NEW YORK — The back-to-school season got off to a strong start as discounts and high temperatures in July drove shoppers to air-conditioned malls. But merchants worry that momentum won’t continue through the remainder of the second-biggest shopping period of the year as the weather gets cold and the deals dry up.

Despite a flow of bad economic news that kept consumer confidence shaky, a number of retailers reported July revenue on Thursday that beat Wall Street estimates, including discounter Target, department store Macy’s and luxury chain Saks. The International Council of Shopping Centers’ preliminary tally of retailers’ revenue at stores open at least a year — a key indicator of a merchant’s health — was up 4.6 percent, a slower pace than June’s 6.9 percent gain but in line with forecasts.

While the numbers offer encouraging signs for the start to the back-to-school shopping period, which runs roughly from mid-July through September, there are concerns that shoppers will stick to the habits of the Great Recession by focusing on necessities and waiting for sales. That could be a big problem for retailers, which are raising prices to offset rising fuel, labor and other production costs.

“Early going, July looks like it’s shaping up to be a solid month despite all the economic headwinds,” said Ken Perkins, president of RetailMetrics LLC., a research firm.”

The back-to-school season is important for retailers because it accounts for 16.1 percent of annual retailers’ revenue, according to the International Council of Shopping Centers. It’s also an opportunity for retailers to gain insight into consumers’ shopping habits heading into the biggest shopping season of the year, which starts on the day after Thanksgiving.