AMNA NAWAZ: Welcome to the "News Hour."

President Trump returned to the White House this afternoon after a weeklong trip to Asia that ended with a meeting early today with Chinese President Xi Jinping.

GEOFF BENNETT: That summit in South Korea comes as the U.S.

and China are engaged in a tough trade war.

After the meeting, the two leaders agreed to something of a trade truce, each side making some concessions, but mostly freezing the business battle lines in place.

Nick Schifrin starts our coverage.

DONALD TRUMP, President of the United States: We're going to have a very successful meeting.

I have no doubt.

NICK SCHIFRIN: President Trump arrived to the highest-stakes meeting of his second term predicting success.

DONALD TRUMP: It's a great honor to be with a friend of mine really for a long time now.

NICK SCHIFRIN: But Xi Jinping has not back downed to U.S.

pressure this year, and he began today's meeting by recognizing the rivalry, as heard through his translator.

XI JINPING, Chinese President (through translator): We do not always see eye to eye with each other.

And it is normal for the two leading economies of the world to have frictions now and then.

NICK SCHIFRIN: After an hour and 40 minutes, the two leaders did not solve their disputes, but agreed to a pause.

The U.S.

will lower tariffs connected to the fentanyl trade from 20 percent to 10 percent, bringing the U.S.'

combined tariff rate on China down from 57 percent to 47 percent.

DONALD TRUMP: Well, it's going to be very strong at enforcing the fentanyl, everything having to do with fentanyl regulations internally.

But, overall, I guess on the scale from zero to 10, with 10 being the best, I would say the meeting was a 12.

NICK SCHIFRIN: China agreed to pause new restrictions on the export of heavy rare elements and powerful magnets for one year, but kept restrictions imposed earlier this year.

Beijing controls nearly all the world's processing of rare earth magnets, essential for everything from the United States' most advanced fighter jets to electric vehicles.

China also agreed to restart purchases of American soybeans and bring them back to an annual average by next year.

DONALD TRUMP: Large amounts, tremendous amounts of the soybeans and other farm products are going to be purchased immediately, starting immediately.

NICK SCHIFRIN: There was no agreement on whether to give China access to the U.S.'

most advanced computer chips, but President Trump also didn't rule it out.

DONALD TRUMP: We discuss chips.

And he's going to be -- they're going to be talking to Nvidia and others about taking chips.

Look, we make great chips.

Nvidia is the leader.

NICK SCHIFRIN: The president also said today the U.S.

couldn't be the nuclear leader unless it tested nuclear weapons.

The U.S.

regularly tests unarmed nuclear-capable missiles, but hasn't detonated a nuclear device since this one in 1992.

Russia recently tested its own unarmed nuclear-capable missile.

That's apparently what pushed the president to announce the U.S.

would resume testing, but he also envisioned a world without nukes.

DONALD TRUMP: We have more than anybody, but I see them testing.

I say, well, if they're going to test, I guess we have to test.

I'd like to see a denuclearization.

NICK SCHIFRIN: But the president did say he would share with South Korea nuclear-powered submarine technology, what military officials call their crown jewels.

At the end of this week in Asia, there was progress, but no final deal with China, no solution to the overall tensions, just a truce.

For the "PBS News Hour," I'm Nick Schifrin.