South Korea's Kospi rose 1.1%. Hong Kong's Hang Seng Index gained 0.2%.
China's Shanghai Composite Index, though, inched down 0.1%, erasing slight gains in earlier trading.
Japanese markets are closed for a public holiday.
Major stock indexes ended in the green Monday. Futures for the Dow, S&P 500, and the Nasdaq rose Tuesday during Asian trading hours.
Here are some other moves happening in Asian markets at 12:00 p.m. HKT:
The People's Bank of China on Tuesday injected 250 billion yuan ($35 billion) into the banking system. It's the largest cash injection of its kind in nearly five months, according to Reuters. The Chinese central bank says the injection is meant to help meet cash demand during tax season. Markets expect the central bank to continue easing monetary policy, though, using rate cuts and other measures. The Chinese economy recently posted its weakest growth rate in 27 years for the third quarter.
ESR Cayman, an Asian warehouse operator, wants to raise as much as 11.37 billion Hong Kong dollars ($1.45 billion) in an initial public offering in Hong Kong. It would be the second biggest IPO in the Asian financial hub this year. ESR Cayman intends to start trading in November. Shares of 51 Credit Card, an online credit card issuer and manager based in Hangzhou, China, remain suspended from trading Tuesday after plunging 34% Monday. Trading was halted on Monday afternoon. The company acknowledged in a statement that Chinese authorities conducted an investigation at the firm's office.
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