For Dukakis, a battle is won, but war goes on. He still must deal with Jesse Jackson, while GOP turns up the heat

Michael Dukakis can already hear the political cannons of summer. Despite a sweeping four-state victory Tuesday that clinched the Democratic nomination, Governor Dukakis faces major challenges on two fronts leading up to next month's convention in Atlanta.

The governor's most urgent problem involves Jesse Jackson, his main Democratic rival. The Rev. Mr. Jackson will march into the convention with more than 1,100 delegates - enough to make the ground shake in Atlanta.

Mr. Jackson has begun laying down his demands. He wants more spending for education, health care, and farmers. He threatens to fight Mr. Dukakis on the convention floor unless the governor agrees to brand South Africa a ``terrorist state.''

Simultaneously, Republican George Bush is escalating his criticism of Dukakis. In California, he began defining Dukakis as a taxing, spending ``McGovernite'' liberal who is out of touch with most American voters.

``That's nonsense,'' Dukakis retorts.

But Republican attacks are increasing. Gov. George Deukmejian of California, a Bush ally, told a TV interviewer this week that Dukakis is ``Walter Mondale when it comes to taxes ... a Tip O'Neill when it comes to spending ... a Teddy Kennedy when it comes to the issues of crime, and ... a George McGovern when it comes to defense issues.''

Although Dukakis looks strong at the moment, his strength could be fleeting. American voters don't know him well. Larry Hugick, an analyst with the Gallup Organization, says the electorate is so undecided that ``at least half the voters right now could go either way.''

Dukakis's most difficult challenge could be increased scrutiny of his record, says John Chubb, a scholar at the Brookings Institution. Dukakis has soared in the polls on a wave of publicity coming from his victories in the primaries. Now that will end. Mr. Chubb adds:

``Inevitably, when there aren't any victories to write about, there will be more scrutiny. At that point, it is an open question how he will fare.''

Nelson Polsby, a political scientist at the University of California, Berkeley, says Dukakis's problems with Jackson outweigh everything else.

According to Dr. Polsby, the governor's handling of Jackson will be an important indicator for all voters, both black and white. His ability to deal with issues raised by Jackson on their merits, and his strength under pressure, will be critical, Polsby suggests.

Gary Nordlinger, a Democratic consultant in Washington, says that while Dukakis fends off Jackson, he must also quickly lock up the uncommitted delegates to the convention. Any delay in winning over the uncommitteds could give the campaign a bad image, he says.

Political veteran Horace Busby says Dukakis will also need to handle skillfully the interest groups, such as labor, that will now become deeply involved in his campaign. How Dukakis deals with labor, and whether he can maintain his independence, will be a test of his leadership.

Dukakis's lack of foreign policy experience could be his most damaging problem, says G.Donald Ferree, associate director of the Institute for Social Inquiry at the University of Connecticut.

``That needs to be worked on,'' Mr. Ferree says, ``especially since Dukakis wants to make terrorism, Iran-contra, and drugs into big issues for the campaign.''

Richard Brody, chairman of the political science department at Stanford University, agrees with Ferree.

``Foreign policy is one of the key areas where he has to create a credible impression,'' Dr. Brody says.

While experts warn of the dangers ahead for Dukakis, they also sing praises for Dukakis after his primary triumphs.

William Sweeney, a Democratic consultant, observes: ``It's only six months since Michael was one of the `seven dwarfs.' He is no longer a dwarf, he is the Democratic nominee. The challenge now is to put an agenda on the table and give people a reason to listen to him and vote for him.''

Dukakis, who once described himself as the dwarf with the bushy eyebrows, rolled like a political giant through California, New Jersey, Montana, and New Mexico Tuesday. His victories gave him an estimated 2,249 delegates, well above the required 2,081 for the nomination.

He did that without creating deep divisions in the party. Even Jackson supporters, by 36 percent to 32 percent, have a favorable impression of Dukakis, according to a Los Angeles Times survey of California voters. Overall, California Democrats were favorable toward Dukakis by a huge 69-to-16 percent margin, which bodes well for November.

As Dukakis savored his victories, there was increasing talk of confidence - and even cockiness - in the Dukakis camp. Some of that confidence comes from the political polls, which show the governor pulling far out in front of Bush, both nationally and in some states, including California.

Yet Mr. Busby scoffs at the current polls. `You should disregard them,'' he says. ``Outside of Massachusetts, the country does not know Mike Dukakis well enough to mean anything. These polls are ridiculous.''

Mr. Sweeney points out: ``People forget how far Jimmy Carter was ahead of Gerald Ford in 1976 and how close Ford came.''

For Busby and many others, the fascinating question is how Jackson will play his hand at Atlanta, and how Dukakis will respond.

``Jackson himself has become much more hard-line than I thought he was going to be,'' Busby says. ``He has been shut out. And he has his pride.''

Polsby says Jackson's main power now lies in his ability to hurt the party.

Brad O'Leary, president of the largest Republican fund-raising organization, says that once Dukakis gets beyond the Jackson problem, he must grapple with the logistics of running a national campaign as a candidate about whom voters know very little.

Like previous Democratic nominees, Mr. O'Leary says, Dukakis will draw on special-interest groups for help, especially big labor. He must do that at a time that he will be appealing to middle-of-the-road independents and Democrats, which could be a delicate balancing act.

Meanwhile, Jackson - after a two-day rest with his family - will continue campaigning for uncommitted delegates until the convention. Jackson was upbeat, despite his losses. He observed that he won a total of more than 7 million votes during the primaries - more than Jimmy Carter in his winning campaign of 1976, or Walter Mondale in 1984.

Jackson gives mixed signals about his relationship to Dukakis, but vows that they can ``compete without conflict, and differ without division. ... I salute him for a fine campaign.''