Responsibility for planning is split between the County Council, district and borough councils, and South Downs National Park Authority. You can find guidance on the planning process on the Planning Portal.
South Downs National Park Authority
Responsible for planning within the national park.
District and borough councils
They deal with all other planning applications: building regulations and building control; listed buildings; conservation areas; tree preservation orders; local planning policy; community infrastructure levy (CIL) and neighbourhood plans.