PHOENIX — The showdown over Arizona’s immigration law played out in court and on Phoenix’s sun-splashed streets on Thursday, as the state sought to reinstate key parts of the measure and angry protesters chanted that they refused to “live in fear.” Dozens were arrested.

A federal judge’s decision a day earlier to block the strict law’s most controversial elements didn’t dampen the raging immigration debate.

The judge has been threatened. Protesters rallied in cities from Los Angeles to New York. The sheriff of the state’s most populous county vowed to continue targeting illegal immigrants. Lawmakers or candidates in as many as 18 states say they still want to push similar measures.

Along the U.S.-Mexico border, life continued as before, with officials sending back people who were captured while attempting to cross.

In Phoenix, hundreds of the law’s opponents massed at a downtown jail, beating on the metal door and forcing sheriff’s deputies to call for backup. Officers arrested at least 32 people, and dozens more were detained elsewhere throughout the day.

Activists focused their rage at Maricopa County Sheriff Joe Arpaio, the 78-year-old ex-federal drug agent known for his immigration sweeps.

Outside his downtown office, marchers chanted “Sheriff Joe, we are here. We will not live in fear.” One was dressed in a papier-mache “Sheriff Joe” head and prison garb. Arpaio said he’d continue with a Thursday sweep.

“I’m not going to be intimidated and stopped,” he said. “If I have to go out and get in the car, I’ll do it.”

Activists, armed with video cameras and aided by others listening to police scanners, roamed the county’s neighborhoods, saying they were ready to document any deputies harassing Hispanics.

In Tucson, between 50 and 100 people on both sides of the issue gathered at a street corner. About 200 protesters blocked a busy Los Angeles intersection, with police arrested about a dozen who were linked with plastic pipes and chains.

In New York, about 300 immigrant advocates rallied near the federal courthouse in lower Manhattan.

“It’s one step closer for us, but I think the fight is still ahead,” said Adelfa Lugo, a 56-year-old Mexican-born Brooklyn resident who joined the protest. “If we don’t fight this in Arizona, this anti-immigrant feeling will spread across the country.”

Since Wednesday’s ruling, U.S. District Judge Susan Bolton has received thousands of phone calls and e-mails. Some were positive, but others were “from people venting and who have expressed their displeasure in a perverted way,” said David Gonzales, the U.S. Marshal for Arizona.

Gonzales said his agents are taking some of the threats to Bolton seriously. He wouldn’t say how many there were or whether any threats were coming from recognized hate groups. He refused to discuss any extra security measures, which U.S. marshals routinely provide federal judges.

The protests came as Gov. Jan Brewer appealed Bolton’s ruling to the 9th U.S. Circuit Court of Appeals in San Francisco.

Arizona has more than 400,000 illegal immigrants, and its border with Mexico is awash with smugglers who funnel narcotics and immigrants throughout the U.S.