Holomorphic functions with growth conditions

Petersen, Bent E.

doi:10.1090/S0002-9947-1975-0379879-8

Holomorphic functions with growth conditions
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by Bent E. Petersen

Trans. Amer. Math. Soc. 206 (1975), 395-406

DOI: https://doi.org/10.1090/S0002-9947-1975-0379879-8

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Abstract:

Let $P$ be a $p \times q$ matrix of polynomials in $n$ complex variables. If $\Omega$ is a domain of holomorphy in ${{\mathbf {C}}^n}$ and $u$ is a $q$-tuple of holomorphic functions we show that the equation $Pv = Pu$ has a solution $v$ which is a holomorphic $q$-tuple in $\Omega$ and which satisfies an ${L^2}$ estimate in terms of $Pu$. Similar results have been obtained by Y.-T. Siu and R. Narasimhan for bounded domains and by L. Höormander for the case $\Omega = {{\mathbf {C}}^n}$.

References

Lars Hörmander, $L^{2}$ estimates and existence theorems for the $\bar \partial$ operator, Acta Math. 113 (1965), 89–152. MR 179443, DOI 10.1007/BF02391775
Lars Hörmander, An introduction to complex analysis in several variables, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1966. MR 0203075
Saunders Mac Lane, Homology, Die Grundlehren der mathematischen Wissenschaften, Band 114, Academic Press, Inc., Publishers, New York; Springer-Verlag, Berlin-Göttingen-Heidelberg, 1963. MR 0156879
Raghavan Narasimhan, Cohomology with bounds on complex spaces, Several Complex Variables, I (Proc. Conf., Univ. of Maryland, College Park, Md., 1970) Springer, Berlin, 1970, pp. 141–150. MR 0279337
Bent E. Petersen, On the Laplace transform of a temperate distribution supported by a cone, Proc. Amer. Math. Soc. 35 (1972), 123–128. MR 298414, DOI 10.1090/S0002-9939-1972-0298414-9
Yum-tong Siu, Holomorphic functions of polynomial growth on bounded domains, Duke Math. J. 37 (1970), 77–84. MR 251244
Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
R. G. Swan, Algebraic $K$-theory, Lecture Notes in Mathematics, No. 76, Springer-Verlag, Berlin-New York, 1968. MR 0245634