Some groundcovers for underneath oaks include Mondograss (monkeygrass), Liriope, Aztecgrass (looks like liriope except is white with green stripes), Cast Iron Plant (taller than most groundcovers but very tough), and Asian jasmine (best for shady locations beneath Live Oaks).
Perennials for sun is a very long list. I would consider the many type of Salvia including Mystic Spires and Mexican Bush Sage (Salvia leucantha). Also perennial types of hibiscus, perennial grasses such as Maiden Grass (Miscanthus) and Fountain Grass (Pennisetum), and lantanas.
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