(Adds Trump comments, updates prices)

* Ceasefire strained as Iran claims Strait of Hormuz

closed

* Oil prices bounce after Wednesday's steep slide

* Nikkei eases, S&P and EU share futures dip

* Dollar steadies, awaiting US core price data

By Wayne Cole

SYDNEY, April 9 (Reuters) - Asian share markets were in

a sober mood on Thursday as cracks quickly began to appear in

the fragile Gulf truce, nudging oil prices back up and reminding

investors the inflationary fallout would last a long time yet.

Crucially, there was scant sign that the Strait of Hormuz was

open in any meaningful way, with Iran flexing its control over

the vital oil artery and demanding tolls for safe passage.

President Donald Trump took to social media to declare U.S.

forces would remain in the Gulf until a deal was reached and

compiled with, otherwise the shooting would begin again.

Meanwhile, Israel carried out its heaviest strikes on Lebanon

since its conflict with Iran-backed Hezbollah militia began last

month, killing more than 250 people on Wednesday.

"You have a fifth of the world's oil supply moving through a

corridor that is still effectively under the influence of one of

the parties to the conflict," said Nigel Green, CEO at deVere

Group. "That's not stability."

"You don't need a full blockade to move oil markets sharply

higher again," he added. "Missiles are still being launched in

the Gulf, Israel is still engaged on another front, and yet

markets are behaving as though the region has normalised."

As a result, prices for U.S. crude futures bounced

3.1% to $97.33 a barrel and Brent rose 2.1% to $96.86.

Japan's Nikkei dithered either side of flat, after

jumping 5.4% the previous session. South Korea dipped

0.4%, following a leap of 6.8%.

Chinese blue chips slipped 0.6%, while MSCI's

broadest index of Asia-Pacific shares outside Japan

eased 0.7%.

On Wall Street, S&P 500 futures and Nasdaq futures

were both off 0.2% as Wednesday's surge petered out.

For a mixed Europe, EUROSTOXX 50 futures eased

0.1%, while DAX futures fell 0.5% and FTSE futures

rose 0.4%.

INFLATION IS INEVITABLE

With oil prices still around 40% higher than pre-conflict,

an inflationary spike is about to show up in the hard data

across the globe.

Figures on U.S. core prices for February due later Thursday

are expected to show a chunky 0.4% rise for a second month, and

that was before the surge in energy costs.

Minutes from the Federal Reserve's last policy meeting showed a

growing number of members felt a rate hike might be needed to

contain inflation, though many hoped the next move would still

be a cut.

That tempered a rally in Treasuries, which proved modest

compared to the big gains seen in European debt markets. Yields

on U.S. 10-year notes sat at 4.296%, compared to

3.96% before the attack on Iran.

Fed fund futures imply only 6 basis points of easing

for the rest of this year, having given up on 50 basis points of

cuts since the end of February.

"The committee broadly agreed that it was too early to act,

suggesting the Fed will likely remain on hold this year, in line

with our view," said analysts at JPMorgan in a note.

They also saw risks shifting to just one rate hike from the

European Central Bank this year, rather than two.

The shifting outlook for rates saw the dollar pare some of

its knee-jerk losses, with the euro flat at $1.1669

and off a top of $1.1721.

The dollar steadied at 158.68 yen, having fallen

as far as 157.89 at one stage on Wednesday.

In commodity markets, gold was a shade firmer at $4,721 an

ounce, after bouncing as high as $4,777 overnight.