Research Problems in Function Theory — Problem 7.72

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.

Write $U=\{z:|z|<1\}$ and $dA=dx\,dy$. Let \[N=\{v\in L^\infty(U):\int_Uv\phi\,dA=0 \text{ for every }\phi\in\mathcal O(U)\text{ with }\int_U|\phi|\,dA<\infty\}. \] The pairing contains no complex conjugation. Extend $v$ by zero off $U$. Its Beurling transform, defined almost everywhere as a singular integral, is \[(Bv)(z)=-\frac1\pi\lim_{\varepsilon\downarrow0} \int_{U\setminus B(z,\varepsilon)}\frac{v(\zeta)}{(z-\zeta)^2}\,dA(\zeta). \] Let $P$ be all finite complex linear combinations of $z^j\bar z^k$, $j,k\ge0$. Norms for $v$ are essential suprema on $U$. For $v\in N$ the displayed transform vanishes off $\overline U$, by expanding its analytic kernel in $\zeta$, so the essential supremum of $Bv$ on the plane or on $U$ is the same.

Statement recorded in the supplied source.

Is there a universal finite constant $C$ such that every $v\in N$ satisfies $Bv\in L^\infty$ and $\|Bv\|_\infty\le C\|v\|_\infty$? Separately, does this uniform assertion at least hold for every $v\in N\cap P$? [S2]

Short English statement

Is the Beurling transform uniformly bounded on bounded functions annihilating all integrable holomorphic tests, or at least on the polynomial subclass?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2307072 (AMR-022-7072). 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 7.72, its complete update and relevant preceding definitions. https://arxiv.org/abs/1809.07200.