In this paper, we develop the \(L^2\) theory for Riemannian and Hermitian foliations on manifolds with basic boundary. We establish a decomposition theorem, various vanishing theorems and a twisted duality theorem for basic cohomology groups. As a corollary, we derive an extension theorem for basic forms of the induced Riemannian foliation on the boundary. We also prove the complex analogues for Hermitian foliations by extending Morrey’s basic estimate to a foliated version, and obtain its geometric characterization.
