Optimal weighted Wente's inequality

We prove $L^{\mathrm{\infty }}$ and $W^{1,2}$ weighted Wente’s inequalities. We prove in particular the critical case: for the $\mathrm{\mid }x{\mathrm{\mid }}^2$ weighted Wente’s estimate the optimal weight is $\mathrm{\mid }x{\mathrm{\mid }}^2\log \mathrm{\mid }x\mathrm{\mid }$.