Scalar exterior inverse from exhaustion and uniform annular estimates

For $f\in Y_{\beta }$ solve ${\mathrm{\Delta }}_h{u}_T=f$ on the smooth truncation $M_T$, with zero values on both its original and outer boundary, using the compact-domain lemma. Let $F=\mathrm{\parallel }f{\mathrm{\parallel }}_{2,\alpha ,\beta -2}$. The norm definition gives $\mathrm{\mid }f\mathrm{\mid }\le C_{0}F{\rho }^{\beta -2}$. Comparing $\pm u_T-C_{1}FB$ using the barrier, with $C_1$ large enough, gives $\mathrm{\mid }{u}_T\mathrm{\mid }\le C_{1}FB$ independently of $T$. The comparison works at its seam by the derivative-jump argument, and on both boundary parts the barrier is positive.

Cover any annulus $A_R$ by a fixed finite number of balls of radius proportional to $R$, using larger annuli $\{R\mathrm{/}2<r<4R\}$ for the estimates. Under $x=Ry$ the equation becomes $(h^{ij}(Ry){\mathrm{\partial }}_{ij}+R{b}^i(Ry){\mathrm{\partial }}_i)u_T(Ry)=R^{2}f(Ry)$. These coefficients have uniformly bounded $C^{2,\alpha }$ norms and uniform ellipticity for large $R$, since $h-\delta \in C_{-1}^{3,\alpha }$. The fourth-order estimate of the compact-domain lemma, here using its interior argument, therefore gives $\mathrm{\parallel }{u}_T(R\cdot ){\mathrm{\parallel }}_{4,\alpha ;A_1}\le C{R}^{\beta }F$ when $4R<T$. A finite ball cover controls close pairs in the annular Hölder seminorm; separated pairs are controlled by the derivative supremum divided by their fixed rescaled separation. The larger annulus data norm is bounded by the given norm using finitely many overlapping annuli. On each fixed compact core the boundary estimate has constants independent of $T$.

Exhaust $T\to \mathrm{\infty }$. On each fixed compact region the fourth-order estimates give, by Ascoli on derivatives and a diagonal subsequence, convergence in $C^4$ (and $C^{4,\gamma }$ for $\gamma <\alpha$) to a zero-Dirichlet solution $u$. Hölder difference quotients pass to the limit for each pair of points, preserving the full exponent $\alpha$ and all weighted bounds from step 2.1. Thus $u\in X_{\beta }$ and ${\mathrm{\Delta }}_{h}u=f$. For uniqueness a homogeneous solution tends uniformly to zero at infinity because $\beta <0$. Apply the scalar maximum principle on truncations and let their outer boundary supremum tend to zero, to $u$ and $-u$. The maximum principle itself follows by perturbing by a strictly subharmonic local test or by the positive-part energy comparison; its zeroth-order term is zero. Hence $u=0$.

The spaces here are Banach: a norm-Cauchy sequence converges on every compact chart in the ordinary Hölder Banach space; the limits agree on overlaps, and passing each annular norm inequality to the limit preserves the global weighted norm and Cauchy convergence. The zero trace subspace is closed. Product differentiation in ${\mathrm{\Delta }}_h=h^{ij}{\mathrm{\partial }}_{ij}+b^i{\mathrm{\partial }}_i$ shows boundedness from $X_{\beta }$ to $Y_{\beta }$. The bound from steps 1.1 and 2.1 passes to the limit and gives a bounded inverse. This proves the Statement.