Scale invariant elliptic operators with singular coefficients

Abstract

We show that a realization of the operator $L=|x|^\alpha\Delta +c|x|^{\alpha-1}\frac{x}{|x|}\cdot\nabla -b|x|^{\alpha-2}$ generates a semigroup in $L^p(\mathbb {R}^N)$ if and only if $D_c=b+(N-2+c)^2/4 > 0$ and $s_1+\min\{0,2-\alpha\}<N/p<s_2+\max\{0,2-\alpha\}$, where $s_i$ are the roots of the equation $b+s(N-2+c-s)=0$, or $D_c=0$ and $s_0+\min\{0,2-\alpha\} \le N/p \le s_0+\max\{0,2-\alpha\}$, where $s_0$ is the unique root of the above equation. The domain of the generator is also characterized.