  We prove a local gradient estimate for positive eigenfunctions of $
\mathcal{L} $-operator on conformal solitons given by a general conformal
vector field. As an application, we obtain a Liouville type theorem for $
\mathcal{L} u = 0 $, which improves the one of Li--Sun (Acta Math. Sin. (Engl.
Ser.), 37(11): 1768--1782, 2021.). We also consider applications where
manifolds are special conformal solitons. Especially in the case of
self-shrinkers, a better Liouville type theorem is obtained.
