* Euro zone "core" bond yields broadly lower
* Bloc's business activity gloomy in Nov
* Euro zone periphery govt bond yields http://tmsnrt.rs/2ii2Bqr (Updates with fresh quote)
LONDON, Nov 22 (Reuters) - Most euro zone government bond yields fell on Friday after data showed euro zone business growth almost ground to a halt this month while activity in the dominant services industry rose at a much weaker pace than expected.
In her first policy speech as ECB President, Christine Lagarde meanwhile said the euro zone needs to create more of its economic growth at home if it is to withstand weakness abroad and become more balanced internally.
IHS Markit's flash composite Purchasing Managers' Index, seen as a reliable guide to economic health, was 50.3, down from October's 50.6 and creeping towards the 50 mark separating growth from contraction.
After a relatively lackluster week in regional debt markets, driven by headlines related to U.S.-China trade relations, the focus returned to the euro zone economy.
"The PMIs were a bit of a mixed bag, some elements were constructive and others disappointing," said Andy Cossor, rates strategist at DZ Bank.
"If you take the euro zone as a whole, manufacturing was a bit better but the services sector was weaker, which on balance is a sign of an economy that is not firing on all cylinders."
Ten-year bond yields across the euro zone's higher-rated bond markets were down 3-4 basis points each .
Germany's benchmark 10-year bond yields was last down 3 bps at -0.35%, nearing almost three-week lows hit earlier this week. It was set for its biggest one-day fall in just over a week.
Analysts said Lagarde's speech did not offer any new hints on the monetary policy outlook, while continuing to stress the need for fiscal policy to play a greater role.
Lagarde did not discuss monetary policy, merely saying the central bank would continue to do its part to support the euro zone economy.
"For an ECB president who's trying to be consensus driven, it was as expected and didn't contain anything new," said Peter Chatwell, head of rates strategy at Mizuho.
"But one reason bonds have rallied is not just because of poor data today but also because Lagarde has made reference to monetary easing being the ECB's main tool."
In southern Europe, borrowing costs were little changed on the day, with Italy's 10-year bond yield hovering around 1.29%, after being up 3 bps in early trading.
(Reporting by Dhara Ranasinghe; Additional reporting by Saikat Chatterjee Editing by Mark Heinrich and Giles Elgood)