Usage patterns
〈場所〉の覇権を握るto seize hegemony over a region
That country is trying to seize hegemony over Asia.
覇権を争うto contend for supremacy
The two great powers are competing for supremacy.
覇権を確立するto establish hegemony
The empire established hegemony over the region.
覇権を維持するto maintain hegemony
They are maintaining their hegemony through economic power.