Japan's benchmark Nikkei 225 lost 0.6% to 21,546.25 in afternoon trading. Australia's S&P/ASX 200 lost nearly 0.1% to 6,646.40. South Korea's Kospi added 0.3% to 2,089.34 and Hong Kong's Hang Seng rose 0.2% to 28,613.72, while the Shanghai Composite shed 0.2% to 2,937.00.
"The lack of impetus for Asia markets has kept prices trading in a lackluster fashion this morning," said Jingyi Pan, market strategist for IG in Singapore.
One factor weighing on sentiment is the impasse between the U.S. and China over trade.
The White House's repeated threats to raise tariffs further in retaliation for the U.S. trade deficit and policies that critics say are unfair has made companies hesitant to invest and has hurt trade internationally. They're a big reason China on Monday reported its weakest quarter of economic growth in at least 26 years.
U.S. Treasury Secretary Stephen Mnuchin told reporters in Washington on Monday that he has held "several conversations" with Chinese officials and expects another phone call this week.
The aim is to restart high-level talks that collapsed in May.
"To the extent that we make significant progress I think we'll go there," he said.
Wall Street had a wobbly day of trading with gains that were just enough to nudge benchmarks further into record territory.
The S&P 500 rose 0.53 points to 3,014.30 after drifting between a gain of 0.1% and a loss of 0.2% earlier in the day. The Dow Jones Industrial Average gained 0.1%to 27,359.16 and the Nasdaq composite added 0.2% to 8,258.19.
U.S. stocks have jumped since early June on increasing expectations that the Federal Reserve will cut interest rates to help the economy, and investors are virtually certain that it will happen at the next Fed meeting at the end of this month. The only question, investors say, is how deeply the Fed will cut when it lowers rates for the first time in a decade.
Until then, the main drivers for the market will likely be the hundreds of earnings reports scheduled to come from big companies, showing how much profit they made from April through June.
Several economic reports are also on the schedule this week, including updates on retail sales, the housing industry and shoppers' confidence. The U.S. economy has generally remained solid, but investors don't expect this week's reports to alter the direction of the Fed, which has already given hints about rate cuts given weakening economic trends around the world.
ENERGY: Benchmark U.S. crude fell 8 cents at $59.50 a barrel. It fell 63 cents on Monday. Brent crude, the international standard, was unchanged at $66.48 a barrel.
CURRENCIES: The dollar inched up to 108.02 Japanese yen from 107.98 late Monday. The euro rose to $1.1260 from $1.1277.
.