On any compact Riemannian manifold with no boundary, the integral of the squared Hessian of sqrt(u) is bounded by a universal constant times the integral of u times the squared Hessian of log u.
Stavrov , Curvature of Space and Time, with an Introduction to Geometric Analysis, Student Mathematical Library, vol
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A functional inequality between Hessians in spaces with non-zero curvature
On any compact Riemannian manifold with no boundary, the integral of the squared Hessian of sqrt(u) is bounded by a universal constant times the integral of u times the squared Hessian of log u.