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**On the characteristics for convolution equations in tube domains.**
*(English)*
Zbl 0967.45002

Considering the kernel as a nonzero ultradistribution, the author [Publ. Res. Inst. Math. Sci. 30, No. 2, 167-190 (1994; Zbl 0808.46057)] studied the surjectivity of convolution operators on the space of hyperfunctions. Also the author with R. Ishimura [Bull. Soc. Math. Fr. 122, 413-433 (1994; Zbl 0826.35144)] considered the convolution equations with hyperfunction kernels and studied the existence and the continuation of holomorphic solutions in tube domains. They treated both problems in a unified setting of complex \( {\mathcal S}\) on the purely imaginary space \(\sqrt {-1}{\mathbb R}^{n}\) generated by the convolution operator \(\mu^{*}\) which operates on the space of holomorphic functions in tube domains and whose characteristics are defined in terms of the exponential behaviour of the total symbol of \(\mu^{*}\).

In this paper the author gives another definition of characteristics in terms of zeros of the total symbol \(\widehat{\mu} (y) ={\langle \mu(z), e^{z\zeta } \rangle}_{z}\) and proves its equivalence to the previous one. He also proves in addition that the new definition of characteristics coincides with the micro-support of the complex \({\mathcal S}\). The previous definition of characteristics of \(\mu^{*}\) is based on the growth estimates of \(\widehat{\mu}\), whereas the author’s new concept of characteristics is based on the zero set of \(\widehat{\mu}\). Using the condition \(\text (S)\) by Kawai which \(\widehat{\mu}\) always satisfies and the Harnack-Malgrange-Hörmander condition in Lemma 2.4, the author deduces that for all \(\varepsilon > 0\) there exists an open cone \(\Gamma_{\varepsilon}\) and positive constants \(N_{\varepsilon}'\), \(C_{\varepsilon}'\) such that \[ |\widehat{\mu}(\zeta)|\geq C_{\varepsilon}' \exp(-\varepsilon|\zeta|) \quad \text{on } \zeta \in \Gamma_{\varepsilon}\cap \{|\zeta|> N'_{\varepsilon}\} \] which implies the equivalence of the previous characteristic set of \(\mu^{*}\) to the author’s new definition.

In this paper the author gives another definition of characteristics in terms of zeros of the total symbol \(\widehat{\mu} (y) ={\langle \mu(z), e^{z\zeta } \rangle}_{z}\) and proves its equivalence to the previous one. He also proves in addition that the new definition of characteristics coincides with the micro-support of the complex \({\mathcal S}\). The previous definition of characteristics of \(\mu^{*}\) is based on the growth estimates of \(\widehat{\mu}\), whereas the author’s new concept of characteristics is based on the zero set of \(\widehat{\mu}\). Using the condition \(\text (S)\) by Kawai which \(\widehat{\mu}\) always satisfies and the Harnack-Malgrange-Hörmander condition in Lemma 2.4, the author deduces that for all \(\varepsilon > 0\) there exists an open cone \(\Gamma_{\varepsilon}\) and positive constants \(N_{\varepsilon}'\), \(C_{\varepsilon}'\) such that \[ |\widehat{\mu}(\zeta)|\geq C_{\varepsilon}' \exp(-\varepsilon|\zeta|) \quad \text{on } \zeta \in \Gamma_{\varepsilon}\cap \{|\zeta|> N'_{\varepsilon}\} \] which implies the equivalence of the previous characteristic set of \(\mu^{*}\) to the author’s new definition.

Reviewer: Kim Dohan (Seoul)

### MSC:

45E10 | Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type) |

32A07 | Special domains in \({\mathbb C}^n\) (Reinhardt, Hartogs, circular, tube) (MSC2010) |

46F15 | Hyperfunctions, analytic functionals |

32A45 | Hyperfunctions |