Research Problems in Function Theory — Problem 8.9

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.

Let $\mathbb D=\{z\in\mathbb C:|z|<1\}$ and $\mathbb T=\partial\mathbb D$. For $p>0$, the Bergman space is $A^p(\mathbb D)=\{f\text{ analytic}:\|f\|_{A^p}=(\int_{\mathbb D}|f|^p\,dA)^{1/p}<\infty\}$, where $dA=dx\,dy$ is ordinary area measure. On $A^1$ define the complex linear functional \[\Lambda(f)=\int_{\mathbb D}f(z)\operatorname{sgn}(\operatorname{Re}z)\,dA(z),\] where the sign is $1$, $-1$ or $0$ according as the real part is positive, negative or zero. Its norm is \[\|\Lambda\|=\sup_{\|f\|_{A^1}\le1}|\Lambda(f)|.\]

Statement recorded in the supplied source.

Determine the exact value of $\|\Lambda\|$. The source records that it is strictly below $1$ and that the supremum is attained. [S2]

Short English statement

Compute the exact norm of the Bergman linear functional obtained by integrating against the sign of the real part.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2308009 (AMR-022-8009). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Walter K. Hayman and Edward F. Lingham, Research Problems in Function Theory (2018), arXiv:1809.07200. Problem 8.9, complete original question, all following updates, and the relevant chapter definitions. https://arxiv.org/abs/1809.07200.