Lower bound of the supremum of ergodic averages for ${\mathbb{Z}^d}$ and ${\mathbb{R}^d}$-actions

For ergodic ${\mathbb{Z}^d}$ and ${\mathbb{R}^d}$-actions, we obtain a pointwise lower bound for the supremum of ergodic averages. For ${\mathbb{Z}^d}$-actions in the case when the supremum is taken over multi-indices exceeding $\vec{n}$ located in a certain sector, the resulting inequality is not improvable over $\vec{n}$ in the class of all averaging integrable functions.