Abstract
This paper comprises two parts. We first investigate an
{L^{p}}
-type of limiting absorption principle for Schrödinger operators
{H=-\Delta+V}
on
{\mathbb{R}^{n}}
(
{n\geq 3}
), i.e., we prove the ϵ-uniform
{L^{{2(n+1)}/({n+3})}}
–
{L^{{2(n+1)}/({n-1})}}
-estimates of the resolvent
{(H-\lambda\pm i\epsilon)^{-1}}
for all
{\lambda>0}
under the assumptions that the potential V belongs to some integrable spaces and a spectral condition of H at zero is satisfied. As applications, we establish a sharp Hörmander-type spectral multiplier theorem associated with Schrödinger operators H and deduce
{L^{p}}
-bounds of the corresponding Bochner–Riesz operators. Next, we consider the fractional Schrödinger operator
{H=(-\Delta)^{\alpha}+V}
(
{0<2\alpha<n}
) and prove a uniform Hardy–Littlewood–Sobolev inequality for
{(-\Delta)^{\alpha}}
, which generalizes the corresponding result of Kenig–Ruiz–Sogge [20].
Spectral multiplier theorem for $H^1$ spaces associated with some Schrödinger operators
