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Nonradial Hörmander algebras of several variables and convolution operators. (English) Zbl 0992.46020
For a certain type of weight function \(p\) on \(\mathbb{C}^N\), \(N\geq 1\), the Hörmander algebra \(A_p\) consists of all entire functions \(f\) such that, for some \(k\in\mathbb{N}\), \[ |f|_k= \sup_{z\in \mathbb{C}^N}|f(z)|\exp(- kp(z))< \infty. \] In the case of a radial weight \(p\) it is well-known that each principal ideal in \(A_p\) is closed, but this is not true in general in the non-radial case. For general non-radial Hörmander algebras and \(N= 1\), S. Momm [Arch. Math. 58, No. 1, 47-55 (1992; Zbl 0804.46066)] had characterized the closed principal ideals. In the present article, a characterization of the closed principal ideals in non-radia Hörmander algebras of holomorphic functions of several variables is obtained in terms of the behavior of the generator.
Let \({\mathcal E}_{(\omega)}(\mathbb{R}^N)\) denote the space of ultradifferentiable functions of Beurling type associated to a (quasianalytic or non-quasianalytic) weight \(\omega\) on \(\mathbb{R}_+\). Via Fourier-Laplace transform \(F\), the convolution algebra \({\mathcal E}_{(\omega)}'(\mathbb{R}^N)\), endowed with its strong topology, is topologically algebra isomorphic to \(A_p\) for the non-radial weight function \(p\), \(p(z)= \omega(|\text{Re }z|)+ |\text{Im }z|\) for \(z\in \mathbb{C}^N\). And for \(\mu\in{\mathcal E}_{(\omega)}'(\mathbb{R}^N)\) the convolution operator \(T_\mu:{\mathcal E}_{(\omega)}(\mathbb{R}^N)\to{\mathcal E}_{(\omega)}(\mathbb{R}^N)\) is surjective if and only if the principal ideal \(\widehat\mu A_p\) is closed in \(A_p\), where \(\widehat\mu= F(\mu)\). Hence, as a corollary, the authors obtain that \(T_\mu\) is surjective if and only if \(\mu\) is “slowly decreasing for \((\omega)\)”, a result which for non-quasianalytic weight \(\omega\) had already been proved by J. Bonet and A. Galbis [Glasgow Math. J. 38, No. 1, 125-135 (1996; Zbl 0861.46025)].
Next, the authors treat the question when for two weights \(\omega\leq\sigma\) the range of every convolution operator \(T_\mu\) or the range of every ultradifferential operator \(G(D)\) of class \((\omega)\) on \({\mathcal E}_{(\omega)}(\mathbb{R}^N)\) contains \({\mathcal E}_{(\sigma)}(\mathbb{R}^N)\). If \(\omega= o(\sigma)\), the same question is treated for the Roumieu class \({\mathcal E}_{\{\sigma\}}(\mathbb{R}^N)\) instead of \({\mathcal E}_{(\sigma)}(\mathbb{R}^N)\). The limit case when the Roumieu class is replaced by the space of real analytic functions is studied at the end of the article. Bonet and Galbis, in the article quoted above, had shown that, for non-quasianalytic \(\omega\), the range of each convolution operator on \({\mathcal E}_{(\omega)}(\mathbb{R}^N)\) contains the space of real analytic functions. Using results of R. Sigurdsson [Math. Scand. 59, 235-304 (1986; Zbl 0619.32003)], the authors now present a quasianalytic weight \(\sigma\) and an ultradistribution \(\mu\in{\mathcal E}_{(\sigma)}'(\mathbb{R})\) such that the range of \(T_\mu:{\mathcal E}_{(\sigma)}(\mathbb{R})\to {\mathcal E}_{(\sigma)}(\mathbb{R})\) does not contain the space of real analytic functions.

MSC:
46E25 Rings and algebras of continuous, differentiable or analytic functions
46E10 Topological linear spaces of continuous, differentiable or analytic functions
46F05 Topological linear spaces of test functions, distributions and ultradistributions
32A38 Algebras of holomorphic functions of several complex variables
42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
46F10 Operations with distributions and generalized functions
35R50 PDEs of infinite order
32A15 Entire functions of several complex variables
46A11 Spaces determined by compactness or summability properties (nuclear spaces, Schwartz spaces, Montel spaces, etc.)
46A13 Spaces defined by inductive or projective limits (LB, LF, etc.)
46A22 Theorems of Hahn-Banach type; extension and lifting of functionals and operators
46H10 Ideals and subalgebras
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