The AU force is being deployed to help the interim government bring peace to the country and replace Ethiopian troops who helped mount a successful offensive against the Islamic Courts Union in December that had controlled most of southern Somalia for the previous six months.
Uganda is one of a handful of African countries that have made a definitive commitment to the force.
The AU has only enough pledges for half the number of troops it has said it requires.
Uganda has promised 1,500 troops, Burundi 1,700 and Nigeria 850, while Malawi and Ghana are also expected to contribute.
Tuesday's formal ceremony to welcome the Ugandan troops at Mogadishu airport was attended by officials from the government of Abdullai Yusuf, the Somali president, and warlords who had once controlled the capital.
Two more planes were expected later in the day, Somali officials said.
The Ugandans are due to patrol Mogadishu, one of the world's most dangerous and heaving weapon-infested cities.
Guerrillas who mount almost daily attacks on joint government-Ethiopian forces in Mogadishu have threatened to attack any peacekeepers or government allies.
The guerrillas are suspected to be a mix of Islamists and clan militiamen fighting for control of the city.
It is the first international peacekeeping venture in Somalia since Operation Restore Hope, the ill-fated UN-backed, US-led peace mission launched in December 1992 one year after the removal Mohamed Siad Barre, the country's former dictator.