  In this paper, we prove an $L^2$ extension theorem with optimal estimate in a
precise way, which implies optimal estimate versions of various well-known
$L^2$ extension theorems. As applications, we give proofs of a conjecture of
Suita on the equality condition in Suita's conjecture, the so-called
$L-$conjecture, and the extended Suita conjecture. As other applications, we
give affirmative answer to a question by Ohsawa about limiting case for the
extension operators between the weighted Bergman spaces, and we present a
relation of our result to Berndtsson's important result on
log-plurisubharmonicity of Bergman kernel.
