Weighted inequalities for negative powers of Schrödinger operators

Abstract

In this article we obtain boundedness of the operator $(-\Delta~+~V)^{-\alpha/2}$ from $L^{p,\infty}(w)$ into weighted bounded mean oscillation type spaces $BMO_\LL^{\beta}(w)$ under appropriate conditions on the weight $w$. We also show that these weighted spaces also have a point-wise description for $0 < \beta < 1$. Finally, we study the behaviour of the operator $(-\Delta~+~V)^{-\alpha/2}$ when acting on $BMO_\LL^{\beta}(w)$.


Published: J. Math. Anal. Appl. 348 (2008), no. 1, 12--27.