Morrey spaces for Schrödinger operators with nonnegative potentials, fractional integral operators and the Adams inequality on the Heisenberg groups

Abstract

Let L=−ΔHn+V be a Schrödinger operator on the Heisenberg group Hn, where ΔHn is the sublaplacian on Hn and the nonnegative potential V belongs to the reverse Hölder class RHs with s∈[Q/2,∞). Here Q=2n+2 is the homogeneous dimension of Hn. For given α∈(0,Q), the fractional integral operator associated with the Schrödinger operator L is defined by Iα=L−α/2. In this article, the author introduces the Morrey space Lρ,∞p,κ(Hn) and weak Morrey space WLρ,∞p,κ(Hn) associated with L, where (p,κ)∈[1,∞)×[0,1) and ρ(⋅) is an auxiliary function related to the nonnegative potential V. The relation between the fractional integral operator and the maximal operator on the Heisenberg group is established. From this, the author further obtains the Adams (Morrey-Sobolev) inequality on these new spaces. It is shown that the fractional integral operator Iα=L−α/2 is bounded from Lρ,∞p,κ(Hn) to Lρ,∞q,κ(Hn) with 0<α<Q, 1<p<Q/α, 0<κ<1−(αp)/Q and 1/q=1/p−α/Q(1−κ), and bounded from Lρ,∞1,κ(Hn) to WLρ,∞q,κ(Hn) with 0<α<Q, 0<κ<1−α/Q and 1/q=1−α/Q(1−κ). Moreover, in order to deal with the extreme cases κ≥1−(αp)/Q, the author also introduces the spaces BMOρ,∞(Hn) and Cρ,∞β(Hn), β∈(0,1] associated with L. In addition, it is proved that Iα is bounded from Lρ,∞p,κ(Hn) to BMOρ,∞(Hn) under κ=1−(αp)/Q, and bounded from Lρ,∞p,κ(Hn) to Cρ,∞β(Hn) under κ>1−(αp)/Q and β=α−(1−κ)Q/p.