Linearity, locality, and commutation of weak derivatives

Let $v_j=D^{\alpha }{u}_j$ weakly, and fix $a,b\in \mathbb{C}$ and a test $\phi \in C_c^{\mathrm{\infty }}(\mathrm{\Omega })$. By the weak identities in [F1] and linearity of the integral, ${\int }_{\mathrm{\Omega }}(a{u}_1+b{u}_2)D^{\alpha }\phi dx=(-1{)}^{\mathrm{\mid }\alpha \mathrm{\mid }}{\int }_{\mathrm{\Omega }}(a{v}_1+b{v}_2)\phi dx\mathrm{.}$ Finite sums of locally integrable functions remain locally integrable. By [F2] and [F3], the pairings and derivative classes do not depend on the chosen representatives. Since the test was arbitrary, this proves the stated complex-linearity.

Let $\phi \in C_c^{\mathrm{\infty }}(V)$. Its extension by zero to $\mathrm{\Omega }$ is a smooth compactly supported test on $\mathrm{\Omega }$: its support is compactly contained in $V$, so it vanishes in a neighborhood of $\mathrm{\Omega }\setminus V$. Apply the weak identity for $D^{\alpha }u=v$ on $\mathrm{\Omega }$ to this extension. The test and its derivatives vanish outside $V$, so the resulting integrals are exactly the weak identity on $V$ for $u{\mathrm{\mid }}_V$ and $v{\mathrm{\mid }}_V$. Thus $v{\mathrm{\mid }}_V=D^{\alpha }(u{\mathrm{\mid }}_V)$ weakly.

Write $T_f$ for the regular distribution of a locally integrable class $f$. By [F1], the assumptions $D^{\alpha }u=v$ and $D^{\beta }u=w$ give ${\mathrm{\partial }}^{\alpha }{T}_u=T_v,{\mathrm{\partial }}^{\beta }{T}_u=T_w\mathrm{.}$ By [F4], ${\mathrm{\partial }}^{\beta }{\mathrm{\partial }}^{\alpha }{T}_u={\mathrm{\partial }}^{\alpha +\beta }{T}_u={\mathrm{\partial }}^{\alpha }{\mathrm{\partial }}^{\beta }{T}_u\mathrm{.}$ All equalities here are distributional; they assert no locally integrable representative until one of the three weak derivatives in the Statement is assumed to exist.

Suppose first that $g=D^{\beta }v$ exists in $L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1(\mathrm{\Omega })$. Then [F1] and step 1.3 give $T_g={\mathrm{\partial }}^{\beta }{T}_v={\mathrm{\partial }}^{\beta }{\mathrm{\partial }}^{\alpha }{T}_u={\mathrm{\partial }}^{\alpha +\beta }{T}_u={\mathrm{\partial }}^{\alpha }{T}_w\mathrm{.}$ The regular-distribution/weak-derivative equivalence in [F1] says that this same locally integrable $g$ is both $D^{\alpha +\beta }u$ and $D^{\alpha }w$. If instead $h=D^{\alpha }w$ exists, then $T_h={\mathrm{\partial }}^{\alpha }{T}_w={\mathrm{\partial }}^{\alpha }{\mathrm{\partial }}^{\beta }{T}_u={\mathrm{\partial }}^{\alpha +\beta }{T}_u={\mathrm{\partial }}^{\beta }{T}_v,$ so $h$ represents both $D^{\alpha +\beta }u$ and $D^{\beta }v$. Finally, if $r=D^{\alpha +\beta }u$ exists, then $T_r={\mathrm{\partial }}^{\alpha +\beta }{T}_u={\mathrm{\partial }}^{\beta }{T}_v={\mathrm{\partial }}^{\alpha }{T}_w,$ so $r$ represents both iterated weak derivatives. These three implications prove equivalence of the existence conditions, with no assumption that an unrepresented distributional derivative is a function.

In each of the three cases, the named representatives satisfy the same weak derivative identities for the same base function and multi-index. By [F5] their locally integrable value classes are equal almost everywhere. This identifies all three values whenever any one existence condition holds. Countable Choice is used only through the representative and uniqueness interfaces [F3] and [F5]; the test-identity calculations, restriction, and distributional commutation use no choice. If $\alpha =0$ or $\beta =0$, the order-zero test identity in [F1] and uniqueness [F5] identify that derivative with the original class, so the same argument includes these cases. If both multi-indices are zero, all three derivatives are simply $u$.