On generalization of Paley-Wiener theorem for weighted Hardy spaces

We consider the Hardy space $H^p_\sigma(\mathbb{C}_+)$ in the half-plane with an exponential weight. In this space we study the analytic continuation from the boundary. In the previous works for the case $p \in (1, 2]$ a result on analytic continuation from the imaginary axis was obtained, and it was a generalization of Paley–Wiener theorem. But for many applications the case $p = 1$ is more interesting. For this case in the paper we obtain estimates for a function satisfying certain standard conditions.