Analysts have disagreed about the effect of the inauguration: Some believe it would inject optimism into the global markets, while others think the threat of tariffs would dampen the Asian economies.
The concern about the effects of Trump's China policy has, however, eased somewhat, since both sides have promised to work towards improved relations.
By lunchtime on Tuesday, Japan's important Nikkei 225 had risen 0.2 percent, while the broader Topix index was hovering just below zero.
The composite indexes of Shanghai and Shenzhen were both up 0.2 percent. Hong Kong's Hang Seng index was up 1 percent.
