Norm estimates for the $\bar{\mathrm{\partial }}$-equation on a non-reduced space

We study norm-estimates for the $\bar{\mathrm{\partial }}$-equation on non-reduced analytic spaces. Our main result is that on a non-reduced analytic space, which is Cohen-Macaulay and whose underlying reduced space is smooth, the $\bar{\mathrm{\partial }}$-equation for $(0,1)$-forms can be solved with $L^p$-estimates.