Judge Christopher S. Brasher, of the Fulton County Superior Court, said in a 14-page ruling that he believed state and county officials were complying with the Georgia elections standards that govern voter registration.

Judge Brasher, who sits in Atlanta, issued his decision four days after he heard oral arguments in the litigation filed by the NAACP and the New Georgia Project, an officially nonpartisan group founded by the state House’s Democratic leader, Stacey Abrams.

The organizations contended in their lawsuit that the Georgia secretary of state, Brian P. Kemp, and certain county election boards had not properly handled applications from more than 40,000 residents, many of them minorities and young people.

The groups argued that regulators did not quickly process applications and had not contacted prospective voters who submitted incomplete forms.

The organizations alleged that the decisions essentially amounted to a lockout from the polls for thousands of Georgians.

But Judge Brasher wrote: “Not only is there no proof that the respondents have failed to fulfill their duties, but there is also affirmative proof to the contrary. The secretary of state and the county registrars…