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We present an $L^2$-extension theorem with an estimate depending on theweight functions for domains in $\mathbb{C}$. When the Hartogs domain definedby the weight function is strictly pseudoconvex, this estimate is strictlysharper than known optimal estimates. When the weight function is radial, weprove that our estimate provides the $L^2$-minimum extension.

On sharper estimates of Ohsawa-Takegoshi $L^2$-extension theorem