Research Problems in Function Theory — Problem 5.22

Definitions and mathematical statement

Fix $0<p<1$ and let $\mathbb D=\{z\in\mathbb C:|z|<1\}$. The analytic Hardy space $H^p$ consists of holomorphic functions $f$ on $\mathbb D$ for which \[ \|f\|_{H^p}^p=\sup_{0<r<1}\frac1{2\pi} \int_0^{2\pi}|f(re^{i\theta})|^p\,d\theta<\infty. \] This is a complete quasi-normed space; for $p<1$ the displayed $p$-th-root expression is not generally a norm. A complex sequence $\lambda=(\lambda_n)_{n\ge0}$ is a coefficient multiplier of $H^p$ into itself if, for every $f(z)=\sum_{n\ge0}a_nz^n\in H^p$, the series \[ T_\lambda f(z)=\sum_{n\ge0}\lambda_na_nz^n \] converges on $\mathbb D$ and defines another member of $H^p$. Characterize exactly all such sequences for each $p\in(0,1)$. The desired characterization should be an intrinsic condition on $\lambda$, rather than a restatement that the operator works on every input function. Coefficientwise multiplication is intended; this is different from multiplication of functions, which convolves their Taylor coefficients. Neither mere boundedness of $\lambda_n$ nor conditions adequate for multipliers into a different target space may be presumed to answer the self-multiplier problem.

Short English statement

Which sequences can multiply Taylor coefficients of every $H^p$ function, for $0<p<1$, without leaving that same Hardy space?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2305022 (AMR-022-5022), 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 5.22. Original source indexed by AMR-022-5022. https://arxiv.org/abs/1809.07200.