Estimates of some selfadjoint operators in the neighborhood of $L_2$

We consider some class of operators, selfadjoint in (vectorial) $L_2$ and defined in its neighborhood in scale of Lebesgue spaces. Typical representative of our class be the operator $D\Delta^{-1}div$. The norm of operator is log-convex by Riesz-Thorin interpolation theorem and so, Lipschitz-continuous in interpolation parameter. No higher smoothness is possible in general situation.
We establish that $L_2$ be the more smooth point of scale: growth of norms of our operators is of second degree of smallness near $L_2$.