Riesz potential on the Heisenberg group and modified Morrey spaces

In this paper we study the fractional maximal operator M_α, 0 ≤ α < Q and the Riesz potential operator ℑ_α, 0 < α < Q on the Heisenberg group in the modified Morrey spaces L͂_p,λ(ℍ_n), where Q = 2n + 2 is the homogeneous dimension on ℍ_n. We prove that the operators M_α and ℑ_α are bounded from the modified Morrey space L͂_1,λ(ℍ_n) to the weak modified Morrey space WL͂_q,λ(ℍ_n) if and only if, α/Q ≤ 1 - 1/q ≤ α/(Q - λ) and from L͂_p,λ(ℍ_n) to L͂_q,λ(ℍ_n) if and only if, α/Q ≤ 1/p - 1/q ≤ α/(Q - λ).

In the limiting case we prove that the operator M_α is bounded from L͂_p,λ(ℍ_n) to L_∞(ℍ_n) and the modified fractional integral operator Ĩ_α is bounded from L͂_p,λ(ℍ_n) to BMO(ℍ_n).

As applications of the properties of the fundamental solution of sub-Laplacian Ը on ℍ_n, we prove two Sobolev-Stein embedding theorems on modified Morrey and Besov-modified Morrey spaces in the Heisenberg group setting. As an another application, we prove the boundedness of ℑ_α from the Besov-modified Morrey spaces BL͂^s_pθ,λ(ℍ_n) to BL͂^s_pθ,λ(ℍ_n).