President Bush is being warmly welcomed at each stop on his South American trip this week. Everywhere his proposals for expanded hemispheric trade are being applauded, and in each country he is hearing calls for closer economic ties with the United States. But behind the bravos and abrazos are some real worries.

The most serious -- and one that exacerbates all the others -- is Latin America's deep and persistent economic crisis. Per capita income in the region has declined again in 1990, poverty is growing worse, massive debt burdens remain a block to recovery, staggering rates of inflation and unemployment continue to plague many countries (notably, Brazil and Argentina) and investment capital is in desperately short supply.

At the same time, and despite Latin America's impressive advances toward democracy, democratic governments in many countries are fragile. In some places, political opening has stopped at the ballot box, and repeated elections have not led to a significant expansion of fundamental freedoms. In others, democratic institutions are being challenged not only by the economic crisis but also by political and criminal violence, deep social and economic inequities, conflicts between military and civilian authorities and an erosion of public confidence.

The second short-lived revolt by army dissidents in Argentina a few days ago is an indicator of how unsettled and unpredictable the situation is in a number of countries. These challenges can only be met by sustained national efforts. But Latin leaders expect the president to recognize that there are a number of things the United States can do to help:

Work closely with the countries of Latin America in order to transform President Bush's Enterprise for the Americas Initiative into a common strategy. The initiative provides official debt relief to some of Latin America's weakest economies. More important, the proposed negotiations toward freer hemispheric trade can offer the right incentives -- i.e., increased access to U.S. markets and investment capital -- to encourage economic reform.

Accelerate the reduction of Latin America's commercial debt burdens. Few countries have yet qualified for the debt relief promised when the administration announced the Brady Plan in early 1989. Most have fallen into deep arrears on their bank loans, undermining business confidence and frightening potential investors. Adequate debt reduction is essential for Latin economic recovery.

Work with the other countries of the hemisphere in promoting negotiated solutions to Latin America's remaining guerrilla insurgencies in Colombia, Peru, El Salvador and Guatemala.

Take the lead in inter-American cooperation to address narcotics trafficking and environmental deterioration.

No one expects President Bush to reshape U.S.-Latin American relations during his one-week visit. But he is demonstrating understanding of the problems facing the hemisphere and the need to work together to forge cooperative solutions.

The writer, a Washington lawyer, is cochairman of the Inter-American Dialogue and was U.S. ambassador to the Organization of American States and co-negotiator of the Panama Canal treaties.