Presidential assistant Stuart E. Eizenstat strongly appealed to United Mine Workers leaders last week to stick with President Carter for a brighter future.

Eizenstat said Carter is the only presidential candidate who has demonstrated a commitment to shifting the nation's floundering oil-fueled economy to a greater dependence on coal.

The administration is backing coal-conversion programs at many electric generating plants that relied on oil. Carter also is fighting hard for a coal-based synthetic fuels program, Eizenstat said.

"We will not be timid" on the matter of coal-based synthetic fuels, he said.

"We are serious . . . We are going all out on synthetic fuels."

All of this, of course, means more jobs -- thousands of them -- for the nation's coal miners, said Eizenstat. He insisted that Carter can and will deliver -- that, unlike other unnamed Democrats chasing the presidency, Carter has the power to implement his rhetoric.

"Now I'd like to answer your questions," Eizenstat said after his nearly-20-minute discourse on the administration's coal-use initiatives.

There was a momentary silence in the Senate Room of the Capital Hilton Hotel, where some 42 members of the UMW's international executive board and district presidents group had gathered to discuss the union's upcoming constitutional convention in Denver.

Many of the union leaders appeared consumed with thought of problems like that in West Virginia, where nearly 10,000 miners have been laid off because of the slumping coal market.

And so, not many in the group were surprised when the short, rotund, baldin and tough-talking Lou Antal, a UMW district president from Pittsburgh, got up to tell Eizenstat he didn't believe him.

"I don't mean to criticize you, Mr. Eizenstat," Antal said. "But everybody's talking about the president's energy policy, and where is it?

"We're here talking about what we need now. And you're talking about what's going to be done 10 years from now. I don't see anything coming, Mr. Eizenstat."

Eizenstat did his best to counter that blast, saying that the president is working for prompt results, for a "massive" change from oil to coal. But the doubters remained.

"Hey," said a district representative from Alabama, pulling a reporter's coat sleeve. "You don't think that fella Eizenstat came here for political reasons, do ya? Ya do believe he cares about us, don't ya?"

He laughed.