Sufficient conditions for Liouville theorem Recently I was reading the theory of elliptic operators and there was a statement like this: given an elliptic partial differential operator $L$ on $C^2(\mathbb{R}^n)$ the Liouville theorem accounts to the strong maximum principle (SMP). Unfortunately I cannot find now such place (and a paper in fact) - so I am curious what was the meaning of this statement. 
That's why I wonder: is SMP sufficient for the Liouville theorem to hold for an elliptic operator $L$? From my side I tried to prove Liouville theorem using only SMP - but I haven't succeed. 
I assume that SMP and Liouville theorem are known, but I can provide definitions if needed.
 A: In general I don't think Liouville's theorem follows from the SMP (at least I cannot work out a proof at the moment). The SMP is of the formal form that for a certain class of functions $\mathcal{F}$ defined over $\mathbb{R}^d$

If $f\in \mathcal{F}$, let $\Omega$ be bounded with (say) smooth boundary, then for any $\Omega' \Subset \Omega$ you have $\sup_{\Omega'} f \leq \sup_{\partial\Omega} f$ with equality only when $f$ is constant.

Using just this it is clearly impossible to conclude Liouville's theorem, since $f(x) = \tan^{-1}(x_1)$ satisfies the above "abstract" theorem but not Liouville, which says that

If $f\in \mathcal{F}$, and if $|f|$ is bounded, then $f$ is constant.

As far as I know, to actually obtain Liouville's theorem from the general sorts of argument you get for uniformly elliptic operators, you need to appeal to something more quantitative (in particular a way to put a bound on the growth rate). The usual arguments require a use of Harnack's inequality. Now, Harnack's inequality can also be used (as a starting point to derive Hopf-like lemmas) to establish the strong maximum principle, which maybe where you saw the connection drawn between SMP and Liouville.
