Research Problems in Function Theory — Problem 8.1

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 $L=(L_k)_{k\ge0}$ be nonnegative and nondecreasing, with $\sum_{k\ge0}L_kr^k<\infty$ for every $0<r<1$. For $f(z)=\sum_{k\ge0}a_kz^k$ holomorphic in $\mathbb D$ and a polynomial $P_n(z)=\sum_{k=0}^np_kz^k$ of degree $n$, the Hadamard product is $(P_n*f)(z)=\sum_{k=0}^np_ka_kz^k$. Write $\|P\|_\infty=\sup_{|z|<1}|P(z)|$. Define $\mathcal P_L$ by the existence of $A<\infty$ with \[\|P_n*f\|_\infty\le A L_n\|P_n\|_\infty\quad(n\ge0,\ \deg P_n=n), \] and use the least such $A$ as its norm. This formulation also specifies the restrictions on coefficients when some $L_n=0$. For $p>0$, $H^p$ is the class with $\sup_{0<r<1}\int_0^{2\pi}|f(re^{i\theta})|^p\,d\theta<\infty$.

Statement recorded in the supplied source.

Relate familiar holomorphic function spaces to $\mathcal P_L$, in particular the Hardy spaces $H^p$ for $0<p<1$: determine inclusions, identifications and associated norm comparisons as the admissible sequence $L$ varies. The source programme concerns these general multiplier-defined spaces, not just one chosen sequence. [S2]

Short English statement

Relate the holomorphic spaces defined by degree-dependent Hadamard-product bounds to familiar spaces, especially Hardy spaces with exponent below one.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2308001 (AMR-022-8001). 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 8.1, its complete update and relevant preceding definitions. https://arxiv.org/abs/1809.07200.