Just for the sake of completeness, I begin defining the Sobolev space $H^m(\mathbb{R}^n), \; m \in \mathbb{N}$, as the following set: $H^m(\mathbb{R}^n) = \{u \in L^2 : P^{\alpha} F u \in L^2,\; \forall |\alpha| \leq m\}$, where $P^{\alpha}(x) = x^{\alpha}$, $\alpha$ is an $n$-multiindex and $Fu$ is the Fourier transform of $u$. We defined the weak derivative of an element $u \in H^m$ as follows: $\partial^{\alpha} u = F^{-1}(P^{\alpha}F u)$ (this was motivated by the validity of this formula in the Schwartz space).
Well, the problem arises when I try to prove the consistency of this definition in the case where both the weak and the strong (classic) derivative exist.
More precisely, let $u \in C^m$. If further I have the strong derivatives $D^{\alpha} u \in L^2$ for all $|\alpha| \leq m$, then a theorem states that $u \in H^m$ and $D^{\alpha} u$ is almost everywhere equal to $F^{-1}(P^{\alpha}F u)$. Very good, so far.
But what if the second hypothesis fails? i.e. what happens when it exists $|\alpha| \leq m$ for which $D^{\alpha}u \not \in L^2$? My question is if, in this case, I can state that $u \not \in H^m$, or equivalently if $u \in C^m \cap H^m$ implies the pointwise almost-everywhere equality of strong and weak derivatives. This should be a "good behaviour" that I expect, but I'm not sure of its validity.
Any elucidation is appreciated!