Elliptic regularity for Dolbeault harmonic forms

In a holomorphic chart $z=x+iy$ and frame $e$ with $g=\rho (d{x}^2+d{y}^2)$ and $h(e,e)=\psi >0$, [F2] shows that the principal part of either block is $-\frac{1}{2\rho }({\mathrm{\partial }}_x^2+{\mathrm{\partial }}_y^2)$. Set $a^{ij}=(2\rho {)}^{-1}{\delta }_{ij}$; moving the derivatives of the smooth weights $\rho ,\psi$ into the first- and zero-order coefficients writes each block as $-{\mathrm{\partial }}_i(a^{ij}{\mathrm{\partial }}_j\cdot )+b^i{\mathrm{\partial }}_i+c$. On every relatively compact chart patch, $\rho$ has positive minimum and all metric/frame coefficient derivatives are bounded, so this matrix is uniformly elliptic and all coefficients are smooth.

If $u=u_0+u_1\in \ker {\mathrm{\Delta }}^{\mathrm{\prime }\mathrm{\prime }}$, the energy identity [F1] is a sum of two nonnegative squared norms and equals zero, so $\bar{D}{u}_0=0$ and ${\bar{D}}^{\ast }{u}_1=0$. Conversely, if these two terms vanish, then their zero outputs lie in the opposite operator domains, so $u$ lies in the block domain of ${\mathrm{\Delta }}^{\mathrm{\prime }\mathrm{\prime }}$ and ${\mathrm{\Delta }}^{\mathrm{\prime }\mathrm{\prime }}u=0$. The same argument in each degree gives the stated degreewise characterization.

Let $v\in \mathrm{im}\bar{D}\cap C^{\mathrm{\infty }}(X,{\mathrm{\Lambda }}^{0,1}{T}^{\ast }X\otimes E)$ and choose $u\in \mathrm{dom}\bar{D}$ with $\bar{D}u=v$. By [F3], $u\in H^1$. The defining weak identity for $\bar{D}$ in [F1] says that the local coefficient derivative of $u$ is the corresponding ${\bar{\mathrm{\partial }}}_E$ expression in distributions. Since $v$ is smooth, [F2] gives $v\in \mathrm{dom}{\bar{D}}^{\ast }$ with the stated smooth local formula for ${\bar{D}}^{\ast }v$; therefore $u\in \mathrm{dom}{\mathrm{\Delta }}_0^{\mathrm{\prime }\mathrm{\prime }}$ and ${\mathrm{\Delta }}_0^{\mathrm{\prime }\mathrm{\prime }}u={\bar{D}}^{\ast }v$ is exactly the local scalar block equation in distributions. Integration by parts against compactly supported tests gives the local weak identity of [F10]. Apply the smooth-data corollary [F6] in each chart. Its smooth representatives agree on chart overlaps because they represent the same section $u$ almost everywhere and are continuous, so they glue to a global smooth section $\tilde{u}$. Weak differentiation depends only on the almost-everywhere class by [F7], hence $v={\bar{\mathrm{\partial }}}_E\tilde{u}$. Conversely, every smooth section belongs to $\mathrm{dom}\bar{D}$ and its Hilbert derivative is ${\bar{\mathrm{\partial }}}_E$ by [F1, F2]. This proves the equality of smooth exact representatives. AC is used by [F6] as recorded in [F9].

If ${\mathrm{\Delta }}_q^{\mathrm{\prime }\mathrm{\prime }}u=f$ distributionally, the local coefficient equation from step 1.1 holds against every compactly supported smooth test. Integration by parts in the divergence term gives precisely the local weak identity of [F10]; the coefficients and the datum belong to its stated classes. Thus each local coefficient of $u$ is a local weak solution of a uniformly elliptic divergence-form equation.

A harmonic component belongs to $\mathcal{V}$ by the block domain in [F1], and hence to $H^1$ by [F3]. Its local equation has smooth coefficients and smooth datum $f=0$ by step 1.1; the local weak-solution definition [F10] applies, and [F6] gives a smooth representative in each chart. These representatives agree on overlaps because they represent the same global $L^2$ form almost everywhere and are smooth, so they give a global smooth representative. Conversely, [F1, F2] put every smooth degree-zero form in $\mathrm{dom}\bar{D}$ and every smooth degree-one form in $\mathrm{dom}{\bar{D}}^{\ast }$. If its corresponding first-order derivative vanishes, the zero output lies in the other operator's domain, so step 1.2 puts the form in the Hilbert kernel. AC is used by [F6] as recorded in [F9].

Fix $k\ge 0$ and ${\mathrm{\Omega }}^{\mathrm{\prime }}⋐{\mathrm{\Omega }}^{\mathrm{\prime }\mathrm{\prime }}⋐\mathrm{\Omega }$. Cover $\overline{{\mathrm{\Omega }}^{\mathrm{\prime }}}$ by finitely many chart patches $U_j^{\mathrm{\prime }}$ and choose larger chart patches $U_j^{\mathrm{\prime }\mathrm{\prime }}$ with $\overline{U_j^{\mathrm{\prime }}}\subset U_j^{\mathrm{\prime }\mathrm{\prime }}⋐{\mathrm{\Omega }}^{\mathrm{\prime }\mathrm{\prime }}$; choose domains compactly contained in $\mathrm{\Omega }$ that contain each $\overline{U_j^{\mathrm{\prime }\mathrm{\prime }}}$. The local $H^1$ hypothesis and equation restrict to those domains by [F7, F10]. Apply [F5] with the smooth coefficients from step 1.1 on each nested chart domain. Summing the finite estimates, with the smooth metric and frame weights bounded above and below on the compact supports, gives the stated $H^{k+2}({\mathrm{\Omega }}^{\mathrm{\prime }})$ estimate and local regularity. AC supplies the countable-choice hypothesis of [F5] by [F9].

The harmonic space $\mathcal{H}=\ker {\mathrm{\Delta }}^{\mathrm{\prime }\mathrm{\prime }}$ is closed: if $h_n\in \mathcal{H}$ and $h_n\to h$ in $L^2$, then ${\mathrm{\Delta }}^{\mathrm{\prime }\mathrm{\prime }}{h}_n=0\to 0$, and closedness of the self-adjoint operator [F1] gives $h\in \mathrm{dom}{\mathrm{\Delta }}^{\mathrm{\prime }\mathrm{\prime }}$ with ${\mathrm{\Delta }}^{\mathrm{\prime }\mathrm{\prime }}h=0$. Thus [F8] defines the orthogonal projection $P_{\mathcal{H}}$. For any smooth total form $w$, $P_{\mathcal{H}}w\in \mathcal{H}$, and step 2.2 shows that every element of $\mathcal{H}$ is smooth. Therefore $P_{\mathcal{H}}$ maps smooth forms to smooth forms, without using the later Hodge decomposition. AC supplies the projection theorem's countable-choice assumption by [F9].

Steps 1.1 and 2.1 establish the local scalar equations and weak formulation; step 3.1 proves the nested $H^{k+2}$ gain; steps 1.2 and 2.2 prove the harmonic characterization and smoothness; and steps 1.3 and 3.2 prove smooth exact preimages and smoothness of the harmonic projection. Full AC is used through [F6], with AC${}_{\omega }$ for [F5] and [F8], as stated in [F9].