World leaders attending the G-20 summit sent a strong message to Europe on Friday that it must do more to manage its spiraling debt and financial problems but offered no explicit help, saying the region’s fate was in its own hands.

After two days of talks dominated by the crisis in Greece, the most tangible outcome was an agreement by the cash-strapped government of Italy to allow its economy to be intensely monitored by the International Monetary Fund. Group of 20 leaders also said they would begin studying ways the IMF might provide faster help around the world in a crisis.

Underlying those announcements was impatience about how long it is taking the 17-nation euro region to make key decisions — for example, about how to guarantee that Italy will not be abandoned by global investors and need a costly bailout.

“We . . . urge rapid elaboration and implementation” of recent rescue programs that remain to be fleshed out, the group’s final communique states.

For the Obama administration, the summit amounted to small victories surrounded by two days of frustration. Despite months of prodding by the White House and Treasury Secretary Timothy F. Geithner, European leaders have not moved with the convincing force U.S. officials sought.

“Having heard from our European partners over the past two days, I am confident that Europe has the capacity to meet this challenge,” President Obama said in a news conference at the summit’s conclusion. “I know it isn’t easy, but what is absolutely critical, and what the world looks for in moments such as this, is action.”

He added, “Make no mistake — there’s more hard work ahead and more difficult choices to make.”

The G-20 had kind words for Obama’s approach to U.S. economic troubles, saying his $447 billion proposal to spur job growth, which has been blocked in Congress, represents the kind of step that could invigorate the global economy in the short run. Deficit reduction, which the administration has set as a longer-run goal, is also important for heavily indebted countries, the group said.

China, meanwhile, agreed to move “more rapidly” to float its exchange rate and open up its financial system. Administration officials say the language represents a more serious recognition by China that its own economy is at risk from Europe’s problems. With demand flagging in the developed world, “China will rebalance . . . towards domestic consumption by implementing measures to strengthen social safety nets, increase household income and transform the economic growth pattern,” a separate summit action plan states.

The focus of the conference was on Europe, where political turmoil in Greece has raised the risk of a fracture in the euro currency union. Seven of the G-20’s members are European — six countries plus the European Union — but the crisis on the continent has come to represent a threat to the global economy.

Concerns have been rising that political paralysis in Italy could soon put the euro region’s third-largest economy in jeopardy. Investors have been demanding higher interest rates to lend Italy money, and the cost could soon become unsustainable for the government in Rome.

Amid high-level discussions about Italy’s plight, Italian premier Silvio Berlusconi “invited” the IMF to complete quarterly reports about his government’s performance on controlling spending and carrying out economic reform. That level of oversight is more typical for countries under an IMF emergency loan program.

IMF Managing Director Christine Lagarde said the aim of those steps is to rebuild confidence that Berlusconi will follow through on promises to cut spending and make economic changes.

“The problem is a lack of credibility” that the plans announced by Berlusconi will be implemented, Lagarde said. The quarterly IMF reports will be about “verification, certification and implementation,” she said, adding that “it is now the IMF’s credibility that is a little bit on the line.”

Though G-20 leaders made no promises about providing outside support for the euro region — whose $14 trillion regional economy rivals the United States — the group did direct the IMF to expand the help it could provide in a crisis.

Lagarde said one new proposal will be submitted to the IMF board soon: the creation of a “precautionary liquidity line” that would allow fast, six-month loans to countries considered “innocent bystanders” to problems elsewhere.

However, IMF officials said the program would probably not apply to countries such as Italy that face problems of their own making.

Coming into the summit, world leaders had discussed raising money from outside the euro zone — for instance from China, Russia and Brazil — to help pay for Europe’s emergency rescue program. But the suggestion got a cool reception. A senior U.S. official said G-20 leaders ultimately felt Europe needed to first “put more force and clarity” behind its own decisions.