With the U.S. visit of Egyptian President Anwar Sadat beginning today, the Carter administration is entering a critical period in its relations with the Arab nations.

The decisions the United States makes in the coming weeks, Arab officials and U.S. analysts of Middle Eastern affairs say, could have a major impact on Washington's interests in the Arab world.

The most urgent question, and the one that will be at the center of Sadat's talks with President Carter, is what to do about the rising tide of demands by Egypt and Saudi Arabia for strong American action that would force Israel to yield on issues that have deadlocked the Middle East peace negotiations.

The Americans have already made it clear that they are not going to be stampeded by Arab warnings that cendancy of moderate Arab leaders could be jeopardized if Washington fails to act in some decisive way. Those arguments have been heard many times before.

Sadat reportedly has been told that his visit is not going to produce what he and other Arabs would like most, an unequivocal U.S. threat to cut off aid to Israel if the Israelis continue with policies the United States opposes.

But there is a substantial body of informed Arab and American opinion that is the absence of such a major policy shift the United States will have a difficult time giving the Arabs enough satisfaction on other issues to hold together the relationships it has developed with such countries as Saudi Arabia, Egypt, Sudan, Jordan and even Syria.

The disillusionment of the Egyptians, the impatience of the Saudis, the discomfiture of the Jordonians are beginning to show. Whether the United States can coddle them along by selling them sophisticated weapons or backing an Arab role in the war on the Horn or Africa or other gestures is problematical.

The Arabs want the United States to force Israel to give up the occupied territories and grant political rights to the Palestinians. So warnings of dire consequences from failure to do so suit Arab purposes. But there is no doubt that the present situation carries the seeds of real trouble for U.S. - Arab relations.

Sadat's peace initiative, a prominent American student of the Arab world wrote privately, "has given us a challenge. Perhaps he had to. Anyway, he has. What we choose to do, and avoid doing, in the coming days and weeks will be the real event."

Secretary of State Cyrus Vance has said that the failure to resolve the Palestinian question once and for all would leave a "continuing, festering sore that will worsen as time goes by."

But the challenge for the Carter administration lies not just in trying to help Egypt and Israel negotiate a peace settlement. It lies also in explaining to other Arab governments that are essentially pro-American why the United States is neither willing nor able to impose its will on the Jewish state, no matter how great its interests in the Arab world.

In this decade, U.S.-Arab relations, on balance, have added up to what one high ranking Egyptian calls "a big success story." The complete break after the 1967 war has faded into history, giving way to close ties with some Arab states, joint economic commissions with several, and regular diplomatic consultation.

The once dominant influence of the Soviet Union among the hundred million Arabs had diminished considerably, and the United States is enjoying at least temporary stability in oil prices and supply levels, despite the upheavel of the 1973 war.

The argument being advanced by the Egyptians, and other Arabs, is that all this could be jeopardized if the United States fails to move Israel toward peace terms the Arabs could accept.

In simplest terms, the Egyptians argue that since their terms for peace and the state policies of the Carter administration are not very different, the Americans would be acting in their own self-interest if they turned up the heat under the Israelis.

A corollary to this is the argument - also certain to be raised during Sadat's sojourn in Washington - that it is in the best interest of Israel too to take this opportunity to achieve peace on terms the Arabs can accept as honorable. If the Israelis are too shortsigted to realize that themselves, the Arabs argue, it is time to put into practice what is known here as the "George Ball Theory." That refers to the argument of former Under Secretary of State George Ball that Israel's current policies will ultimately prove self-destructive and so the United States should "save Israel in spite of itself."

Sadat apparently believed that these arguments, all raised in the past could be left behind after his trip to Jerusalem. When the Israelis failed, or refused to respond in kind to his leap of faith, he turned once again to Washington to bring Israel around - not just to the administration but to the American public and especially the American Jewish community, hoping they would be more prepared to see things his way than the Carter administration, which he viewed as timorous.

One reason Sadat broke off direct negotiations with the Israelis, according to authoritative sources, was his displeasure at finding the Americans participating in the talks as brokers between two sides whose views they treated with equal deference, rather than as advocates for the Egyptian view.

This sour view of the U.S. performance has been communicated to the Saudis, Kuwaitis and other Persian Gulf states that were already dubious about American sincerity, and it was one of the elements that weighed in Sadat's decision to take his case directly to the United States at this time, according to Egyptian officials.

Sadat does not actually have the option of taking Egypt back into the Soviet orbit; for example, now is another oil embargo likely except in the event of war.

The Sadudi oil minister, Sheikh Zaki Yamani, created little stir when he warned again last week that Saudi Arabia's response to increased American demand for Saudi oil would depend on the establishment of "a just and durable peace" in the Middle East. The impact of his words is diminished because he has said similar things in the past.

The Americans are also resisting what they see as an essentially simplistio Arab case because they believe the Carter administration has already put considerable distance between itself and Israel on many issues.