COMMENTS ON THE BACKGROUND FIELD METHOD IN HARMONIC SUPERSPACE - NONHOLOMORPHIC CORRECTIONS IN N=4 SYM

We analyze the one-loop effective action of N = 4 SYM theory in the fr amework of the background field formalism in N = 2 harmonic superspace . For the case of on-shell background N = 2 vector multiplet we prove that the effective action is free of harmonic singularities. When the lowest N = I superspace component of the N = 2 vector multiplet is swi tched off, the effective action of N = 4 SYM theory is shown to coinci de with that obtained by Grisaru et al. on the base of the N = 1 backg round field method. We compute the leading non-holomorphic corrections to the N = 4 SU(2) SYM effective action.