Estimates for the difference of the fractional powers of self-adjoint operators under unbounded perturbations

For positive self-adjoint operators $A_0$, $A_1$ on Hilbert spaces $\mathcal{H}_0$, $\mathcal{H}_1$ and for an operator $\mathcal{J}: \mathcal{H}_0\to\mathcal{H}_1$, the following estimate is obtained:
$$ |\alpha^{-1}(A_1^\alpha\mathcal{J}-\mathcal{J}A_0^\alpha)|_\gamma\leqslant A_1^{-\delta}(A_1\mathcal{J}-\mathcal{J}A_0)A_0^{-\delta},\quad 2\delta=1-\alpha,\quad-1<\alpha<1. $$
Here $|\cdot|_\gamma$ denotes the norm in some symmetric-normed operator ideal $\gamma$. Some generalizations of this estimate are presented too. Applications to the differential operators are discussed.