Let $\mathcal L$ be a differential operator with constant coefficients and $\mathcal{L} f=0$ for some $f\in C^{\infty}(\mathbb{R}^n).$ Under what conditions on $\mathcal {L}$ the function $f$ extends to an entire function in $\mathbb{C}^n$?
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$\begingroup$ For the main question should we expect solutions to be automatically entire ?or that is too good to be true? $\endgroup$– BigMCommented Dec 8, 2014 at 4:57
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$\begingroup$ Is not "entire" the part of your question? Can you decide what are you asking? $\endgroup$– Alexandre EremenkoCommented Dec 8, 2014 at 5:02
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$\begingroup$ Laplace operator is an example of an operator for which the answer is "yes". $\endgroup$– Alexandre EremenkoCommented Dec 8, 2014 at 5:04
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1$\begingroup$ Solutions of Laplace equation extend to entire functions in $C^n$. $\endgroup$– Alexandre EremenkoCommented Dec 8, 2014 at 5:32
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3$\begingroup$ If $f$ admits an extension to an entire function, then $f$ must be real analytic. Requiring $\mathcal{L}$ to be elliptic will guarantee real analyticity. However you need to phrase your question more carefully. Should we expect all functions in $\ker \mathcal{L}$ to extend or only some of them? $\endgroup$– Liviu NicolaescuCommented Dec 8, 2014 at 9:25
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1 Answer
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If that property is satisfied, then "hypoelliptic analyticity" holds, which means that $\mathcal L f$ analytic implies $f$ analytic. For constant coefficient operators that property is equivalent to ellipticity, a sharp contrast with the $C^\infty$ case, where hypoellipticity was characterized by L. Hörmander and holds for instance for the heat equation (which is not elliptic).
As a reminder, the operator $$P=\sum_{\vert \alpha\vert\le m}a_\alpha D^\alpha$$ is elliptic means that $ \sum_{\vert \alpha\vert= m}a_\alpha \xi^\alpha\not=0\text{ for $ \xi\in\mathbb R^n\backslash\{0\}$}. $ The Laplace equation or $\overline{\partial}$ equation are examples of elliptic operators.
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$\begingroup$ thanks.I did not know about hypoelliticity at all. Now assuming the operator is elliptic should we expect that solutions extend to entire functions?entirety is of course much stronger than analyticity. $\endgroup$– BigMCommented Dec 8, 2014 at 13:02
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2$\begingroup$ Could you give the exact reference on Hormander where he proves that analyticity holds in the WHOLE $C^n$. I only found "in a complex neigborhood" of the real subspace. $\endgroup$ Commented Dec 8, 2014 at 15:27
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2$\begingroup$ I should have made my answer more precise, since it deals indeed with real analyticity (I have modified my answer). Of course, ellipticity alone does not imply the sought property globally in $\mathbb C^n$: take for instance for $n=1$, $\frac{\partial}{\partial z}$ and the elliptic equation $\frac{\partial u}{\partial z}=0$, which has no (non-trivial) holomorphic function as a solution. $\endgroup$– BazinCommented Dec 8, 2014 at 19:59