Next Step : And Now for the Tough Part . . . : Japan’s new Prime Minister Morihiro Hosokawa has won over the public. Next, he must carry through on his promises.

There he was on national television, pointer in hand, beside tables and charts filled with facts and figures on his emergency package to stimulate the limping economy. Japanese Prime Minister Morihiro Hosokawa was at it again: skillfully using the media last week to reach the people, giving them plain talk about a complex situation.

But there was something new in the reaction that followed this appearance of the man who is leading the first government to break the Liberal Democratic Party’s 38-year stranglehold on power--an eclectic coalition of seven parties ranging from the leftist Socialists to the religious Clean Government Party to the hawkish Renewal Party.

“The vintage Hosokawa histrionics were not bad,” said the Asahi newspaper, “but in contrast to the conspicuous manner of the new plan’s announcement, the newly decided policies do not measure up to the ‘responsible changes’ that the prime minister has often proclaimed.”

After six weeks in office, score a resounding success for Hosokawa’s style. He’s fresh, frank and open. He embodies change. He listens well. Despite his noble upbringing and boyhood butlers, he does such proletariat things as pour his own coffee.

All of which has contributed to whopping public approval ratings, ranging from 65% to 79% in polls earlier this month--historic highs.

But as he prepares to meet President Clinton in New York on Sept. 27, it is clear that Hosokawa’s honeymoon is winding down and his stylistic flourishes are no longer surprising. From bureaucrats to the business sector, from his political allies to the public, the chorus is getting louder: Substance, please.

“The new government, bringing political change after decades of stagnation, is off to a remarkably good start,” said Minoru Morita, a political commentator. “Having an appealing, frank leader like Hosokawa helps--but it won’t be enough if the government fails to be effective.”

Consider the mounting tasks:

* U.S.-Japan relations: Swollen trade surplus. Closed markets. Greater international contributions. Sound familiar? They should. The same issues that have riddled the relationship between the world’s two economic superpowers for years still do. But Hosokawa and Clinton will confront them again when both visit New York for the U.N. General Assembly meeting next week.

Personally, the two leaders are likely to hit if off--both are former governors with reformist agendas whose victories marked a political changing of the guard. But policywise, the two sides seem intractably stalemated: The Americans want numerical targets to measure progress in trade talks. The Japanese don’t. No one expects Clinton to hammer Hosokawa in their first meeting, however. Hosokawa is expected to outline his economic program to shave the politically nettlesome $49.6-billion bilateral trade surplus. Clinton is likely to listen politely and encourage--not demand--further measures.

Aside from economics, the two will probably also discuss such issues as the Mideast peace accord--particularly Japan’s contribution to the multinational aid effort promised to support the framework agreement.

“The government recognizes Hosokawa is sitting on a rather fragile coalition, has been in office all of six weeks, has other issues like political reform on his mind and can’t bring that much with him that soon,” a U.S. official said.

* Economic woes: Japan remains gripped by a serious recession. The economy contracted at an annual rate of 2% in the latest April-to-June quarter. Export volume is down, battered by the soaring yen. Personal spending is down. So is corporate investment. And looming on the horizon are widespread layoffs--what the Japanese call “restructuring.”

Hosokawa’s emergency package--$57 billion worth of new spending, deregulatory measures and consumer benefits--was widely dismissed as too weak to jolt the economy back to life. Officials, however, bill it as merely the first step. They hint at other measures, including a possible overhaul of the tax system that would reduce levies on income and hike the consumption tax.

* Political reform: Hosokawa has wagered his entire political fortunes on an issue that has brought down the last two prime ministers. They pledged to clean up the system of “money politics,” failed and were promptly ousted from power. But that did not deter Hosokawa from publicly pledging that if he fails at reform by year’s end, he’ll take responsibility--to wit, resign.

His coalition commands a slim majority in Parliament and recently worked out a compromise plan. But Hosokawa’s ministers are not called “the glass Cabinet” for nothing. The deal could still be easily shattered by coalition members who don’t fully back it, such as some of the Socialists.

* Budget priorities: Hosokawa pledged to enrich the lives of ordinary people by reordering the nation’s spending priorities from producers to consumers. The first budget draft released earlier this month showed little change from LDP blueprints. Some analysts, however, say that it will take as long as two budget cycles for the administration to put its own stamp on the spending plan.

* Foreign relations: Hosokawa made headlines when he called World War II in the Pacific a “war of aggression” and “a mistaken war,” the strongest words ever for a Japanese prime minister. Now the question is whether he’ll back his mea culpas with money. Payments to those who suffered at the hands of the Japanese aren’t legally required, but they may be granted as a gesture of goodwill and compassion.

So far, Hosokawa hasn’t triggered substantive breakthroughs on any of the issues. Supporters say it’s too early to expect more. And, in trying to manage a fragile and sometimes fractious coalition, he can’t afford to move too hastily--even though his personal style is to jump into action.

How Hosokawa successfully manages these areas will depend not only on his political skills but also on how well he continues to work with Shinseito (Renewal Party) leader Ichiro Ozawa, the coalition’s master tactician and political horse-trader.

In recent days, however, fissures in the coalition have begun appearing. Ozawa, for instance, recently had a public spat with Hosokawa’s chief coalition ally, Masayoshi Takemura, over the shape of future political realignment. There also are many news stories about a gradual splintering of the coalition, with the conservative Japan Renewal Party and Buddhist-backed Clean Government Party on one side and Hosokawa’s progressive Japan New Party and Takemura’s New Party Harbinger on the other.

But for now, analysts say that mutual needs still outweigh individual ambition. They give the Hosokawa government a better than even shot at bringing some measure of the change for which the long-suffering Japanese public is clamoring.

“As long as Hosokawa maintains a two-thirds public approval rating, he will have substantial authority,” one Western political analyst said. “The question is whether he can manage the political process, and that’s what Ozawa provides.

“Ozawa doesn’t have public support. Hosokawa doesn’t have the political know-how. For now, they really need each other.”