Compact potential multiplication without a Dirichlet restriction

The target weighted space is complete: a target-norm Cauchy sequence is Cauchy in each bounded chart Hölder norm and hence converges there. The local limits agree on overlaps. Pass the Cauchy inequalities to the limit in each derivative supremum and Hölder difference quotient and then take the annular supremum. This gives a global target element and convergence in the weighted norm. Thus the local Banach completeness patches to the given target norm.

Given a bounded sequence $u_j$ in the full input space, take the locally convergent subsequence of the local-compactness lemma. Multiplication by $V$ preserves compact $C^{2,\alpha }$ convergence, by the same product inequality as in the tail lemma. For any tolerance first make the common annular tails small by choosing $T$ large, then make the subsequence products close on the compact truncation through $r=2T$. Annuli with $R<T$ lie inside that truncation and have uniformly bounded weight factors because $R\ge r_0>0$. Hence the products form a target-norm Cauchy subsequence and converge by target completeness.

Every sequence in the image of a bounded input set therefore has a convergent subsequence in the target. The metric compactness criterion gives relative compactness, which is the definition of compactness here; equivalently the same tail and compact-truncation argument supplies finite nets for the image. Boundedness and linearity were supplied by the product lemma. A bounded sequence in the Dirichlet subspace is also a bounded sequence in the full space, proving the restricted assertion.