First, there's Mahmoud Abbas, the Palestinian Authority's new prime minister. He says he is committed to ending violence and reopening peace talks with Israel. But Abbas' position as an underling of Yasser Arafat makes him weak, and many top figures in the mainstream Palestinian Fatah movement view him as an American-Israeli puppet whom they refuse to follow.

Then there's Ariel Sharon, Israel's prime minister. He now says he wants to crown his career with a major peacemaking achievement. But he faces growing opposition from right-wingers in his coalition government and his own Likud Party. They have set up a new lobby group to support the West Bank and Gaza settlers and hope to encourage American Jews to oppose a peace deal.

Does this mean the "road map" is destined to fail, as have all previous initiatives since the eruption of the Palestinian intifada in September 2000?

Not necessarily.

On the face of it, conditions are ripe for a new chapter in Israeli-Palestinian relations.

The American victory in Iraq has changed the regional balance of power, and the road map plan drafted by the United States, European Union, United Nations and Russia marks a new, widely accepted international peace plan.

However, prospects for new negotiations were jeopardized Tuesday night when yet another suicide bomber attacked a popular Tel Aviv nightspot, killing at least three people and wounding 35.

Abbas' own Fatah movement claimed responsibility, along with Hamas.

Nevertheless, the United States finally presented its much-vaunted road map to both Sharon and Abbas on Wednesday.

What the road map has that the Mitchell (April 2001), Tenet (June 2001) and Zinni (March 2002) plans lacked is a clear linkage between a cease-fire and the political endgame. The equation is simple: Palestinian quiet equals Palestinian statehood, guaranteed by the international community.

After 2-1/2 years of terror and retaliation, both sides are showing signs of fatigue, both bruised societies are looking for change, and both recognize the link between ongoing violence and economic hardship.

But to break out of the cycle of terror and response, both sides will have to make allowances. The Palestinians will have to make genuine efforts to stop the violence before Israel withdraws from Palestinian territories; and Israel may have to allow the Palestinian effort to be judged by its resolve, not its success.

As political pundit Chemi Shalev wrote in the Ma'ariv newspaper: "In one of the refugee camps, near one of the bomb-making workshops, sits the man who holds the real key to the road map. Not George Bush or Colin Powell, not Ariel Sharon or Shaul Mofaz, and not even Yasser Arafat or Abbas, but the lone terrorist, the potential suicide bomber, who with perfect timing will blow himself, his surroundings and the faint hope for change to smithereens.''

Shalev's article was written before Tuesday's bombing, but it hit on Israel's dilemma: Should it respond strongly after any attacks, or give Abbas a grace period to prove his resolve against terror?

To give peace a chance, says Shalev, Sharon will have to take back the veto power he gave radical Palestinian terrorists when he demanded absolute quiet before making any peace moves.

But talk of a cease-fire that could lead to the reopening of negotiations won't come easily. Both Sharon and Abbas face powerful domestic opposition to peacemaking.

Still, because of the changed conditions, it's too early to discount the road map's chances of success.

On Sunday, the head of Israel's Shin Bet security service, Avi Dichter, outlined to the Cabinet what he saw as Abbas' main difficulties.

First and foremost, Arafat, the Palestinian Authority president, will still be around, breathing down Abbas' neck and trying to trip him up. There is a real danger of a bifurcated Palestinian leadership emerging, with Abbas pulling in one direction and Arafat in the other, Dichter said.

Moreover, many top figures in the mainstream Palestinian Fatah movement don't accept the new prime minister's authority.

Abbas' security chief and strongman, Mohammed Dahlan, has real clout only in Gaza, not in the West Bank. In Dichter's view, the upshot might well be that Abbas will show toughness toward Israel to win domestic support — which could work against progress in peacemaking.

The Palestinian legislature confirmed Abbas on Tuesday by a vote of 51-18 with three abstentions.

Just before the vote, however, Arafat made two major moves to shore up his position that may hamper Abbas' room to maneuver.

First, he set up a National Security Council that will have ultimate control over Palestinian security organizations and will report directly to Arafat. This was a clear, 11th-hour attempt to undermine Dahlan, who is the minister responsible for security affairs in Abbas' government.

Dahlan sees as his main goal the creation of a single, unified Palestinian armed force under his control. The Israelis and Americans see that as vital for progress toward peace.

Arafat, through the National Security Council, could try to prevent it — and the outcome of that power struggle could be crucial.

Second, Arafat appointed Saeb Erekat as head of negotiations with Israel within the Fatah movement, after having forced Abbas to make Erekat chief negotiator with Israel in his government. Erekat is an unreserved Arafat loyalist.

Through the National Security Council and through Erekat, Arafat hopes to retain control of the two keys to future relations with Israel: security affairs and peacemaking. Life will not be easy for Abbas.

As for Sharon, he faces growing opposition from his own allies. Members of his own party are dedicated to undermining the road map and preventing the establishment of a Palestinian state.

Eighteen of the 40 Likud legislators, including Public Security Minister Tzachi Hanegbi, already have joined the pro-settler lobby group, dedicated to undermining the road map and preventing the establishment of a Palestinian state. One of the founding members was the Likud's Yehiel Hazan, who says "the road map in its current form will lead to the destruction on the state of Israel."

They also are calling on right-wing elements in the American Jewish community to oppose the road map and to make their views known to U.S. officials

The settlers, too, have not been idle. The Yesha settler council has produced an "alternative road map" that leaves most of the West Bank in Israeli hands, does not dismantle settlements and rules out the establishment of a Palestinian state.

Sharon himself seems to be torn: He knows it would be foolhardy to obstruct American-led peace efforts, but he is loath to give up settlements that he himself was instrumental in setting up over the years.

Sharon believes one of the keys to peacemaking will be Abbas' handling of the fundamentalist Hamas and Islamic Jihad militias. He insists that they be disarmed — by force, if necessary.

That, indeed, is perhaps Abbas' greatest dilemma: whether to confront the radicals head-on or try to reach a hudna, or temporary cease-fire, agreement with them. Under such a cease-fire, the radicals would keep their arms but wouldn't use them against Israel — as long as peace talks continue, that is.

Sharon fears the radicals would simply use a hudna to regroup for another round of terrorism against Israel, or even perhaps to turn on the Palestinian Authority and seize power by force.

Abbas, however, fears a clash could be tantamount to civil war.

Abbas looks at things another way.

While Arafat hoped to pressure Israel into major concessions through terror, Abbas argues that if the Palestinians refrain from terror — which has failed — international pressure on Israel will accomplish their goals instead.

But political pundits say any future success will depend on how both Abbas and Sharon handle those closest to them — the right-wing fringe that will do whatever it can to ensure the failure of both prime ministers.