Mixed-Parabolicity and Mixed-Liouville Property for Products of Riemannian Manifolds

Let $p_1,p_2\in (1,\mathrm{\infty })$ and $M=M_1\times M_2$ be the product of two geodesically complete Riemannian manifolds. In this paper, the authors first develop an anisotropic potential-theoretic framework adapted to the Green operator $G^M$ and the mixed-norm Lebesgue space $L^{p_2}(L^{p_1})(M)$, and then demonstrate that the classical equivalence among \emph{parabolicity}, \emph{Green function integrability}, and \emph{Liouville property} persists in this genuinely anisotropic setting. More precisely, the authors establish the following equivalence: $M$ is $L^{p_2}(L^{p_1})$-parabolic if and only if the Green function $G^M(x;\cdot )$ fails to belong to $L^{p_2^{\mathrm{\prime }}}(L^{p_1^{\mathrm{\prime }}})(M\setminus B(x,r))$, which is in turn equivalent to the $L^{p_2^{\mathrm{\prime }}}(L^{p_1^{\mathrm{\prime }}})$-Liouville property, where $p_i^{\mathrm{\prime }}$ denotes the conjugate exponent of $p_i$. Under a weak radial Harnack-type inequality – in particular, under Li–Yau heat kernel estimates, and hence for products of manifolds with nonnegative Ricci curvature – these conditions are further equivalent to the divergence of the nonlinear mixed-potential ${\mathcal{G}}_{p_1,p_2}(f)$ for every nonzero nonnegative $f\in {\mathcal{C}}_c^{\mathrm{\infty }}(M)$. A key feature of this anisotropic theory is its sensitivity to the geometry of each factor $M_i$, rather than merely to that of the total manifold $M$. In contrast to the isotropic case, where parabolicity and the classical Liouville property holds on ${\mathbb{R}}^n$ precisely when $n\le 2$, the anisotropic setting exhibits a refined threshold: the $L^{p_2}(L^{p_1})$-parabolicity and the $L^{p_2^{\mathrm{\prime }}}(L^{p_1^{\mathrm{\prime }}})$-Liouville property holds on ${\mathbb{R}}^{n_1}\times {\mathbb{R}}^{n_2}$ if and only if $D_{\mathrm{e}\mathrm{f}\mathrm{f}}:=\frac{n_1}{p_1}+\frac{n_2}{p_2}\le 2.$ This effective dimension $D_{\mathrm{e}\mathrm{f}\mathrm{f}}$ captures the anisotropic interplay between the exponents $p_1,p_2$ and the geometries of $M_1,M_2$.