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.