Research Problems in Function Theory — Problem 5.73

Definitions and mathematical statement

Fix $0<\alpha<1$. For a finite positive Borel measure $\mu$ on $\mathbb R$, define its Riesz potential by \[ I_\alpha\mu(x)=\int_{\mathbb R}|x-t|^{-\alpha}\,d\mu(t), \] allowing the value $+\infty$. Here "finite" means $\mu(\mathbb R)<\infty$; neither compact support nor absolute continuity is required. In particular atoms are allowed, and the potential diverges at any atom. Characterize the nonnegative measurable functions $g$ on $\mathbb R$ for which there exists such a measure satisfying \[ g(x)\le I_\alpha\mu(x)\qquad\text{for every }x\in\mathbb R. \] This pointwise formulation follows the stated domination question; domination only almost everywhere is a distinct possible variant and should be identified if used. A useful criterion should be expressed directly in terms of $g$, for instance through integral, covering, or capacity conditions, and should be both necessary and sufficient. The total mass of the witnessing measure may depend on $g$, and the normalization of the kernel can be absorbed into that mass. The problem does not require $g$ itself to be a potential, only to lie below one. This difference permits highly irregular measurable functions while preserving the global finite-mass constraint on their majorants.

Short English statement

Which nonnegative measurable functions on the line admit a pointwise majorant that is a Riesz potential of a finite positive measure?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2305073 (AMR-022-5073), 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 5.73. Original source indexed by AMR-022-5073. https://arxiv.org/abs/1809.07200.