Strongly elliptic operators with distributional coefficients

We study operators of the form $$L=\sum _{\mathrm{\mid }\alpha \mathrm{\mid },\mathrm{\mid }\beta \mathrm{\mid }\le n}{D}^{\alpha }{c}_{\alpha ,\beta }{D}^{\beta },x\in {\mathbb{R}}^n$$ provided that the coefficients of the main symbol corresponding to the indices $\mathrm{\mid }\alpha \mathrm{\mid }=\mathrm{\mid }\beta \mathrm{\mid }=m$ are continuous while the other ones are distributions. Assuming that the main symbol defines the strongly elliptic operator we find sufficient conditions for the coefficients $c_{\alpha ,\beta }(x)$ which guarantee that the operator $L$ is well defined. In particular, if $c_{\alpha ,\beta }(x)$ belong to the spaces of multipliers from $H_2^{m-\mathrm{\mid }\alpha \mathrm{\mid }}$ to $H_2^{\mathrm{\mid }\beta \mathrm{\mid }-m}$ then $L$ defines a maximal sectorial operator in $L_2({\mathbb{R}}^n)$. We also describe the spaces of multipliers.