Restricting Riesz-Morrey-Hardy potentials

To strengthen the weak restriction principle discovered in both [2, Theorem 5.1] for λ=κ and [32, Lemma 2.1] for λ<κ, this paper shows that the Riesz operator Iα of order α maps the Morrey-Hardy space LHp,κ to the Morrey-Radon space Lμq,λ continuously if and only ifsup(x,r)∈Rn×(0,∞)⁡μ(B(x,r))/rβ<∞provided{0<λ≤κ≤n;0<ακ>0 then the iff-statement fails. Quite remarkably, such an optimization confirms affirmatively the conjecture in [7, Remark 3.3]: ifα<αp<κ<λ=n&q=pβ/(κ−αp) then there is no continuity of Iα:Lp,κ→Lμq,n for an arbitrary Radon measure μ with ⦀μ⦀β<∞.