In this entry, we prove the following interesting result
Let be a smooth function. Then
for each fixed.
For simplicity, let us write . Then
Proof. We recall from this entry that for each matrix ,
This and the fact that
give us after differentiating with respect to
Summing with respect to we get
Again, it is not hard to see that
which completes the proof.
Source: L. Evans, Partial differential equations, AMS, 1998.