KANSAS CITY, Mo. (AP) _ A federal judge has ordered $10.7 million in tax refunds for property owners in the Kansas City School District, but only to the 9,000 taxpayers who filed formal protests.

The refund involves a property tax increase ordered by U.S. District Judge Russell G. Clark in September 1987 to help finance school desegregation.

The U.S. Supreme Court in April declared the 1987 increase unconstitutional. Its unanimous ruling said Clark exceeded his legal authority in directly raising the district's tax rate from $2.05 to $4 on each $100 assessed value. But it also said federal judges may order local officials to raise property taxes to help desegregate public schools.

The money to be refunded has been held in an escrow account by Jackson County and will not come out of the district's current budget or from the state.

''I thought that was good for the district, because we don't have any money,'' said school board President Julia Hill. She applauded Clark's order implementing the refund. ''That's what the school board asked for. ... We're quite pleased about that.''

Clark's refund order, which was dated Oct. 29 but arrived from his office in Springfield on Wednesday, said refunds would go only to the taxpayers who followed the protest process required by Missouri law.

Homeowners due refunds will receive $37.05 for every $10,000 of property value.

Arthur Benson II, lawyer for the plaintiffs in the desegregation case, said that limiting refunds to the people who protested was appropriate. He did not protest his property taxes and will receive no refund.

He said he was sympathetic to those who failed to protest their tax payments. ''I'm sorry for them, and I'm sorry they won't get a refund,'' Benson said. ''But people have an obligation to protest their rights.''

Some of Kansas City's largest businesses should receive a big portion of the refunds. Hallmark Cards Inc., for example, protested more than $1 million in tax payments.

Thousands of people also paid their 1988 and 1989 property taxes under protest, but none of those payments will be refunded, Clark ruled.

The Supreme Court ruling held that the method used to set the district's tax rate in 1988 and 1989, even though it did not rely on a vote of the people, was constitutional. Because of that, Clark said, there was no cause to refund those payments.

On Monday, the Supreme Court let stand a ruling that boosted Clark's power to order predominantly white suburban schools to accept inner-city black students. The justices, without comment, rejected an appeal by three suburban school districts facing such a desegregation order.