Minimal $L^2$ integrals for the Hardy spaces and the Bergman spaces

In this article, we consider the minimal $L^2$ integrals for the Hardy spaces and the Bergman spaces, and we present some relations between them, which can be regarded as the solutions of the finite points versions of Saitoh’s conjecture for conjugate Hardy kernels. As applications, we give optimal $L^2$ extension theorems for the Hardy spaces, and characterizations for the holding of the equality in the optimal $L^2$ extension theorems.