A distributional Hessian is the second covariant derivative of a function interpreted by integration
by parts against test vector fields and functions.
In the weak calculus of an RCD space, a locally 
function 
has the evaluation
 |
where 
, 
is a compactly supported
bounded globally Lipschitz function in the
Laplacian domain with 
, and 
is a compactly supported Lipschitz function. On a smooth
manifold this agrees with integrating the ordinary Hessian
applied to 
twice.
