Strongly elliptic operators with distributional coefficients

Abstract

We study operators of the form $$ L=\sum_{|α|,|β|\le n} D^αc_{α,β} D^β, x\in\mathbb R^n $$ provided that the coefficients of the main symbol corresponding to the indices $|α|=|β|=m$ 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 $c_{α,β}(x)$ which guarantee that the operator $L$ is well defined. In particular, if $c_{α,β}(x)$ belong to the spaces of multipliers from $H_2^{m-|α|}$ to $H_2^{|β|-m}$ then $L$ defines a maximal sectorial operator in $L_2(\mathbb R^n)$. We also describe the spaces of multipliers.