To answer the initial question, it wasn't really subject to Duke William's authority. The first time we see Normans in Southern Italy is on a pilgrimage, if the primary source Amatus of Montecassino is to be believed we see reference to Norman pilgrims being in Southern Italy before the year 1000 who help defend Salerno from Muslim attack. The first major and consistent appearance of the Normans in Southern Italy were as mercenaries, examples include the sons of Tancred of Hauteville from 1035 onwards and Ranulf slightly earlier. These fought for Guaimar of Salerno in the early 11th century and occasionally the Byzantines, for example in 1038 when the Normans first got their taste of Sicily. Why did they end up in Southern Italy? Ranulf was allegedly exiled by Robert I, Duke of Normandy, and the sons of Tancred left Normandy due to Hauteville having little land for them to own and that they had heard tales of great wealth in Southern Italy from the earlier mercenary work of Normans like Ranulf. It was these reasons rather than the request of Duke William that we see Normans in Sicily, with the Drengots (Ranulf) and the Hautevilles being the two dominant Norman forces in Southern Italy. We also already have Normans in Southern Italy before William even becomes Duke in 1035.

The conquest of Sicily in 1061 was undertaken by two of Tancred's sons, Robert Guiscard and Roger, after they had already secured a foothold in Apulia and attempting to get one in Calabria. We've already established why the clan Hauteville appear in Southern Italy, little to no land of their own in Normandy and the appealing tales of wealth in Southern Italy from their brothers. There is little to no evidence of them being under the authority of Duke William who was undertaking his invasion of England during this time frame. Roger did marry niece of Duke William however this doesn't seem to have led to any major communication between Roger and William.

So essentially we have evidence of Normans already in Southern Italy, as pilgrims and mercenaries, before William became Duke of Normandy who were seeking fortune and power in the fractured area that was Southern Italy and Sicily. They received reinforcements in the form of other Normans who had heard of tales of wealth in Southern Italy and occasionally the backing of the papacy who saw the Normans as the lesser evil over the Byzantines and Muslims. They also didn't need major reinforcements due to the fact that the Normans brought military tactics and equipment that hadn't really been seen by the occupiers of Southern Italy and Sicily.

Sources -

Amatus of Montecassino - 'History of the Normans' (Primary) Goffredo Malaterra - 'The Deeds of Count Roger of Calabria and Sicily and of Duke Robert Guiscard his brother.' (Primary)

G. A Loud, 'The Age of Robert Guiscard: Southern Italy and the Norman Conquest'. (Secondary)
J. J Norwich, The Normans in the South 1016-1130. (Secondary)