Strongly elliptic operators with distributional coefficients

Abstract

We study operators of the form $$ L=\sum_{|\alpha|,|\beta|\le n} D^\alpha c_{\alpha,\beta} D^\beta, x\in\mathbb R^n $$ provided that the coefficients of the main symbol corresponding to the indices $|\alpha|=|\beta|=m$ 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 $H_2^{m-|\alpha|}$ to $H_2^{|\beta|-m}$ then $L$ defines a maximal sectorial operator in $L_2(\mathbb R^n)$. We also describe the spaces of multipliers.