The falling dollar has been an elixir for American companies that have come to rely on sales overseas. For example, Nike, which now gets 60 percent of its sales outside of the United States, said last month that the weaker dollar had increased its worldwide revenue by 3 percent in the past quarter.

But what is giving a lift to American multinationals has become a millstone for European companies competing in the United States.

Take BMW, which said in August that its second-quarter earnings fell 4.6 percent as sales in the United States were eroded by the weaker dollar. The German automaker competes against companies in Japan, whose currency has not risen as much as the euro has, as well as against Detroit. BMW has not recently raised its prices in the United States, but it may have to as its profit margins come under increasing pressure.

With the euro having recently risen to 1.40 to the dollar, the exchange rate has "attained a pain threshold for European companies,'' according to the president of the European business lobbying group, BusinessEurope. Other European currencies have been climbing against the dollar as well. Measured against an index of major-trading-partner currencies, the dollar has reached a 15-year low. Against some currencies, the move is even more extreme; last month the dollar reached parity with the Canadian dollar, known as the loonie, for the first time in 31 years.

Here is the value of the dollar against the euro over the last 12 months:

The problem for European companies is twofold: They must remain competitive with U.S. companies abroad, and they must continue to woo consumers within the U.S, a country that is not only the European Union’s single largest market but also one battling sluggish growth.

''The United States is a huge market and the last thing they want to do is lose too much market share, because it is hard to get it back,'' says Steve Hoch, a marketing professor at the University of Pennsylvania's Wharton School. ''A lot of manufacturers will take it; they’d rather subsidize in the short run than try to go back later and buy back share.”

It's not easy to hold the line on prices, as any seller of French wine, Italian suits, German cars, or Spanish olives can attest. Economic conditions call for a rise in prices of European goods, but that collides with the need to remain competitive. That tension can be seen at any store that has European imports.

''We have a $10-and-under table,'' says Matt DeVriendt, manager of Smith & Vine wine store in Brooklyn. ''Some [European] wines that were $6 or $7 are now priced at $11 or $12. That makes a big difference in the volume we sell.''

Now consider a jump of that magnitude on an airplane-size scale. EADS, the parent company of Airbus, has been locked in a bruising competition with Chicago-based Boeing to sell jets to the world’s airlines.

That would be a tough enough battle under any circumstances, but it is compounded by a weak dollar interceding on Boeing’s behalf. EADS, which is cutting 10,000 jobs as part of a sweeping reorganization, recently noted that every 10-cent drop in the value of the dollar against the euro costs it an additional $1.4 billion in restructuring costs.

If the dollar is “trading at a 23 percent discount to the average currency around the world, that automatically makes our products more attractively priced,” says Sam Stovall, chief investment strategist at Standard & Poor’s Equity Research.

There is an upside to a robust currency, as inhabitants of the 13 European nations that share the euro, as well as Britons, with their strong pound, and Swedes, with their krona, have learned. They have greater buying power, and securities denominated in their currencies are more attractive to the world’s investors. The governor of the French central bank, Christian Noyer, noted that the strength of the euro proves that the currency has become “a safe haven in our globalized financial world.”

The costs of a strong euro, on the other hand, will become increasingly visible as big European companies report results over the next few weeks.

Unilever, the giant British-Dutch consumer products company, reports earnings on November 1. Europe’s largest drugmaker, GlaxoSmithKline, reports on October 23. And the giant French tiremaker, Michelin Group, reports its earnings on October 29. GlaxoSmithKline earns about half its revenue in the U.S., while Michelin Group takes in about a third of its sales there.

For many European companies, “the combination of higher interest rates and a stronger euro will combine to see [earnings per share] growth in low single digits at best,” says Ian Harnett, co-founder of the Absolute Strategy Research firm in Britain.

Already, some companies are warning of lower earnings. Tate & Lyle, the British maker of Splenda sweetener, angered shareholders after saying that a drop in the dollar would hit second-half pretax profits. And Europe's largest papermakers, Stora Enso and UPM-Kymmene of Finland, took a hit after Deutsche Bank downgraded their stocks to ''sell,'' saying that the weak dollar would crimp their export volumes.

European stock prices have so far remained buoyant, but a recent survey found that investor confidence in the euro nations had sunk to its lowest level in two years.

The combination of a weak dollar and a slowing U.S. economy ''is kind of a double whammy for some European export companies,'' says Kurt Umbarger, portfolio specialist for international equities at T. Rowe Price. ''The Unilevers of the world, we just don’t own.''

Yet it is not as simple as that. Many European companies sell goods in nondollar markets overseas. Those that are heavily involved in emerging markets will probably do well, as will those whose main market is still Europe.

Atlas Copco, a Swedish mining-equipment company active in emerging markets, has such strong demand for its products that it ''will more than compensate for any currency headwind [it] will face,'' says Umbarger. Another Swedish company, H&M, has actually benefited from the weak-dollar trend because the majority of the clothing retailer’s goods are made in Asia, where many currencies closely track the dollar.

Even companies dependent on the U.S. market have been able to soften the blow by hedging some of their exposure to currency risk through financial derivatives (which is only a short-term solution), and by diversifying where they buy their raw materials and where they construct their products. Both LM Ericsson of Sweden and EADS use currency hedges. Many foreign manufacturers, including Volkswagen and BMW, use ''natural hedges'' as well. BMW, for instance, has plants in both the United States—its largest export market—and in China. It plans to increase production in both countries.

And even though some big European companies will take hits, Europe is less dependent on the American market than it once was.

''The whole world is less sensitive to the U.S., so the impact of the dollar is less pronounced,” explains S.&P.'s international equity strategist, Alec Young, predicting that the persistent strength in the euro is only ''a moderate headwind for European sales and earnings.''