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The purpose of this paper is to study basic property of the operator  $$\mathcal{L}_{\mu_1,\mu_2} u=-\Delta +\frac{\mu_1 }{|x|^2}x\cdot\nabla+\frac{\mu_2 }{|x|^2},$$ which generates at the origin due to the criticalgradient and the Hardy term, where $\mu_1,\mu_2$ are free parameters. Thisoperator arises from the critical Caffarelli-Kohn-Nirenberg inequality. Weanalyze the fundamental solutions in a weighted distributional identity andobtain the Liouville theorem for the Lane-Emden equation with that operator, byusing the classification of isolated singular solutions of the related Poissonproblem in a bounded domain $\Omega \subset \mathbb{R}^N$ ($N \geq 2$)containing the origin.

Qualitative properties for elliptic problems with CKN operators