Non-improvability of Sharp Endpoint Estimates
Zdeněk Mihula, Luboš Pick, Armin Schikorra
TL;DR
The paper addresses whether the sharp endpoint target for the Riesz potential $I_\gamma$ can be improved when the domain is restricted from $L^1(\mathbb{R}^n)$ to a Lorentz endpoint space $L^{1,q}(\mathbb{R}^n)$ with $q\in(0,1)$. It develops two abstract methods: (i) a compatibility- and simple-approximation framework showing the optimal target cannot be improved under domain shrinkage along the fundamental level, and (ii) a Calderón-endpoint approach using extremal operators $R_\sigma$ and $S^0_\sigma$ within Fatou-representable rearrangement-invariant lattices to establish when the endpoint target $L^{q,\infty}$ remains optimal for a broad class of operators. The main consequences are a negative answer for the Riesz potential in this Lorentz setting and general, structurally motivated conditions guaranteeing endpoint optimality for Calderón-able operators, highlighting why sharper endpoint targets are unattainable in this general framework. The results connect the endpoint theory to traces, Hausdorff measures, and Fatou-representability, offering a robust, operator-agnostic blueprint for nonimprovability in sharp endpoint estimates.
Abstract
For an integer $n$ and the parameter $γ\in(0,n)$, the Riesz potential $I_γ$ is known to take boundedly $L^1(\mathbb{R}^n)$ into $L^{\frac{n}{n-γ},\infty}(\mathbb{R}^n)$, and also that the target space is the smallest possible among all rearrangement-invariant Banach function spaces. We study the natural question whether the target space can be improved when the domain space is replaced with a (smaller) Lorentz space $L^{1,q}(\mathbb{R}^n)$ with $q\in(0,1)$. The classical methods cannot be used because the spaces $L^{1,q}(\mathbb{R}^n)$ are not equivalently normable. We develop two new abstract methods, establishing rather general results, a particular consequence of each (albeit achieved through completely different means) being the negative answer to this question. The methods are based on special functional properties of endpoint spaces. The results can be applied to a wide field of operators satisfying certain minimal requirements.