For a function \(f\) on the real line (or the unit circle), the existence of extension to the complex \(F\) such that \(\overline\partial F\) tends to zero in a prescribed manner when approaching the real line (respectively the unit circle) (this \(F\) is called an almost holomorphic extensions) is related to regularity properties of the function \(f\). The authors extends this sort of result in the case of several variable. To achieve this, they review, in the proper setting, the results of \textit{A. Beurling} [Acta Math. 128, 153--182 (1972; Zbl 0235.30003) and Lecture Notes, Stanford 1961] and \textit{E. M. Dyn'kin} [Math. Sb., n. Ser. 89(131), 182--190 (1972; Zbl 0251.30033), J. Anal. Math. 60, 45--70 (1993; Zbl 0795.30034)]. More precisely, let \(H\) the class of continuous functions on \(\mathbb{R}\), nonnegative and increasing on \(\mathbb{R}^+\) and concave and decreasing on \(\mathbb{R}\) -- and tending to \(\infty\) for \(t\to\pm\infty\), \({\mathcal A}_h= \{f\in{\mathcal C}(\mathbb{R})(\int|\widehat f| e^h0\}}(h(t)- ty)\) and for \(g\) nonnegative decreasing on \(\mathbb{R}^+\) and increasing on \(\mathbb{R}^-\), \(g^b(t)= \inf_{\{y,yt>0\}}(g(y) +ty)\). One has that \(f^{\#b}\) is the smallest concave majorant of \(h\) and \(g^{b,\#}\), is the largest convex minorant of \(g\). If \(h\in H\) then \(h= h^{\# b}\); if \(g\in G\), \(g= g^{b\#}\). A precise version of Dyn'kin result is proved: Let \(h\in H\), \(f\in{\mathcal C}(\mathbb{R})\). Then \(f\) can be extended to a function \(F\) in \({\mathcal C}(\mathbb{C})\) such that \[ \sup_n \int_{\mathbb{R}} \Biggl|{\partial F(x\sin)\over\partial\overline z}\Biggr|\, e^{g(-y)} dy\leq {1\over 2} \int_{\mathbb{R}}|\widehat f(t)|\;e^{h(t)}\,dt \] with \(g= h^\#\) (and if \(h\in {\mathcal C}^2\) an uniform estimate of \(|{\partial F\over\partial\overline z}| e^{g(-y)}\) in terms of \(|\widehat f|\) and \(e^h\) is given) and also a partial converse which states that if \(f\) has an extension \(\overline\partial F\) small near the real line, the \(f\) must decrease rapidly at infinity. Moreover, if \(g\in G\) is of class \(C^2\), one can control the support of the extension. Let now \(\Omega\) be a domain in \(\mathbb{C}\) such that \(\Omega\cap \mathbb{R}\neq\emptyset\) and \(\omega\) a holomorphic \((1,0)\) form on \(\Omega\setminus\mathbb{R}\) satisfying \(\int_\Omega |\omega| e^{-g} 0\) for any \(a\in S^{n-1}\). Now \((k^a)^\#\) and \((k^a)^b\) make sense. For \(h\) any nonnegative function on \(\mathbb{R}\{0\}\) define now \[ g^b(t)= \sup_{a\in S^{n-1}} (g^a)^b(a\cdot t),\;h^\#(y)= \inf_{a\in S^{n-1}} (h^a)^\#(a\cdot y). \] The authors are able to prove: Let \(h\) a weight function concave on rays, and \(g= h^\#\). If \(f\in S'(\mathbb{R})\) such that \(\int_{\mathbb{R}^n} |\widehat f(t)|\,e^{h(t)}\,dt 1)\) and \(\omega\) a \(\overline\partial\)-closed \((n, n 1)\) form in \(\Omega\setminus\mathbb{R}^n\) such that \(\int|\omega|\, e^{-g(-y)}< \infty\), where \(g\) satisfies the nonquasianalyticity condition and \((***)\) the following are equivalent: (i) \(\overline\partial\omega= 0\) across \(\mathbb{R}^n\). (ii) For any \(\Omega'\subset\subset\Omega\) , there is a \((n, n- 2)\) form in \(\Omega'\setminus\mathbb{R}^n\) such that \(\overline\partial u= \omega\) and \(\int| u|\, e^{-g'(-y)}< \infty\) for some weight function \(g'\) that satisfies \((**)\). (iii) For any \(\Omega'\subset\subset\Omega\) there is a \((n, n- 2)\) form in \(\Omega'\setminus \mathbb{R}^n\) with \(\overline\partial u= w\). This is proved using Cauchy-Fantappiê kernels. As a corollary it follows that if \(E\) is the space of exact forms in \(\Omega\setminus \mathbb{R}^n\), and \(g(y)= g(| y|)\) then \(E\) is closed in \(L^1_{\text{loc}}(e^{- g})\) iff \(g\) satisfy \((**)\).