Weak derivatives persist under local Lp limits

Fix $\phi \in C_c^{\mathrm{\infty }}(\mathrm{\Omega })$ and let $K=\mathrm{supp}\phi$. By [F5], $K$ has finite measure. Both $\phi$ and $D^{\alpha }\phi$ are bounded and supported in $K$, so they belong to every finite-exponent $L^r(K)$ and to $L^{\mathrm{\infty }}(K)$. For $1<r<\mathrm{\infty }$, Hölder with the constant function $1$ shows that every $L^r(K)$ class is integrable; at $r=1$ this is the definition, and at $r=\mathrm{\infty }$ its integral is bounded by the essential supremum times $\mathrm{\mid }K\mathrm{\mid }$. By [F10], the assumed local Lp classes and weak derivative data have locally integrable representatives, independent of their choice in the test identity. Thus all data in the weak identities are locally integrable, including when $p$ or $q$ is an endpoint.

For each $j$, [F1] gives ${\int }_{\mathrm{\Omega }}{u}_j{D}^{\alpha }\phi dx=(-1{)}^{\mathrm{\mid }\alpha \mathrm{\mid }}{\int }_{\mathrm{\Omega }}{v}_j\phi dx\mathrm{.}$ By [F10], these test identities and pairings are independent of the chosen locally integrable representatives. Hölder on $K$, using [F3] for real values and [F4] for complex values, gives $\mid {\int }_{\mathrm{\Omega }}(u_j-u)D^{\alpha }\phi dx\mid \le \mathrm{\parallel }{u}_j-u{\mathrm{\parallel }}_{L^p(K)}\mathrm{\parallel }{D}^{\alpha }\phi {\mathrm{\parallel }}_{L^{p^{\mathrm{\prime }}}(K)}⟶0,$ and $\mid {\int }_{\mathrm{\Omega }}(v_j-v)\phi dx\mid \le \mathrm{\parallel }{v}_j-v{\mathrm{\parallel }}_{L^q(K)}\mathrm{\parallel }\phi {\mathrm{\parallel }}_{L^{q^{\mathrm{\prime }}}(K)}⟶0.$ These estimates also hold at $p=1,\mathrm{\infty }$ and $q=1,\mathrm{\infty }$ with the conjugate endpoint exponents. Passing to the limit in the identity therefore gives ${\int }_{\mathrm{\Omega }}u{D}^{\alpha }\phi dx=(-1{)}^{\mathrm{\mid }\alpha \mathrm{\mid }}{\int }_{\mathrm{\Omega }}v\phi dx\mathrm{.}$

Since $\phi$ was arbitrary, the last identity is exactly the weak $\alpha$-derivative definition, so $v=D^{\alpha }u$ weakly. By [F2] this locally integrable value is unique as an almost-everywhere class. The statement about local Sobolev sequences is the same conclusion applied to their derivative classes from [F6]; no convergence of derivatives other than the indicated $D^{\alpha }{u}_j$ is asserted.

If $\alpha =0$, the test identity and uniqueness identify $v$ with the class $u$, so the argument covers order zero. In dimension $n=1$ the same test and compact-support estimates apply with the single coordinate. If $K=\mathrm{\varnothing }$, then $\phi =0$ and both sides vanish. On the empty domain, or when all sequence and limit classes are zero, the identity is zero on both sides. The arguments for $p=1,\mathrm{\infty }$ and $q=1,\mathrm{\infty }$ were included in step 2.1.