Gårding estimates for the Dolbeault Laplacian on a compact Riemann surface

Let $w\in C_c^{\mathrm{\infty }}({\mathbb{R}}^2;\mathbb{C})$. Expanding with [F4] gives $4\mathrm{\mid }{\mathrm{\partial }}_{\bar{z}}w{\mathrm{\mid }}^2=\mathrm{\mid }{w}_x{\mathrm{\mid }}^2+\mathrm{\mid }{w}_y{\mathrm{\mid }}^2+i(w_y\overline{w_x}-w_x\overline{w_y})$. Integration by parts shows $\int w_x\overline{w_y}=\int w_y\overline{w_x}$, so this integral is real and the cross term integrates to zero. Hence $\mathrm{\parallel }\mathrm{\nabla }w{\mathrm{\parallel }}_{L^2}=2\mathrm{\parallel }{\mathrm{\partial }}_{\bar{z}}w{\mathrm{\parallel }}_{L^2}$; the identical calculation gives $\mathrm{\parallel }\mathrm{\nabla }w{\mathrm{\parallel }}_{L^2}=2\mathrm{\parallel }{\mathrm{\partial }}_{z}w{\mathrm{\parallel }}_{L^2}$. Now let $w\in L^2({\mathbb{R}}^2)$ have compact support and distributional ${\mathrm{\partial }}_{\bar{z}}w=h\in L^2$. Choose a nonnegative smooth bump supported in the unit ball and positive near zero by [F13], and normalize its positive integral to one. Mollify $w$ with its rescalings ${\rho }_{ϵ}$. Testing the weak derivative against the translated smooth kernel gives ${\mathrm{\partial }}_{\bar{z}}{w}_{ϵ}=h\ast {\rho }_{ϵ}$. Both convolutions converge in $L^2$ by the $k=0$ case of [F6]: for $ϵ\le 1$ their supports lie in one fixed compact set, so local convergence is global. The smooth identity uniformly bounds each ${\mathrm{\partial }}_j{w}_{ϵ}$. For every test $\phi$, integration by parts and Cauchy–Schwarz therefore bound $\phi \mapsto -\int w{\mathrm{\partial }}_j\phi$ by $2\mathrm{\parallel }h{\mathrm{\parallel }}_2\mathrm{\parallel }\phi {\mathrm{\parallel }}_2$. Density [F3] and Riesz [F3] represent each functional by an $L^2$ function (conjugating the representative for the bilinear weak-derivative convention); thus $w\in H^1$ and $\mathrm{\parallel }\mathrm{\nabla }w{\mathrm{\parallel }}_2\le 2\sqrt{2}\mathrm{\parallel }h{\mathrm{\parallel }}_2$. Conjugation gives the same conclusion when ${\mathrm{\partial }}_{z}w\in L^2$. For a local $L^2$ coefficient with derivative $h$, apply this compact-support result to $\eta w$ extended by zero, retaining ${\mathrm{\partial }}_{\bar{z}}(\eta w)=\eta h+({\mathrm{\partial }}_{\bar{z}}\eta )w$; both terms are $L^2$ on the compact support.

Expanding [F1] in $x,y$, each local Laplacian block has principal part $-\frac{1}{2\rho }({\mathrm{\partial }}_x^2+{\mathrm{\partial }}_y^2)$; derivatives of $\rho$ and $\psi$ contribute only smooth lower-order terms. Put $a^{ij}=(2\rho {)}^{-1}{\delta }_{ij}$. Then the block is $-{\mathrm{\partial }}_i(a^{ij}{\mathrm{\partial }}_j\cdot )+b^i{\mathrm{\partial }}_i+c\cdot$ for smooth $b^i,c$, after absorbing the derivatives of $a^{ij}$ into $b^i$. On every relatively compact chart subdomain, positivity of $\rho$ gives a positive lower bound for $a^{ij}{\xi }_j\overline{{\xi }_i}\mathrm{/}\mathrm{\mid }\xi {\mathrm{\mid }}^2$, and all coefficient derivatives are bounded there. The distributional equation in the Statement and integration by parts against compactly supported tests make each local coefficient a weak solution in the sense of [F9].

In a chart/frame let $u_0=fe$ and $u_1=ad\bar{z}\otimes e$. The weak maximal-domain identity [F2] gives ${\mathrm{\partial }}_{\bar{z}}f\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^2$. For $u_1\in \mathrm{dom}{\bar{D}}^{\ast }$, test the adjoint identity against compactly supported smooth sections $\varphi e$. The $L^2$ pairing [F12] and the formal adjoint formula [F1], interpreted distributionally by integration by parts, give ${\bar{D}}^{\ast }{u}_1=-2(\rho \psi {)}^{-1}{\mathrm{\partial }}_z(\psi a)$; hence ${\mathrm{\partial }}_z(\psi a)\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^2$. Choose a partition cutoff $\chi$ with compact support in the chart, using [F8]. The distributional product rule [F5] gives ${\mathrm{\partial }}_{\bar{z}}(\chi f)=\chi {\mathrm{\partial }}_{\bar{z}}f+({\mathrm{\partial }}_{\bar{z}}\chi )f$ and ${\mathrm{\partial }}_z(\chi \psi a)=\chi {\mathrm{\partial }}_z(\psi a)+({\mathrm{\partial }}_z\chi )\psi a$. Apply step 1.1 to $\chi f$ and, by conjugation, to $\chi \psi a$. Since $\rho ,\psi$ and their inverses and first derivatives are bounded on the compact support, [F1, F12] then bounds the local $H^1$ norms of both coefficients by their local $L^2$ norms and the corresponding coefficients of $\bar{D}{u}_0$ and ${\bar{D}}^{\ast }{u}_1$.

Fix $k\ge 0$ and let $({\chi }_j)$ be the fixed partition used in the finite-chart norm of the Statement. Choose nested chart subdomains $U_j^{\mathrm{\prime }}⋐U_j^{\mathrm{\prime }\mathrm{\prime }}⋐U_j$ with $\mathrm{supp}{\chi }_j\subset U_j^{\mathrm{\prime }}$; the $U_j^{\mathrm{\prime }}$ cover $X$ because ${\sum }_j{\chi }_j=1$. Apply the interior estimate [F9] to each local equation from step 1.2, for both $q=0,1$. It gives $H^{k+2}(U_j^{\mathrm{\prime }})$ regularity and bounds each local norm by $C_j(\mathrm{\parallel }f{\mathrm{\parallel }}_{H^k(U_j^{\mathrm{\prime }\mathrm{\prime }})}+\mathrm{\parallel }u{\mathrm{\parallel }}_{L^2(U_j^{\mathrm{\prime }\mathrm{\prime }})})$. To compare these local norms with the fixed global norms at any finite order $r$, write each coefficient as the finite sum of the partitioned coefficients in the other frames. Repeated chain [F14] and Leibniz rules express each derivative through order $r$ as a finite sum of transformed derivatives through order $r$, multiplied by smooth transition derivatives. On compact overlaps those factors and coordinate Jacobians are bounded, with the Jacobians bounded away from zero. Changing variables therefore bounds each local $H^r$ norm on a relatively compact set by the global $H^r$ norm; restriction and [F7] give the converse bounds for partitioned coefficients. These inequalities extend from smooth forms to the completions by testing weak derivatives. Apply this with $r=k$ to control $\mathrm{\parallel }f{\mathrm{\parallel }}_{H^k(U_j^{\mathrm{\prime }\mathrm{\prime }})}$, and use [F7] to bound ${\chi }_{j}u$ in $H^{k+2}$ by the local estimate. Each such coefficient is a compactly supported $H^{k+2}$ class; by [F6], approximate it smoothly in its chart, multiply by a cutoff equal to one near its support from [F13], extend by zero and sum. The comparison just proved makes these global smooth forms converge in the defining $H^{k+2}$ norm, placing $u$ in that completion. Summing the finite estimates now proves the asserted global bound, including $k=0$. The same norm comparison in order $r$ gives a continuous injective inclusion $H^r\to L^2$: if a smooth Cauchy sequence has zero $L^2$ limit, testing every local derivative against compactly supported tests forces all its derivative limits to zero.

Choose a finite holomorphic chart/frame cover and a subordinate partition of unity as in [F8]. Summing the finitely many local estimates of step 2.1, and using equivalence of the positive smooth metric and volume weights with Euclidean norms on each compact support, gives $\mathrm{\parallel }u{\mathrm{\parallel }}_{H^1}\le C(\mathrm{\parallel }\bar{D}{u}_0{\mathrm{\parallel }}_{L^2}+\mathrm{\parallel }{\bar{D}}^{\ast }{u}_1{\mathrm{\parallel }}_{L^2}+\mathrm{\parallel }u{\mathrm{\parallel }}_{L^2})$. The local formulas [F1] also give the reverse bound of the graph norm by the $H^1$ norm.

Step 2.1 shows every $u\in \mathcal{V}$ has local $H^1$ coefficients, in the finite-chart norm of [F11]. For each of the finitely many partitioned coefficients, Meyers–Serrin [F6] gives smooth approximants in its chart; multiplying them by a compactly supported cutoff equal to one near the coefficient's support preserves convergence by [F7]. Converting these compactly supported coefficients back to sections and summing gives smooth global forms converging to $u$ in the finite-chart $H^1$ norm, so $u\in H^1(X,{\mathrm{\Lambda }}^{0,\bullet }\otimes E)$. Conversely, if $u\in H^1$, choose smooth $u_n\to u$ in that norm by its completion definition. The first-order formulas [F1] make $\bar{D}{u}_{n,0}$ and ${\bar{D}}^{\ast }{u}_{n,1}$ converge in $L^2$; closedness of the operators [F2] gives $u\in \mathcal{V}$. Together with step 3.1 this proves equality of the spaces, equivalence and completeness of their norms, and density of smooth forms in the graph norm.

If $u\in \mathrm{dom}{\mathrm{\Delta }}^{\mathrm{\prime }\mathrm{\prime }}$, [F2] gives $u\in \mathcal{V}=H^1$. For every smooth test $v$, the Hilbert-adjoint identities yield $⟨{\mathrm{\Delta }}^{\mathrm{\prime }\mathrm{\prime }}u,v⟩=⟨\bar{D}{u}_0,\bar{D}{v}_0⟩+⟨{\bar{D}}^{\ast }{u}_1,{\bar{D}}^{\ast }{v}_1⟩$; hence the operator equation is the distributional equation used in step 2.2. Taking $f={\mathrm{\Delta }}^{\mathrm{\prime }\mathrm{\prime }}u$ proves the domain corollary. Finally, the first-order form inner product adds $⟨u,v{⟩}_{L^2}$, so it represents $I+{\mathrm{\Delta }}^{\mathrm{\prime }\mathrm{\prime }}$, as claimed.

Steps 1.1–5.1 prove the graph-domain $H^1$ estimate, smooth graph-norm density, and the $H^{k+2}$ estimates for distributional and Hilbert-domain solutions. Full AC is spent only through Sobolev cutoff localisation; the remaining Countable Choice instances are inherited from the cited analytic and Hilbert-space interfaces.