HARTFORD, Conn. — Lawyers for right-wing provocateur Alex Jones and the families who won nearly $1 billion a month ago for being targets of his Sandy Hook school conspiracy theories argued Monday over another potentially extraordinary award of punitive damages for what was described in court as his “evil” and “reprehensible” behavior.

A Waterbury jury awarded the relatives $965 million in October as compensation for the decade of threats and harassment they suffered by members of Jones’ vast broadcast audience who believe his claims that the 2012 elementary school massacre was a hoax and they were actors in a scheme to outlaw gun ownership.

Norm Pattis, an attorney of Infowars founder Alex Jones, urged a judge on Monday to award only nominal, largely symbolic, punitive damages. Bryan Woolston/Associated Press

In its verdict, which Jones’ lawyer called unprecedented in Connecticut and perhaps anywhere else, the jury also awarded punitive damages against Jones after finding the harm his broadcasts caused the relatives of the shooting victims to be especially egregious.

Under state law, Superior Court Judge Barbara Bellis, who presided over the trial, will decide what the victims get in punitive damages after weighing wildly disparate arguments from the parties Monday. Under one calculation given Bellis by lawyers for the relatives, combined punitive damages could approach $3 trillion.

Bellis heard about an hour of argument Monday and did not indicate when she will reach a decision.

Attorney Norman A. Pattis, who represents Jones and his principal business, Free Speech Systems, urged Bellis to award nominal, largely symbolic, punitive damages, saying the $965 million compensatory verdict is so large that it achieves the purposes of punitive damages, which include deterrence.

“If a billion doesn’t do it,” Pattis said, “then a trillion is not going to do it. Or 3 trillion.”

Christopher Mattei, who argued for the relatives, did not recommend specific figures for punitive damages. But in related court filings, he suggests calculations that result in extraordinary sums – one is $2.75 trillion for violations of the state’s unfair trade practices law and about $320 million more for the relatives’ legal fees and costs.

Mattei said the punitive damages should reflect the amount of the unusually large $965 million compensatory award, which he said was fair and reasonable in view of conduct by Jones that he called “reprehensible,” “evil” and “just malicious.”

Infowars founder Alex Jones has already been ordered by a jury to pay Sandy Hook families $965 million for spreading the lie that the 2012 school massacre was a hoax. Tyler Sizemore/Hearst Connecticut Media via AP, file

He said Jones violated the unfair trade practices law, which places no limit on punitive damages, by repeating specious theories that he knew to be false on broadcasts to an audience of millions because he knew that Sandy Hook denial programming caused both his audience and sales at his retail sites to spike.

“It was really unspeakable,” Mattei said.

Fourteen relatives of shooting victims and an FBI agent who was part of the law enforcement response sued Jones and gave gripping accounts while testifying of being tormented by Sandy Hook deniers.

The relatives described strangers appearing at their homes and demanding proof that their children or spouses were in fact dead. There were death threats and threats of other violence. Many were under continual assault on social media.

As punishment for his failure to comply with orders to disclose information about his broadcast and internet businesses, Bellis issued a pretrial default against Jones that restricted his defense, found him liable for defamation and other offenses and ruled that his broadcasts were the cause of the harassment.