Research Problems in Function Theory — Problem 7.60

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For $n\ge1$ let $\mathcal E=\mathcal O(\mathbb C^n)$ be all entire functions. For a multi-index $\alpha=(\alpha_1,\ldots,\alpha_n)\in\{0,1,\ldots\}^n$, put $z^\alpha=\prod z_j^{\alpha_j}$ and $D^\alpha=\prod(\partial/\partial z_j)^{\alpha_j}$. If $P(z)=\sum_\alpha p_\alpha z^\alpha$ is a polynomial, define $P(D)=\sum_\alpha p_\alpha D^\alpha$ and $T_{P,Q}f=P(D)(Qf)$. Entire exponential type means $|f(z)|\le A e^{B|z|}$ for some finite positive $A,B$.Let $m\ge0$, let $P$ have total degree $m$ with nonzero coefficient of $z_1^m$, and set $Q(z)=z_1^m$.

Statement recorded in the supplied source.

Are both $T_{P,Q}:\mathcal E\to\mathcal E$ and $T_{Q,P}:\mathcal E\to\mathcal E$ bijective? The original update answers both questions affirmatively. The first question is related to existence of entire solutions of the noncharacteristic Cauchy problem with entire data on a hyperplane. [S2]

Short English statement

Historical question: for a degree-m polynomial with a nonzero pure first-variable leading term, are both orders of the associated multiplication and differential operators bijective on all entire functions?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2307060 (AMR-022-7060). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Walter K. Hayman and Edward F. Lingham, Research Problems in Function Theory (2018), Problem 7.60, its complete update and relevant preceding definitions. https://arxiv.org/abs/1809.07200.