Let \(\mathbb B_n\) denote the unit ball in \(\mathbb C^n\) and \(d\sigma\) the normalized Lebesgue measure in the unit sphere \(\mathbb S_n\). The Hardy space \(H^p(\mathbb B_n)\) consists of all holomorphic functions \(f\) on \(\mathbb B_n\), for which
\[ \sup_{0\leq r<1} \int_{\mathbb S_n} | f(r\cdot\zeta)| ^p\,d\sigma(\zeta)<\infty. \]
Let \(f(z)=\sum_{| \alpha| =0}^\infty a_\alpha Z^\alpha\) be a holomorphic function in the polydisc \(\mathbb U^n\) or the unit ball \(\mathbb B_n\), the Cesàro averaging operator on \(\mathbb U^n\) and \(\mathbb B_n\) is defined by
\[ \mathcal C^{\vec{\gamma}}(f)(z) = \sum_{| \alpha| =0}^\infty \frac{\sum_{\beta\leq \alpha} a_{\alpha-\beta} \prod_{j=1}^nA_{\beta_j}^{\gamma_j}}{\prod_{j=1}^n A_{\alpha_j}^{\gamma_j +1}} z^{\alpha}. \]
In this paper, the authors give the boundedness of the Cesàro operator on the Hardy spaces. The main result is the following:
Theorem. Let \(0<p<\infty\), \(\gamma=(\gamma_1, \cdots, \gamma_n)\), \(r=(r_1, \dots, r_n)\) such that \(\mathcal R \gamma_j>-1\) and \(0\leq r_j<1\), \(j=1, \dots, n\). Then there exists a constant \(C\) independent of \(f\) and \(r\) such that
\[ \int_{\mathbb S_n} | \mathcal C^{\vec{\gamma}}(f)(r\cdot \zeta)| ^p \,d\sigma(\zeta)\leq C(p,\vec{\gamma},n)\int_{\mathbb S_n} | f(r\cdot \zeta)| ^p \,d\sigma(\zeta) \]
for any holomorphic function \(f\) on \(\mathbb B_n\).