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.
