SINGAPORE, Sept 29 (Reuters) - Asian shares eased away from their lowest levels in 10 months but were on course to clock their worst quarterly performance in a year as worries over elevated interest rates dragged on sentiment, while the dollar held strong.

MSCI's broadest index of Asia-Pacific shares outside Japan was 0.59% higher but not far off the 10-month low it touched on Thursday.

The index is set for 5% drop in the July-September period, its worst quarterly performance since a 13.6% drop in the same period last year.

Japan's Nikkei was 0.10% lower, while Australia's S&P/ASX 200 index rose 0.21%. Hong Kong's Hang Seng Index was up 0.64%. The Chinese markets were closed for a holiday and are on a break next week.

Focus will be on the Chinese property sector after China Evergrande Group said its founder is being investigated over suspected "illegal crimes".

Meanwhile, data showed the U.S. economy maintained a fairly solid pace of growth in the second quarter and activity appears to have accelerated this quarter, but a looming government shutdown and an ongoing strike by auto workers are dimming the outlook for the rest of 2023.

"During the most recent Fed press conference, (Fed Chair Jerome) Powell mentioned that while the Fed doesn’t target levels of real GDP, it evaluates whether it poses a risk to achieving the 2% inflation target," said Ryan Brandham, head of global capital markets, North America at Validus Risk Management.

"From this perspective, the current GDP figure is not seen as a significant threat and may provide some comfort in an otherwise concerning inflationary environment."

Investors will now switch their attention to U.S. personal consumption expenditures price index later on Friday for the latest view on inflation.

The recent rise in Treasury yields to 16-year highs has cast a shadow over the stock market, with the Federal Reserve's hawkish tilt last week also weighing on risk sentiment.

In Asian hours, the yield on 10-year Treasury notes was down 0.8 basis points to 4.589%, easing away from the fresh 16 year peak of 4.688% it touched on Thursday.

Federal Reserve Bank of Richmond President Thomas Barkin said on Thursday the central bank's decision to hold steady on rates earlier this month was the right move, and it’s unclear whether more monetary policy changes will be needed in coming months.

In foreign exchange market, the dollar index eased 0.057% to 106.10 but hovered near the 10 month high of 106.84 it touched earlier this week. The index is up 2.4% this month and set for second straight month of gains.

The Japanese yen was at 149.33 per dollar, perilously close to the 150 level which is viewed as potentially spurring intervention from Japanese authorities.

Core inflation in Japan's capital slowed in September for the third straight month mainly on falling fuel costs, data showed on Friday, suggesting that cost-push pressures are starting to peak, in a relief for the fragile economic recovery.

Oil prices fell in early trade on Friday after a recent rally, as profit-taking and expectations of supply increases by Russia and Saudi Arabia outweighed forecasts of positive demand from China during its Golden Week holiday.

U.S. crude was up 0.09% to $91.79 per barrel and Brent was at $95.17, down 0.22% on the day.

Gold prices were braced for their biggest monthly fall since February, hovering around six-month lows. Spot gold was little changed at $1,864.70 an ounce.

(Reporting by Ankur Banerjee in Singapore; Editing by Lincoln Feast.)