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

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