Attorneys for X Corp. and a research organization that studies online hate speech traded arguments in court Thursday after the social media platform sued the non-profit Center for Countering Digital Hate for documenting the increase in hate speech on the site since it was purchased by Elon Musk.

X, formerly known as Twitter, alleges the center's researchers violated the site's terms of service by improperly compiling public tweets, and that its subsequent reports on the rise of hate speech cost X millions when advertisers fled.

U.S. District Court Judge Charles Breyer appeared skeptical during oral arguments Thursday in San Francisco, questioning X's attorney how the center violated any platform rules simply by reporting on posts that were already publicly available.

"I can't think of anything basically more antithetical to the First Amendment than this process of silencing people from publicly disseminating information once it's been published," Breyer said during back-and-forth with X's attorney.

The case is being watched closely by researchers who study social media and the way it both reflects and shapes public discourse.

In its suit, filed in the Northern District of California, San Francisco-based X alleges that the center's researchers improperly collected a vast amount of data for its analysis, using third-party software to "scrape" the site. Such actions violated the terms of service that all users agree to, said Jon Hawk, an attorney for X.

The company is seeking millions of dollars in damages to compensate for lost advertising, and the staff time it took to look into how the center compiled its reports.

"When they published the report and the advertisers saw the report, then they stopped spending money," Hawk said.

Attorney John Quinn, arguing for the researchers, said they only used automated search tools to analyze posts that were publicly available on the site, and that X's lawsuit is a poorly thought out effort to silence its critics.

"Given the nature of what happened here, the use of a search function to look at tweets, I think that's a hard case to make," Quinn said.

The center is a nonprofit with offices in the U.S. and United Kingdom. It regularly publishes reports on hate speech, extremism or harmful behavior on social media platforms like X, TikTok or Facebook.

The organization has published several reports critical of Musk's leadership, detailing an increase in anti-LGBTQ hate speech as well as climate misinformation since his purchase.

The center is not the only group that has pointed to the rise of hateful material on X since Musk's purchase in October 2022. Last November, several big advertisers including IBM, NBCUniversal and its parent company Comcast, said that they stopped advertising on X after a report from the liberal advocacy group Media Matters said their ads were appearing alongside material praising Nazis. It was yet another setback as X tries to win back big brands and their ad dollars, X's main source of revenue. X has also sued Media Matters.

Later that month, Musk went on an expletive-ridden rant in response to advertisers that halted spending on X in response to antisemitic and other hateful material, saying they are are engaging in "blackmail" and, using a profanity, essentially told them to go away.

Thursday's hearing was called after the center filed a motion to dismiss X's lawsuit. Breyer said he will take the motion under consideration.

Musk is a self-professed free speech absolutist who has welcomed back white supremacists and election deniers to the platform, which he renamed X last year. He initially had promised that he would allow any speech on his platform that wasn't illegal. "I hope that even my worst critics remain on Twitter, because that is what free speech means," Musk wrote in a tweet last year.

Nevertheless, the billionaire has at times proven sensitive about critical speech directed at him or his companies. Two years ago he suspended the accounts of several journalists who covered his takeover of Twitter.