* Italy bond yields fall after Thursday's sharp rise
* Debt relief boosts Greek debt
* Euro zone PMIs stronger than expected
* Banks to repay just fraction of ultra-cheap ECB cash
* Euro zone periphery govt bond yields http://tmsnrt.rs/2ii2Bqr (Updates throughout)
LONDON, June 22 (Reuters) - Italy's borrowing costs fell on Friday as the government bond market regained its footing after a sharp selloff the previous session, while the Greek bond market received a boost from a euro zone debt relief announcement.
Bond yields across broader euro zone government debt markets faced some upward pressure from better-than-expected business activity data from the bloc.
But it was southern Europe that took the spotlight a day after the appointment of two eurosceptics to head key finance committees in Italy's parliament stoked concerns about the anti-establishment coalition's commitment to the euro.
Claudio Borghi, the president of the lower house budget committee, said in a newspaper interview on Friday that Italy's exit from the euro would solve many of the country's problems, but it is not part of the current government's plans.
Despite a calmer tone to markets, analysts said traders remained sensitive to the headlines coming from Italy - the euro zone's third biggest economy.
"Developments in Italy remain the major driver for bond markets," said René Albrecht, a rates strategist at DZ Bank. "Sentiment is still fragile and although yesterday's sell off was not as strong as the one in late May, traders are very sensitive to the noise."
Italy's two-year bond yield fell 12 basis points to 0.80 percent, off a one-week high hit on Thursday at around 0.97 percent but set to end the week 13 bps higher.
Ten-year Italian bond yields fell 7 bps to 2.68 percent -- narrowing the gap over euro zone benchmark issuer Germany.
Greek bond yields meanwhile fell after euro zone finance ministers extended maturities and deferred interest of a major part of their loans to Greece along with a big cash injection to ensure Athens can stand on its own feet after it exits its bailout in August.
Greek 10-year bond yields fell 16 bps to 4.15 percent , its lowest level since mid-May, while five-year bond yields fell 23 bps to 3.27 percent.
"The confirmation of the debt relief deal provides the Greek government with a huge amount of breathing space so the outperformance of Greek bonds now and for a while should be what we expect to see," said Chris Scicluna, head of economic research at Daiwa Capital Markets.
Outside Italy and Greece, euro area bond yields crept up after stronger-than-expected business activity data.
IHS Markit's Euro Zone Composite Flash Purchasing Managers' Index, seen as a good guide to economic health, climbed in June to 54.8 from 54.1 in the previous month.
The European Central Bank meanwhile said on Friday that euro zone banks will repay 11.00 billion euros of ultra cheap funding returning only a fraction of their borrowings two years ahead of schedule.
Banks borrowed 399 billion euros ($465 billion) in a four-year targeted longer-term refinancing operation (TLTRO) in mid-2016, piling into a facility that promised to pay them a small amount of interest if they met their quota of loaning to the real economy.
(Reporting by Dhara Ranasinghe; Graphic by Balazs Koranyi; Editing by Toby Chopra)