When the UN administered Kosovo from 1999 to (de facto) 2008, Kosovo and all of its cities could technically be described as "under UN law." But this is misleading, because the UN expressly preserved all existing laws "insofar as they do not conflict with [international human rights standards and non-discrimination principles]." as described in this document. This was later amended to give full effect to Kosovo law as it was on March 22, 1989, but it was still subject to modifications by UN regulations and to international human rights standards.
At best, this was a mixture of UN law and Kosovo law, but in reality, I would tend to assume that UNMIK (the UN mission responsible for this work) tried to avoid directly legislating any more than absolutely necessary. As far as I can tell, they immediately worked to establish a democratic system of governance, holding local elections in 2000 and national elections the following year, as well as establishing an assembly (legislature) and other organs of democratic government. After that, the UN had at most a supervisory role in Kosovo's administration. Of course, that was the UN's plan all along - to establish a democratic government and then transfer most if not all of UNMIK's administrative responsibilities to that government.
This is illustrative of a broader point. The UN was not designed to function as a world government. While it does have the practical capacity to administer a territory for a time, this is an anomalous state of affairs. The immediate goal of any UN-administered territory with a non-trivial population will always be to establish, re-establish, or reinforce a democratic system of self-government and then transfer responsibilities into that government until the country is fully self-governing. A similar thing happened in Cambodia in 1992-93, and there have been a few other examples as well.
One or two of those examples don't fit this model, because they are essentially (or actually) demilitarized zones surrounded by some conflict, and will (presumably) be ceded to some country once the conflict no longer requires UN involvement (which may take a very long time to happen). It is difficult (and might be counterproductive) to set up a self-governing system for such a small territory, especially when that territory's ultimate fate may be at issue in the broader conflict. The end result is that these territories are under de facto martial law, but areas with more significant civilian population will usually be administered by one side or the other rather than by the UN. For example, as far as I can tell, both parts of Nicosia are administered by "regular" local governments, and it's just a thin strip of land that's officially UN-administered.