Do I have to use sucrose to induce a lac promoter? I'd like to optimize the expression of a Fab fragment in _Escherichia coli_. For induction of the _lac_ promoter on the pAK400 vector I use IPTG and sucrose. Do I optimize the expression in case I would just add IPTG?

You can induce the lac operon by two things: Lactose (or more precisely Allolactose a metabolite of it) and lactose analogons which are not metabolized by the bacteria.

Lactose induces the expression and is metabolized while IPTG is not a target of the $\beta$-Galactosidase and will give you a strong and permanent induction. Sucrose will not have any effect on the promoter.