Research Problems in Function Theory — Problem 5.64

Definitions and mathematical statement

Let $k\ge0$ and let $f$ be holomorphic in $\mathbb D$ with \[ |f(z)|=O((1-|z|)^{-k})\qquad(|z|\uparrow1). \] Such a function has a distributional boundary value. With the source's pairing convention, define the distribution $\Lambda_f$ on smooth functions $\phi$ on $\mathbb T$ by \[ \Lambda_f(\phi)=\lim_{r\uparrow1}\frac1{2\pi} \int_0^{2\pi}f(re^{i\theta})\phi(e^{-i\theta})\,d\theta. \] The reflected argument fixes the Fourier-index convention and does not change distributional order. A distribution has order at most $m\in\mathbb Z_{\ge0}$ if some $C<\infty$ satisfies \[ |\Lambda(\phi)|\le C\max_{0\le j\le m} \sup_\theta\left|\frac{d^j}{d\theta^j}\phi(e^{i\theta})\right| \] for every smooth test function $\phi$. Determine the sharp restrictions on the order of $\Lambda_f$ resulting from the growth exponent $k$. How do these restrictions improve under the stronger assumption $\sup_{|z|=r}|f(z)|(1-r)^k\to0$ as $r\uparrow1$? The problem asks for optimal order information, including endpoint behaviour when $k$ is integral; merely showing that the boundary value has some finite distributional order leaves the quantitative question unresolved.

Short English statement

What is the sharp distributional order of boundary values of analytic functions with prescribed power growth, and does little-oh growth improve it?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2305064 (AMR-022-5064), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Hayman - Research Problems in Function Theory (2018), Problem 5.64. Original source indexed by AMR-022-5064. https://arxiv.org/abs/1809.07200.