On the coercive solvability of differential-operator equations in spaces of the Besov type (Q1593957)

Consider the differential-operator equation \[ L_\lambda u= \sum^n_{k=1} (-1)^{\ell_k} D^{2\ell_k}_k u+ A_\lambda u+\sum_{\alpha, 0<k<1} A_\alpha(x) D^\alpha u= f,\tag{1} \] where \(A\) and \(A_\alpha(x)\) are unbounded operators in the Hilbert space \(H\), \(A_\lambda= A+\lambda I\), and the operator \(-A\) generates an analytic semigroup. Theorem. Assume that \(A_\alpha(x) A^{-(1-k-\mu)}\in L_\infty(\mathbb{R}^n; L(H))\) for \(0\leq k+\mu\leq 1\). Then for any \(f\in B^s_{p,q}(\mathbb{R}^n; H)\) and any \(\lambda\in S(\varphi)\) with \(\varphi\in (\pi/2,\pi]\) such that \(|\lambda|\) is sufficiently large, there is precisely one solution of (1) belonging to the space \(B^s_{p,q}(\mathbb{R}^n; H(A),H)\) and \[ \|u\|_{B^{2\ell+ s}_{p,q}(\mathbb{R}^n;H(A),H)} \leq c\|f\|_{B^s_{p,q}(\mathbb{R}^n; H)}. \] Some estimates on the resolvent of the operator \(L_\lambda\) in the Besov space norm are also obtained.