Families of victims of the 2012 Sandy Hook mass shooting have asked a Connecticut judge to order Alex Jones to pay hefty punitive damages on top of what a jury said the conspiracy theorist owes them for falsely claiming the massacre was a hoax.

The families said in the filing on Friiday that the “historic” scale of Jones’ wrongdoing in the case, his “utter lack of repentance” and clear intention to continue spreading lies about them deserve the highest punishment in the court’s power.

A jury awarded more than a dozen relatives of victims $US965 million ($1.5 billion) in compensatory damages earlier this month after a trial in Waterbury, Connecticut, not far from where a gunman killed 20 children and six staff members at Sandy Hook Elementary School in December 2012.

Jones and his Infowars site claimed for years that the shooting was staged using actors as part of a government plot to seize Americans’ guns. The families say they suffered a decade of harassment and death threats from people who believed the bogus claims.

The Connecticut jury found Jones and Infowars parent company Free Speech Systems LLC should also pay punitive damages, which are set to be determined by a judge after several days of hearings next month.

In their filing Friday, the families offered various methods of calculating punitive damages in similar cases. They said that by one metric, they could be entitled to $US2.75 trillion ($4.35) trillion based on the number of article impressions Infowars’ false stories garnered.

But the plaintiffs left it to Judge Barbara Bellis to determine the appropriate amount, which they said should be the highest in her power.

Also on Friday, Jones’ lawyer asked Bellis to substantially reduce the verdict, saying it is “exorbitant, shocks the sense of justice and was influenced by partiality and prejudice”.

The Connecticut verdict came several months after a jury in Texas awarded two Sandy Hook parents $US49.3 million ($A77.9 million) in damages.

A lawyer for Jones has said he would seek to cut the award substantially before it is finalised, calling it excessive under Texas law.

Free Speech Systems filed for bankruptcy in Texas in July, but experts are doubtful the case will allow Jones to shield his assets and avoid paying the judgments.

– AAP