Their second summit in Hanoi in February ended without any agreement due to disputes over the extent of sanctions relief North Korea should win in return for a limited denuclearization step.
The United States and 25 other countries had accused North Korea of violating UN sanctions by importing far more than the annual limit of 500,000 barrels of refined petroleum products.
While Trump has struck an optimistic tone in public, there are indications Pyongyang is not moving quickly enough towards abandoning its nuclear weapons.
Abe, who one year ago warned at the United Nations that the window for diplomacy with North Korea was closing, took a more open but still cautious tone in his latest address to the world body.
North Korea test fired what appeared to be another intercontinental ballistic missile, hours after the US and Japan moved to step up sanctions against Pyongyang.
The Security Council adopted two sanctions resolutions last year to ramp up pressure on Pyongyang and deny leader Kim Jong-Un the hard currency needed to fund his military programs.