Violence between Israelis and Palestinians has fallen sharply in recent weeks. Both sides have spoken in positive terms about the prospect of peace; both have made gestures, albeit mostly verbal, towards attaining that goal.

Not a little fear attended the talks, too. The fear, for Israelis and Palestinians but also for the many others who yearn for a just end to this interminable conflict, is that without urgent, substantive steps forward - along the lines laid out by the international "road map" - a golden opportunity may be lost.

Mr Bush put a characteristically optimistic spin on his discussions with Mr Sharon and, last week, with the Palestinian prime minister, Mahmoud Abbas. "I think we're making pretty good progress in a short period of time," he said.

He might think that is the case. He might wish it to be so. But there are three basic grounds for challenging Mr Bush's rosy judgment.

The first cause for concern arises from the sight of Mr Sharon, standing alongside the US leader, reiterating in uncompromising terms his preconditions for negotiations on the fundamental issues that separate the two peoples.

If anything, Mr Sharon hardened his position. He made no mention, as he has in the past, of Israel's acceptance of a future Palestinian state; he made no reference, as before, to the unsustainability of the occupation of Palestinian land; and perhaps most ominously of all, he omitted all direct reference to the "road map".

"I wish to move forward with a political process with our Palestinian neighbours," Mr Sharon said. "And the right way to do that is only after a complete cessation of terror, violence and incitement, full dismantlement of terror organisations, and completion of the reform process of the Palestinian Authority."

The key word in this sentence is "after". What Mr Sharon was saying, indeed demanding, was that Mr Abbas disarm, disband, and possibly lock up, leaders and members of militant organisations such as Hamas and Islamic Jihad, before Israel does anything substantive by way of reciprocal measures.

Mr Sharon appeared to anticipate failure, even to expect it. "We are thankful for every hour of increased quiet," he said. "At the same time, we are concerned that this welcome quiet will be shattered any minute as a result of the continued existence of terror organisations which the Palestinian Authority is doing nothing to eliminate."

His remarks, in a prepared statement, also seemed to imply that Mr Abbas must achieve complete political control within the authority, including ousting Yasser Arafat from any position of real influence or executive authority for good, before Israel would act.

Mr Sharon is asking for the impossible, as he must know very well. For the second reason for challenging Mr Bush's optimistic assessment is that Mr Abbas has neither the political nor military power to satisfy these Israeli demands at this stage, even if he were fully minded to do so.

His position remains weak, as is to be expected after only a few, controversial months presiding over a government divided and impecunious after years of intifada. He cannot issue fiats or make demands without risking his own downfall, or worse, an inter-Palestinian conflict. If he makes promises he cannot deliver, his credibility will be undermined among his supporters, opponents, and among the Israeli public.

He has no choice but to tread carefully. He needs real Israeli concessions, not mere gestures. So far they have not been offered.

Mr Abbas is already accused by some of his own people of collaborating with the Israelis, of being a dupe or a stooge. They say his policy of engagement, before and since the Aqaba summit, has brought few tangible results. They say the handful of prisoner releases, the charades over the uprooting of "unauthorised outposts", and the very limited military withdrawal, are proof not of Mr Sharon's good faith but of his duplicity.

They say, in short, that Mr Abbas is being taken for a ride, that the Americans are not really pushing Mr Sharon, and indeed, that Mr Sharon is to a lesser extent taking Mr Bush for a ride, too.

It would be comforting to reject all this and say it is merely the product of years of bloodshed and abiding distrust, that all will be well in the end.

But when Mr Sharon in Washington went on to defy the US president, to his face, over Israel's construction of the West Bank security wall, and to ignore the road map's requirement for a freezing of settlement activities, Palestinian suspicions that he is engaged in the old game of talking peace while seizing more and more Palestinian land understandably deepen.

In terms of the bottom line, all Mr Sharon committed Israel to do was to take unspecified "additional steps ... if calm prevails and we witness the dismantlement of terror organisations". This is no commitment at all. And still Mr Bush kept smiling.

In truth, Mr Bush himself is the third reason why optimism seems misplaced at the end of this week's talks.

He says things are moving forward quickly. But he ignores the fact that he wasted two years after he came into office, during which time the conflict grew ever more embittered and entrenched. The opportunity for action is now very limited, partly as a result.

Mr Bush says he and his advisers are committed to the "road map" and making peace work, in line with the timetable for establishing a Palestinian state by 2005. But in reality, they are massively distracted by Iraq, where problems mount, and by broader domestic controversies that are building as the US election year approaches.

Enforced regime change in Iraq is not facilitating the Arab-Israeli peace process, as Mr Bush has frequently claimed it would. If anything, the controversial US policy is obstructing it, just as it did for different reasons before Saddam Hussein's downfall.

Mr Bush also seems quite happy to be almost blatantly bamboozled by Mr Sharon, who is a much more wily and subtle politician that the former Texas governor will ever be accused of being. The Israeli leader must be privately delighted to have a US counterpart who is so easy to handle.

The way Mr Sharon flatters him so outrageously suggests just a smidgin of an older man's condescension.

But Mr Bush's biggest blind spot stems not from his vanity, but from his utter, simplistic determination to cast the Israeli-Palestinian conflict in the fundamental, black and white, for-us-or-against-us terms of his "war on terror".

"Those who want to destroy the peace process through terrorist activities must be dealt with," he said this week, "There will be no peace if terrorism flourishes... The rise of a peaceful Palestinian state and the long-term security of the Israeli people both depend on defeating the threat of terrorist groups." He went on: "The Palestinian Authority must undertake sustained, targeted and effective operations to confront those engaged in terror."

In other words, Mr Bush seems to have bought, in its entirety, Mr Sharon's Machiavellian proposition that any act of "terrorism", however loosely defined, may constitute justification for more foot-dragging by Israel, or even for a de facto suspension of the entire peace process. On this basis, logically, terrorists who oppose the "road map" process (like those politicians in Israel and the US who also oppose it) will always win.

For Mr Bush, the definition of "terrorist" appears to be almost infinitely expandable in the Israel-Palestine context, as in Iraq and elsewhere.

There is no apparent thought given to notions of legitimate self-defence, or deterrence of prior aggression, or struggle against the most provocative breaches of international law - or simply, against the daily theft of land, liberty and livelihood. His few, supposedly balancing phrases about Palestinian rights and Israeli obligations are hopelessly inadequate.

Mr Bush, it seems, just does not get it. He cannot be bothered to undertake the hard grind or to work the issues, as Bill Clinton did, preferring instead to grandstand, to clutch for credit and compliments and for "leadership moments" that look good on television. As a result, he is dangerously, and sadly, off the pace. For Mr Abbas, his complacency could be fatal. For Mr Sharon, it is a gift.

When a few Palestinian extremists finally run out of patience, or when somebody gets killed by the Israeli army, maybe by accident; and when somebody else retaliates and then, if and when the ceasefire collapses, the two sides turn on each other again, it will not be enough to say it is all the "terrorists'" fault. It will not be enough to shrug and say "we tried".