Scale invariant elliptic operators with singular coefficients

We show that a realization of the operator ({L=|x|^alphaDelta +c|x|^{alpha-1}frac{x}{|x|}cdotnabla -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} < N/p < 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.