AUSTIN, Tex. — In the more than five years since a mass shooting at Sandy Hook Elementary School, the conspiracy theorist Alex Jones has used his Infowars media operation to spread false theories that the massacre was a hoax staged by the government in an effort to tighten restrictions over firearms.

That is not in dispute.

But in a courtroom hearing here, lawyers for the parents of a child killed at Sandy Hook, in Newtown, Conn. — where 20 first graders and six adults were shot and killed in December 2012 — and lawyers for Mr. Jones argued on Wednesday over whether he was maligning the family or the mainstream news media in an April 2017 broadcast titled, “Sandy Hook Vampires Exposed.”

The families of nine Sandy Hook victims — including that of Noah Pozner, who was 6 when he was killed — have filed cases against Mr. Jones, whose operation is based in Austin.

The hearing Wednesday was the first in three defamation cases to reach a courtroom. Among the thornier questions it raises: Does speaking publicly about the killing of a child turn the child’s parents into public figures, making it more difficult to win their case?