Non-improvability of Sharp Endpoint Estimates

2025/10/01 by Zdeněk Mihula, Mihula, Zdeněk, Luboš Pick +3
#26A33 #42B35 #44A35 #46E30 #47B38 #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2510.00792

Abstract

For an integer n and the parameter γ∈(0,n), the Riesz potential I_γ is known to take boundedly L¹(ℝⁿ) into L^{(n)/(n-γ),∞}(ℝⁿ), 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(ℝⁿ) with q∈(0,1). The classical methods cannot be used because the spaces L^1,q(ℝⁿ) 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.

Citations

• Potential trace inequalities via a Calderón-type theorem
• On the properties of rearrangement-invariant quasi-Banach function spaces
• Hardy--Littlewood--Sobolev inequality for p=1
• An Optimal Sobolev Embedding for L¹
• Higher-order Sobolev embeddings and isoperimetric inequalities

Related

• Potential trace inequalities via a Calderón-type theorem
• Embeddings into Orlicz spaces via the modified Riesz potential
• Riesz potential estimates under co-canceling constraints
• An Optimal Sobolev Embedding for L¹
• Optimal weak estimates for Riesz potentials