Necessary and sufficient conditions for the boundedness of the anisotropic Riesz potential in anisotropic modified Morrey spaces

We prove that the anisotropic fractional maximal operator M_α,σ and the anisotropic Riesz potential operator I_α,σ, 0 < α < ∣σ∣ are bounded from the anisotropic modified Morrey space L̃_1,b,σ(Rⁿ) to the weak anisotropic modified Morrey space WL̃_q,b,σ(Rⁿ) if and only if, α/|σ|≤1-1/q≤α/(|σ|(1-b)) and from L̃_p,b,σ(Rⁿ) to L̃_q,b,σ(Rⁿ) if and only if, α/|σ| ≤ 1/p-1/q≤α ((1-b) |σ|). In the limiting case we prove that the operator M_α,σ is bounded from L̃_p,b,σ(Rⁿ) to L_∞ (Rⁿ) and the modified anisotropic Riesz potential operator Ĩ_α,σ is bounded from L̃_p,b,σ(Rⁿ) to BMO_σ(Rⁿ).