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More precisely let $\\overline{\\mathfrak{d}}_{m,0}:L^2\\Omega^{m,0}(reg(X),h)\\rightarrow L^2\\Omega^{m,1}(reg(X),h)$ be an arbitrarily fixed closed extension of $\\overline{\\partial}_{m,0}:L^2\\Omega^{m,0}(reg(X),h)\\rightarrow L^2\\Omega^{m,1}(reg(X),h)$ where the domain of the latter operator is $\\Omega_c^{m,0}(reg(X))$. 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