We introduce p-quasilocal operators and prove that if a sublinear operator T is p-quasilocal and bounded from $L_{\infty }$ to $L_{\infty }$ then it is also bounded from the classical Hardy space $H_p\left(T\right)$ to $L_p$ (0 < p ≤ 1). As an application it is shown that the maximal operator of the one-parameter Cesàro means of a distribution is bounded from $H_p\left(T\right)$ to $L_p$ (3/4 < p ≤ ∞) and is of weak type $(L_1,L_1)$. We define the two-dimensional dyadic hybrid Hardy space $H_1^{♯}\left(T^2\right)$ and verify that the maximal operator of the Cesàro means of a two-dimensional...