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$\mathrm{\cup }\mathrm{\cap }\mathrm{o}\mathrm{r}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{p}\mathrm{o}\mathrm{n}\mathrm{d}\mathrm{i}\mathrm{n}\mathrm{g}$ corresponding could

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$\mathrm{\rfloor }\mathrm{a}\mathrm{t}\mathrm{t}\mathrm{e}\mathrm{r}$ least Let let let let limit

means means metric, $\mathrm{M}$

${\overline{\mathrm{\partial }}}_{\mathrm{\ell }\mathrm{,}\mathit{\omega }}^{\mathrm{*}}\mathit{f}\mathrm{=}\mathrm{0}\mathrm{,}{\overline{\mathrm{\partial }}}_{\mathrm{\ell }\mathrm{,}\mathit{\omega }}^{\mathrm{*}}\mathit{f}\mathrm{=}\mathrm{0}\mathrm{)}$. $\mathrm{(}\mathrm{0}\mathrm{,}\mathrm{}\mathit{n}\mathrm{)}$-forms. $\mathrm{(}\mathrm{0}\mathrm{,}\mathrm{}\mathit{q}\mathrm{)}$-forms, ${\mathit{E}}_{\mathit{\alpha }}{\overline{\mathrm{\square }}}_{\mathit{\alpha }}\mathit{f}\mathrm{=}\mathit{f}\mathrm{-}{\mathit{P}}_{\mathit{\alpha }}\mathit{f}\mathrm{,}{\overline{\mathrm{\square }}}_{\mathit{\alpha }}{\mathit{E}}_{\mathit{\alpha }}\mathit{f}\mathrm{=}\mathit{f}\mathrm{-}{\mathit{P}}_{\mathit{\alpha }}\mathit{f}$,

$\mathrm{\partial }\mathrm{(}\mathrm{1}\mathrm{-}\mathrm{|}\mathit{z}{\mathrm{|}}^{\mathrm{2}}\mathrm{)}\mathrm{\text{'}}\mathit{f}\mathrm{=}\mathrm{0}\mathrm{|}\mathrm{\partial }\log \mathrm{(}\mathrm{-}\mathrm{1}\mathrm{/}\mathit{\rho }\mathrm{)}{\mathrm{|}}_{\mathit{\omega }}\overline{\mathrm{\partial }}\mathrm{:}{\mathit{L}}_{\mathrm{\ell }\mathrm{,}\mathit{\omega }}^{\mathrm{2}}\mathrm{\to }{\mathit{L}}_{\mathrm{\ell }\mathrm{,}\mathit{\omega }}^{\mathrm{2}}\overline{\mathrm{\partial }}\mathrm{:}{\mathit{L}}_{\mathrm{\ell }\mathrm{,}\mathit{\omega }}^{\mathrm{2}}\mathrm{\to }{\mathit{L}}_{\mathrm{\ell }\mathrm{,}\mathit{\omega }}^{\mathrm{2}}\overline{\mathrm{\partial }}\mathrm{:}{\mathit{L}}_{\mathit{\alpha }\mathrm{,}\mathrm{0}}^{\mathrm{2}}\mathrm{\to }{\mathit{L}}_{\mathit{\alpha }\mathrm{,}\mathrm{1}}^{\mathrm{2}}$

${\mathit{E}}_{\mathit{\alpha }}\mathrm{=}{\mathit{K}}_{\mathit{\alpha }}{\mathit{K}}_{\mathit{\alpha }}^{\mathrm{*}}\mathrm{+}{\mathit{K}}_{\mathit{\alpha }}^{\mathrm{*}}{\mathit{K}}_{\mathit{\alpha }}\mathrm{,}$ $\mathit{f}\mathrm{=}\mathrm{(}\mathrm{1}\mathrm{-}\mathrm{|}\mathit{z}{\mathrm{|}}^{\mathrm{2}}{\mathrm{)}}^{\mathrm{-}\mathit{\alpha }}\overline{\mathit{h}}\mathrm{,}{\overline{\mathrm{\partial }}}_{\mathit{\alpha }}^{\mathrm{*}}\mathrm{:}$ ${\mathit{L}}_{\mathit{\alpha }\mathrm{,}\mathrm{1}}^{\mathrm{2}}\mathrm{\to }{\mathit{L}}_{\mathit{\alpha }\mathrm{,}\mathrm{0}}^{\mathrm{2}}{\overline{\mathrm{\square }}}_{\mathit{\alpha }}\mathrm{=}\overline{\mathrm{\partial }}{\overline{\mathrm{\partial }}}_{\mathit{\alpha }}^{\mathrm{*}}\mathrm{+}{\overline{\mathrm{\partial }}}_{\mathit{\alpha }}^{\mathrm{*}}\overline{\mathrm{\partial }}$

${\displaystyle \int }\mathrm{(}\mathrm{1}\mathrm{-}\mathrm{|}\mathit{z}{\mathrm{|}}^{\mathrm{2}}{\mathrm{)}}^{\mathrm{-}\mathit{\alpha }}\mathrm{|}\mathit{h}{\mathrm{|}}^{\mathrm{2}}\mathrm{<}\mathrm{\infty }$.

$\mathit{D}\mathit{D}\mathit{f}\mathit{h}\mathit{h}$ ' ${\mathit{C}}^{\mathrm{1}}{{K}}_{\mathit{\alpha }}{\mathit{E}}_{\mathit{b}}^{\mathrm{*}}{{K}}_{\mathit{\alpha }}$ ; $\overline{\mathrm{\partial }}\mathrm{,}$ $\mathit{q}\mathrm{<}\mathit{n}\mathit{\alpha }\mathrm{\ge }\mathrm{1}$. $\mathit{\alpha }\mathrm{>}\mathrm{1}\mathrm{,}$ $\mathit{\alpha }\mathrm{\to }\mathrm{0}\mathrm{,}$ $\mathit{\alpha }\mathrm{\to }\mathrm{0}$; $\mathrm{(}\mathrm{,}\mathrm{}{\mathrm{)}}_{\mathit{b}}$ $\mathrm{(}\mathrm{,}\mathrm{}{\mathrm{)}}_{\mathit{\alpha }}$

$\mathit{f}\mathrm{\in }{{K}}_{\mathit{\alpha }}^{\mathrm{\perp }}\mathit{f}\mathrm{\in }{\mathit{L}}_{\mathit{\alpha }}^{\mathrm{2}}\mathit{f}\mathrm{-}{\mathit{P}}_{\mathit{\alpha }}\mathit{f}\mathrm{0}\mathrm{<}\mathit{\alpha }\mathrm{<}\mathrm{1}\mathrm{.}\mathrm{0}\mathrm{<}\mathit{\alpha }\mathrm{<}\mathrm{1}\mathrm{;}\mathrm{0}\mathrm{<}\mathit{\alpha }\mathrm{\le }\mathrm{1}$. $\mathit{f}\mathrm{\in }{\mathit{L}}_{\mathit{\alpha }}^{\mathrm{2}}\mathrm{,}$ $\mathit{h}\mathrm{\left(}\mathrm{0}\mathrm{\right)}\mathrm{=}\mathrm{0}{\overline{\mathrm{\partial }}}_{\mathrm{\ell }\mathrm{,}\mathit{\omega }}^{\mathrm{*}}$

$\mathrm{(}$ $\mathrm{=}\overline{\mathrm{\partial }}\overline{\mathit{h}}\mathrm{,}$ $\mathit{f}\mathrm{\in }{{K}}_{\mathit{\alpha }\mathrm{,}\mathrm{1}}\mathrm{,}$ $\mathit{f}\mathrm{\in }{\mathit{L}}_{\mathit{\alpha }\mathrm{,}\mathrm{1}}^{\mathrm{2}}{\mathit{K}}_{\mathit{b}}^{\mathrm{*}}\mathrm{,}{\overline{{\displaystyle \coprod }}}_{\mathit{b}}{\overline{\mathrm{\partial }}}_{\mathit{\alpha }}^{\mathrm{*}}\mathit{f}\mathrm{=}\mathrm{0}{\overline{\mathrm{\partial }}}_{\mathit{\alpha }}^{\mathrm{*}}\mathit{f}\mathrm{=}\mathrm{0}\mathrm{1}\mathrm{\le }\mathit{q}\mathrm{\le }\mathit{n}\mathrm{-}\mathrm{1}$. $\mathit{f}\mathrm{-}{\mathit{P}}_{\mathit{b}}\mathrm{-}{\mathit{S}}_{\mathit{b}}$.

$\mathrm{\ell }\mathrm{=}\mathit{\alpha }\mathrm{+}\mathit{n}\mathrm{-}\mathit{q}\mathrm{.}{\overline{\mathrm{\partial }}}_{\mathit{\alpha }}^{\mathrm{*}}\mathit{f}\mathrm{=}\mathrm{0}$. $\mathrm{\left|}\mathrm{\right|}\mathrm{|}{\mathrm{|}}_{\mathrm{\ell }\mathrm{,}\mathit{\omega }}$. $\mathrm{\left|}\mathrm{\right|}\mathrm{|}{\mathrm{|}}_{\mathrm{\ell }\mathrm{,}\mathit{\omega }}$. $\mathit{f}\mathrm{\in }$ Dom ${\overline{{\displaystyle \coprod }}}_{\mathit{\alpha }}$.

Neumann norm. norms norms norms not not not

obvious of of of of of of of of of of of on one one only only ORTEGA-CERDÀ orthogonal

$\mathrm{L}\mathrm{2}\mathrm{6}\mathrm{-}\mathrm{1}\mathrm{9}\mathrm{9}\mathrm{8}\mathrm{-}{\mathrm{N}}^{\mathrm{O}}\mathrm{2}\mathrm{2}\mathrm{5}\mathrm{4}\mathrm{2}\mathrm{.}\mathrm{2}\mathrm{.}\mathrm{2}\mathrm{.}\mathrm{-}$ $\mathrm{3}\mathrm{.}\mathrm{3}$ $\mathrm{3}\mathrm{.}\mathrm{4}$ $\mathrm{3}\mathrm{.}\mathrm{4}$ $\mathrm{3}\mathrm{.}\mathrm{4}$, 3.7 [8], 'iimit (1.3). (2.4) (3.6) (3.6) (3.7)

$\mathrm{\setminus }\mathrm{\prime }\mathrm{3}\mathrm{.}\mathrm{7}\mathrm{)}$ (as

particular particular, preceding proof Proposition Proposition Proposition Proposition

Proposition prove

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