Strange, USPTO says the Steve Jobs trademark has been abandoned:
http://tmsearch.uspto.gov/bin/showfield?f=doc&state=4802:hxlxjr.2.2
Apple can renew it and apply for priority exclusivity across the world that is a signatory to the trademark/patent/copyright laws. (There are mitigations in the laws that allows companies to override existing trademarks if their market share is large enough that the entire world know it belongs to them, it's how Apple won certain iPad trademarks in South America IIRC despite existing there before Apple had iPad trademarks).
Uh....No.
First, Apple
already lost their opposition to the use of just the name "Steve Jobs" as a trademark - in 2014 their opposition was rejected in its entirety by the European Union, as Apple had no legal grounds on which to claim either infringement or ownership of an existing mark. Apple's further legal action was against the "sign" (i.e., the logo), where they did have some grounds for a claim of infringement of their own signs.
Second, you can't "renew" someone else's trademark. Trademark rights are based on
use in commerce. They don't legally have to be registered (in the US at least), so using someone else's trademark is infringement, whether registered or unregistered. Next, just because a mark is shown as Abandoned in the USPTO doesn't mean the mark is not still in use by the registrant, or that you can use it at all. And finally, in order to trademark even a famous person's name, it's not enough for that person simply to be associated with a particular company, no matter how well known. For trademark protectoin, the person's name has to be specifically associated with
a set of goods or services. Since Apple has never sold goods or services under the "Steve Jobs" name, and has no plans to do so, Apple had no rights to that mark to protect, i.e., no use of the mark in commerce that would qualify for registration.
Third, Apple can't claim priority
or exclusivity with regard to "Steve Jobs".
"Priority" would require they were the first to use the "Steve Jobs" trademark in commerce, prior to whatever date in 2012 the Italian company began using it. But Apple
wasn't first, and as noted also has never used "Steve Jobs" as a mark in commerce, so they can never claim priority.
By "exclusivity" I assume you mean acquired distinctiveness or secondary meaning. As noted above, celebrity and/or association with a well-known company alone are not enough to claim protection or trademark rights for a person's name - and the "Steve Jobs" name was never associated with specific goods or services from Apple. But if you do mean "exclusivity", keep in mind Jobs was not exclusively associated with Apple, as he also founded NeXT, not to mention that a large part of his fame came from being
fired by Apple, i.e., not associated with them. The only entity that could claim exclusivity to the name would be Steve Jobs himself, if he were still alive. "Steve Jobs" was never proposed as an Apple brand or product name even when Jobs was in charge, and Jobs did not sign over the rights to commercial use of his name to Apple or anyone else, either pre- or post-humously. While most people would
associate him with Apple, that doesn't mean that his name acquired the secondary meaning of an actual "mark" that Apple can claim protection for.
The names of dead, historical figures and celebrities have often been used by total strangers for companies and products. And while it's often the case that a commercial enterprise named after a famous person was founded by that person, it's not universal. Likewise, though it's more customary to wait until the person has been dead longer than a year, it doesn't seem to be a legal requirement. If you don't take steps to use your name in commerce while you're still alive, it's pretty much up for grabs after you're gone. For example, look up the famous cases surrounding the use of Bob Marley's and Babe Ruth's names, and you'll see there is plenty of precedent that pretty much leaves Apple without any grounds for opposing the Italian "Steve Jobs" trademark.
Frankly, I think the Italian company is silly, and is unlikely to survive. And that legally they really should have lost on the logo opposition, as their logo is a pretty blatant ripoff.
But Apple absolutely can not "renew" the abandoned US registration, nor can they claim some non-existent "priority exclusivity".