JERUSALEM — Under heavy domestic pressure from critics on the right and the left, Prime Minister Benjamin Netanyahu of Israel threatened on Sunday to wage a “harsh offensive against Palestinian Islamic terrorism” as he returned to a country battered by a wave of deadly violence.

“We are in an all-out war against terrorism and we will wage it aggressively,” Mr. Netanyahu wrote in a Facebook post as he was on his way back from New York, where he spoke last week at the United Nations. He headed straight from the airport to a meeting with his top security officials.

In a brief televised statement after the meeting, he announced a series of measures, including the “speeding up of the process for the demolition of the homes of terrorists.”

But returning to a country in a grim mood on the eve of a Jewish holiday that was meant to bring a festive end to the High Holy Days, Mr. Netanyahu faced a predicament. As Israelis debated whether the recent string of attacks by Palestinians, which appeared to lack any orchestrating group, amounted to a third intifada, or uprising, Israeli political analysts said that he would have to calibrate his response so it would be effective but not spur further escalation.