We propose PUNO, a novel deep operator-based framework for point cloud upsampling, addressing the challenge of reconstructing high-resolution geometries from sparse point clouds. PUNO first uses a lightweight network to achieve vertex displacement and manifold parameterization, forming a coarse geometric representation. It then employs a lifting function to encode coordinates into high-dimensional embeddings, followed by iterative kernel integral approximations in function space and a dimensionality reduction to generate the target coordinates. Unlike prior work, PUNO transforms both in data space and function domain, achieving finer and more continuous results that mitigate the ill-posed nature of sparse data. It also introduces multilayer Galerkin-type attention for nonlocal kernel integrals, benefiting global continuity. Extensive experiments demonstrate superior accuracy, robustness, and generalization.