A note on the boundedness of the Wolff potential on complete noncompact manifolds in Zygmund spaces

In this note, we prove sharp necessary conditions for the boundedness of the manifold-adapted Wolff potential between Zygmund spaces on complete noncompact Riemannian manifolds with nonnegative Ricci curvature. The nonlinear homogeneity of the operator determines the natural form of the norm inequality, while the Bishop comparison theorem and localized ball tests determine the admissible Sobolev scaling. The boundedness assumption forces Euclidean lower volume growth, identifies the relation between the source and target integrability exponents, and gives the critical logarithmic constraint for the Zygmund indices. In the linear case, the conclusions reduce to the corresponding Riesz-potential scaling with the expected logarithmic refinement.