M_\alpha^C \) is simply denoted by \(M^C\). A non-negative measurable function (a weight) \(\omega\) satisfies \(A_{p,a}\), \(-10\) such that \[ \Big (\int_Q \omega(y)dy \Big )^{1/p} \Big (\int_Q\omega^{1-p'}(y)d(y, \partial Q)^{\alpha p'} dy\Big)^{1/p'} \leq C|Q|^{1+\frac{\alpha}{n}}, \] for every cube \(Q\), where \(p'\) is the conjugate exponent of \(p\). Fix a natural number \(m>1\) and take real numbers \(p_i\) and \(\alpha_i\) with \(p_i>1\) and \(-10}\prod_{i=1}^m \frac{1}{|Q(x, R)|^{1+\alpha_i/n}}\int_{Q(x, R)}|f_i(y)|d(y, \partial Q(x, R))^{\alpha_i}dy. \] where \(\overrightarrow{f}=(f_1, \cdots, f_m)\) and \(\overrightarrow\alpha = (\alpha_1, \cdots, \alpha_m).\) The main results can be stated as follows. Let \(\overline{\alpha}=\alpha_1 +\cdots + \alpha_m\), \(p\) be such that \(\frac{1}{p}=\sum_{i=1}^m\frac{1}{p_i}\) and \(v_{\overrightarrow{\omega}} = \prod_{i=1}^m \omega_i^{p/p_i}\). The following statements are equivalent. (i) There is \(C>0\) such that \(||\mathcal{M}_{\overrightarrow{\alpha}}(\overrightarrow{\chi}_{\overrightarrow{E}})||_{L^{p, \infty}(v_{\overrightarrow{\omega}})} \leq C\prod_{i=1}^m ||\chi_{E_i}||_{L^{p_i}(\omega_i)}\) for all measurable set \(E_i, i=1, \cdots, m\). (ii) There is \(C>0\) such that \(||\mathcal{M}^C_{\overrightarrow{\alpha}}(\overrightarrow{\chi}_{\overrightarrow{E}})||_{L^{p, \infty}(v_{\overrightarrow{\omega}})} \leq C\prod_{i=1}^m ||\chi_{E_i}||_{L^{p_i}(\omega_i)}\) for all measurable set \(E_i, i=1, \cdots, m\). (iii) \(\overrightarrow{\omega} \in RA_{\overrightarrow{p}, \overrightarrow{\alpha}}\), that is, there is \(C>0\) such that \[ \Big (\int_Q v_{\overrightarrow{\omega}} \Big )^{\frac{1}{p}} \prod_{i=1}^m \Big (\frac{1}{|Q|^{1+\alpha_i /n}} \int_Q \chi_{E_i}(y)d(y, \partial Q)^{\alpha_i}dy \Big ) \leq C\prod_{i=1}^m \Big (\int_Q \chi_{E_i} \omega_i \Big )^{\frac{1}{p_i}} \] for all cubes \(Q\) and all measurable sets \(E_i, i=1, \cdots, m\). If \(v_{\overrightarrow{\omega}}\) is not \(0\) a.e. and statement (iii) holds then \(p_i \geq \frac{1}{1+\alpha_i} \) for all \(i\). Let \(p_i \geq \frac{1}{1+\alpha_i} \). Then the following statements are equivalent. (i) There is \(C>0\) such that \(||\mathcal{M}_{\overrightarrow{\alpha}}(\overrightarrow{f})||_{L^{p, \infty}(v_{\overrightarrow{\omega}})} \leq C\prod_{i=1}^m ||f_i||_{L^{p_i}(\omega_i)} \;(f_i \in {L^{p_i}(\omega_i)})\). (ii) There is \(C>0\) such that \(||\mathcal{M}^C_{\overrightarrow{\alpha}}(\overrightarrow{f})||_{L^{p, \infty}(v_{\overrightarrow{\omega}})} \leq C\prod_{i=1}^m ||f_i||_{L^{p_i}(\omega_i)} \;(f_i \in {L^{p_i}(\omega_i)})\). (iii) \(\overrightarrow{\omega} \in \mathcal{A}_{\overrightarrow{p}, \overrightarrow{\alpha}}\). (iv) The following conditions hold. (a) \(v_{\overrightarrow{\omega}} \in A_{mp, \frac{\overline{\alpha}}{m}}\), (b) \(\omega_i^{1-p'_i} \in A_{mp'_i, \frac{\overline \alpha - \alpha_i}{m}}\) for all \(i=1, \cdots, m\) and (c) \(\omega_i^{\frac{1}{r_i}} \in A_{\frac{mp_i}{r_i}, \frac{\alpha_i}{m}}\), for all \(i=1, \cdots, m\), where \(r_i = (m-1)p_i + 1\). If \(v_{\overrightarrow{\omega}}\) is not \(0\) a.e. and statement (iii) holds then \(p_i \geq \frac{1}{1+\alpha_i} \) for all \(i\) such that \(\alpha_i \neq 0\).