Malgrange–Zerner theorem

In mathematics, Malgrange–Zerner theorem (named for Bernard Malgrange and Martin Zerner) shows that a function on ${\displaystyle \mathbb {R} ^{n}}$ allowing holomorphic extension in each variable separately can be extended, under certain conditions, to a function holomorphic in all variables jointly. This theorem can be seen as a generalization of Bochner's tube theorem to functions defined on tube-like domains whose base is not an open set.

Theorem[1][2] Let

${\displaystyle X=\bigcup _{k=1}^{n}\mathbb {R} ^{k-1}\times P\times \mathbb {R} ^{n-k},{\text{ where }}P=\mathbb {R} +i[0,1),}$

and let ${\displaystyle W=}$ convex hull of ${\displaystyle X}$. Let ${\displaystyle f:X\to \mathbb {C} }$ be a locally bounded function such that ${\displaystyle f\in C^{\infty }(X)}$ and that for any fixed point ${\displaystyle (x_{1},\ldots ,x_{k-1},x_{k+1},\ldots ,x_{n})\in \mathbb {R} ^{n-1}}$ the function ${\displaystyle f(x_{1},\ldots ,x_{k-1},z,x_{k+1},\ldots ,x_{n})}$ is holomorphic in ${\displaystyle z}$ in the interior of ${\displaystyle P}$ for each ${\displaystyle k=1,\ldots ,n}$. Then the function ${\displaystyle f}$ can be uniquely extended to a function holomorphic in the interior of ${\displaystyle W}$.

History

According to Henry Epstein,[1][3] this theorem was proved first by Malgrange in 1961 (unpublished), then by Zerner [4] (as cited in [1]), and communicated to him privately. Epstein's lectures [1] contain the first published proof (attributed there to Broz, Epstein and Glaser). The assumption ${\displaystyle f\in C^{\infty }(X)}$ was later relaxed to ${\displaystyle f|_{\mathbb {R} ^{n}}\in C^{3}}$ (see Ref.[1] in [2]) and finally to ${\displaystyle f|_{\mathbb {R} ^{n}}\in C}$.[2]

References

1. Epstein, Henry (1966). Some analytic properties of scattering amplitudes in quantum field theory (8th Brandeis University Summer Institute in Theoretical Physics: Particle symmetries and axiomatic field theory). pp. 1–128.
2. Drużkowski, Ludwik M. (1999-02-22). "A generalization of the Malgrange–Zerner theorem". Annales Polonici Mathematici. 38 (2): 181–186. doi:10.4064/ap-38-2-181-186. Retrieved 2021-07-01.
3. Epstein, H. (1963). "On the Borchers class of a free field" (PDF). Il Nuovo Cimento. 27 (4): 886–893. doi:10.1007/bf02783277. S2CID 120708058.
4. Zerner M. (1961), mimeographed notes of a seminar given in Marseilles