Compact resolvent beyond positive curvature

Establish whether the free Laplacian associated with closable free difference quotients has compact resolvent under finite free Fisher information or Lipschitz conjugate variables without a positive curvature assumption.

Background

The paper studies compactness of the resolvent of the ordinary free Laplacian Δ=∂*\bar∂ associated with a finite self-adjoint tuple. Before the positive-curvature theorem, the authors explain that compactness is not known under several strong analytic regularity assumptions, including finite free Fisher information and Lipschitz conjugate variables. The paper resolves the problem only under the additional condition CD(c,∞) with c>0, leaving the broader regularity setting unresolved.