Almost holomorphic extensions of ultradifferentiable functions
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 \(t\to\pm\infty\), \({\mathcal A}_h= \{f\in{\mathcal C}(\mathbb{R})(\int|\widehat f| e^h< \infty\}\) (here \(\widehat f\) is the Fourier transformation of \(f\)) \(\widetilde{\mathcal A}_h= \{f\in{\mathcal C}(\mathbb{R}\mid\sup|\widehat f| e^h< \infty\}\), and \(G\) the class of functions \(g\) nonnegative on \(\mathbb{R}\setminus\{0\}\), convex and decreasing on \(\mathbb{R}^+\), increasing on \(\mathbb{R}^-\), and \(\lim_{y\to\pm 0}g(y)= \infty\). The class of weights \(H\) and \(G\) are related by the Legendre transformation: Let \(h\) be nonnegative, increasing on \(\mathbb{R}^+\) decreasing in \(\mathbb{R}^-\). Define \(h^\# (y)= \sup_{\{t:yt>0\}}(h(t)- ty)\) 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 \(\int_\Omega |\omega| e^{-g}< \infty\). The authors define \(\overline\partial\omega\) as an ultradistribution (i.e. an element in the dual of \({\mathcal A}_h\)) on \(\Omega\cap\mathbb{R}\), such that \(\overline\partial\omega= 0\) iff \(\omega\) extends holomorphically to \(\Omega\) and such that if \(\omega_n e^{-g}\to 0\) in \(L^1_{\text{loc}}(\Omega)\),then \(\overline\partial\omega_n\cdot f\), \(f\to\infty\) for each test function. This is possible only if the weight \(g\) is of the form \(g= h^\#\) with \(h\in H\) satisfying \(\int^1_{-1}\log g(y)\,dy<\infty\). In this situation \(\overline\partial\omega\cdot f= \int \omega\wedge\overline\partial F\) where \(f\) is a test function with compact support which has an extension \(F\) with support in \(\Omega\), such that \(\int|\overline\partial F|\, e^g< 1\). If \(g\) satisfies also the condition \(g''/g\to\infty\) as \(y\to 0\) then, if \(\omega\) is a holomorphic form in \(\Omega\setminus \mathbb{R}\), with \(\int|\omega| e^{-g}< \infty\), \(\overline\partial\omega\) (defined as above) is zero iff \(\omega\) extends holomorphically across \(\mathbb{R}\). In the sequel the authors extend these results in the multidimensional case. To achieve this one has to extend the definition of the Legendre transformation. Let \(k\) be any function in \(\mathbb{R}^n\setminus\{0\}\) and set \(k^a(\eta)= k(\eta a)\), \(\eta> 0\) 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 \(\int_{\mathbb{R}^n} |\widehat f(t)|\,e^{h(t)}\,dt< \infty\), then \(f\) can be extended to a function F satisfying \(\sup\int_{| y|< 1} |\overline\partial F(x+ iy)|\, e^{g(- y)}\leq{1\over 2} \int |\widehat f(t)|\, e^{h(t)}\,dt\) (and for \(h\in C^2\) a uniform estimate holds. As in the case \(n= 1\) there is a partial converse: Let \(g\) be a weight function and \(h= g^\#\) and if \(f\in{\mathcal C}(\mathbb{R})\) can be continuously extended to a function \(F\) such that \[ \sup_y \int_{\mathbb{R}}| y\cdot\overline\partial F(x+ iy)|\,e^{g(-y)}+ F(x+ iy)\,dx< \infty \] then \(|\widehat f(t)| r^{h(t)}<{\mathcal C}\), which is proved reducing into the 1-dimensional case. The authors prove also that if \(g\) is a weight function convex on rays and has convex super level sets then there is a non-trivial \(f\in C^\infty_0(\mathbb{R}^n)\) which can be extended to a function \(F\) in \(\mathbb{C}^n\) satisfying \[ |\overline\partial F(x_i y)|\leq e^{-g(- y)}\tag{\(*\)} \] iff \[ \int^1_0\log g_a(s)\,ds< \infty\tag{\(**\)} \] (here \(g_a(s)= \text{supp}_{y\cdot a=s}g(y)\) for a \(a\in S^{h-1}\)) for all \(a\in S^{n-1}\). In this case there is also a sequence of functions satisftying \((*)\) whose restrictions to \(\mathbb{R}^n\) tends to the Dirac measure at 0. The condition \((**)\) is called the nonquasianalyticy condition. The 1-dimensional condition \(\xi''/g\to \infty\) for \(y\to 0\) is replaced in the multidimensional case by \[ \lim_{y\to 0} \sum{\partial^2 g\over\partial y_i\partial y_k} y_i y_k/| y|^2 g(y)= \infty\tag{\(***\)} \] for \(g\) a weight function convex on rays. Now (Theorem 4.9) the authors prove that if \(\Omega\) is a convex domain in \(\mathbb{C}^n\), \(\Omega\cap \mathbb{R}^n\neq\emptyset\) \((n > 1)\) 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 \((**)\).
scientific article; zbMATH DE number 1997294