Generalized Hardy Operators

Consider the operator on $L^{2}(\mathbb{R}^d)$ \begin{equation*}%\label{defn-La} \La = (-\Delta)^{\alpha/2}+a|x|^{-\alpha} \quad \text{with} \quad 0<\alpha<\min\{2, d\}. \end{equation*} Under the condition $a\ge -\f{2^\alpha\Gamma((d+\alpha)/4)^2} {\Gamma((d-\alpha)/4)^2}$ the operator is non negative and selfadjoint. We prove that fractional powers $\mathcal{L}^{s/2}$ for $s\in(0,2]$ satisfy the estimates \begin{equation*} \|\mathcal{L}_{a}^{s/2}f\|_{L^{p}} \lesssim\|(-\Delta)^{s/2}f\|_{L^{p}}, \qquad \|(-\Delta)^{s/2}f\|_{L^{p}} \lesssim \|\mathcal{L}_{a}^{s/2}f\|_{L^{p}} \end{equation*} for suitable ranges of $p$. Our result fills the remaining gap in earlier results from \cite{K.et.al}, \cite{FMS}, \cite{Merz}. The method of proof is based on square function estimates for operators whose heat kernel has a weak decay.