Convex Functions are $p$-Subharmonic Functions, $p >1$ On $\mathbb{R}^n$ with Applications

Theorem 1.1. A $C^2$ convex function on a Riemannian manifold M is p-subharmonic, for every p > 1. Theorem 1.1 is a real analog of a well-known result in complex geometry due to Greene and Wu [GW]

Theorem 1.2. On a Kähler manifold, every $C^2$ convex function is plurisubharmonic. Recall a $\mathbb{C}^2$ real-valued function f on a complex manifold is said to be plurisub-harmonic if the Levi form Lf of f $$Lf \equiv 4 \sum_{\alpha,\beta} \frac{\partial^2 f}{\partial z^{\alpha} \partial \overline{z}^{\beta}} dz^{\alpha} d\overline{z}^{\beta} \ge 0,$$ where $\{z^{\alpha}=x^{\alpha}+\sqrt{-1}\,y^{\alpha}\}\$ is a local (complex) coordinate system in M, $dz^{\alpha}=dx^{\alpha}+\sqrt{-1}\,dy^{\alpha}$ , $\frac{\partial}{\partial z^{\alpha}}=\frac{1}{2}(\frac{\partial}{\partial x^{\alpha}}-\sqrt{-1}\frac{\partial}{\partial y^{\alpha}})$ , and $d\overline{z}^{\alpha}$ and $\frac{\partial}{\partial \overline{z}^{\alpha}}$ are complex conjugates of $dz^{\alpha}$ and $\frac{\partial}{\partial z^{\alpha}}$ respectively. If f is a submersion, i.e. $|df| \neq 0$ everywhere, then one can extend the range of p:

Theorem 1.3. A $C^2$ convex function on a Riemannian manifold M that is a submersion, is p-subharmonic for every $p \ge 1$ . This result is sharp (cf. Counter-Example 4.1). As immediate consequences of Theorems 1.1 and 1.2, we have

Corollary 1.1. Every $C^2$ concave function on a Riemannian manifold M is p-superharmonic, for any p > 1, and every $C^2$ concave submersive function on M is p-superharmonic, for any $p \ge 1$ .

Corollary 1.2. Let $f_i$ , i=1,2 and p be as in the assumption and conclusion of Theorem 1.1 or 1.2 respectively. Let $\lambda > 0$ , then $\lambda f_1$ , $f_1 + f_2$ , and $\max\{f_1, f_2\}$ are p-subharmonic functions.

Corollary 1.3. Let an increasing sequence of functions $\{f_i\}_{i=1}^{\infty}$ and p be as in the Corollary 1.2. Then $\lim_{i\to\infty} f_i$ is p-subharmonic. <span id="page-4-2"></span>If M is Euclidean space $\mathbb{R}^n$ , then one can drop the $C^2$ assumption on f:

Theorem 1.4. A convex function on $\mathbb{R}^n$ is p-subharmonic, for every p > 1, and a convex function on $\mathbb{R}^n$ with the n-dimensional Lebesgue measure $\mathcal{L}^n(\{x \in \mathbb{R}^n : |df| = 0\}) = 0$ , is p-subharmonic, for every $p \ge 1$ , This result is sharp (cf. Counter-Example 4.1). In this paper we combine the link between convex functions and p-subharmonic functions, and the estimates on the growth of p-subharmonic functions (cf.[WLW2], or $\S 2$ ) to prove Liouville type theorems for convex functions. We recall for a given $q \in \mathbb{R}$ , a function or a differential form or a bundle-valued differential form f has p-balanced growth (or, simply, is p-balanced) if f has one of the following: p-finite, p-mild, p-obtuse, p-moderate, or p-small growth, and has p-imbalanced growth (or, simply, is p-imbalanced) otherwise (cf. [WLW2], or $\S 2$ ). As further applications, we have the following.

Theorem 5.1. [Louiville Type Theorem for Convex Functions] Every p-balanced nonnegative $C^2$ convex function on a complete noncompact Riemannian manifold is constant for p > 1. Corollary 5.1 Every $L^q$ - nonnegative $C^2$ convex function on a complete noncompact Riemannian manifold is constant for q > p - 1 > 0.

Theorem 2.1. ([W6], Theorem 5.4.). For a given $q \in \mathbb{R}$ , a function, or differential form or bundle-valued differential form f is p-moderate (2.4) $\Leftrightarrow$ p-small (2.6) $\Rightarrow$ p-mild (2.2) $\Rightarrow$ p-obtuse (2.3) or equiavalently, $p-acute \quad \Rightarrow \quad p-severe \quad \Rightarrow \quad p-large \quad \Leftrightarrow \quad p-immoderate.$ Hence, for a given $q \in \mathbb{R}$ , f is $$p$$ - balanced $\Rightarrow$ either $p$ - finite (2.1) or $p$ - obtuse (2.3) $$p-\text{imbalanced} \Rightarrow \text{both} p-\text{infinite} \text{ and } p-\text{immoderate}$$ . If in addition, $\int_{B(x_0;r)} |f|^q dv$ is convex in r, then the following four types of growth are all equivalent: f is p-mild, p-obtuse, p-moderate, and p-small (resp. p-severe, p-acute, p-immoderate, and p-large), i.e., f is $$(2.2)$$ $\Leftrightarrow$ $(2.3)$ $\Leftrightarrow$ $(2.4)$ $\Leftrightarrow$ $(2.6)$ for the same value of $q \in \mathbb{R}$ . In particular, we have

Corollary 2.1. ([W6, Corollary 5.1]) Every $L^q$ function or differential form or bundle-valued differential form f on M has p-balanced growth, $p \ge 0$ , and in fact, has p-finite, p-mild, p-obtuse, p-moderate, and p-small growth, $p \ge 0$ , for the same value of q In [WLW 2], among many different types of inequalities on a complete noncompact Reimannian manifold M, we have the following uniqueness property.

Theorem 2.2. (Liouville Property for solutions of $f \operatorname{div}(|\nabla f|^{p-2}\nabla f) \geq 0$ ). Every $C^2$ solution $f: M \to (-\infty, \infty)$ of $f \operatorname{div}(|\nabla f|^{p-2}\nabla f) \geq 0$ is constant provided f is p-balanced, i.e. f is one of the following: p-finite, p-mild, p-obtuse, p-moderate, or p-small, for some q > p - 1. In particular, every $C^2$ , $L^q$ solution f of $f \operatorname{div}(|\nabla f|^{p-2}\nabla f) \geq 0$ is constant for any q > p - 1.

Theorem 5.1. (Louiville Type Theorem for Convex Functions). Every p-balanced nonnegative $C^2$ convex function on a complete noncompact Riemannian manifold M is constant for p > 1. Proof of Theorem 5.1. Since f is a $C^2$ convex function on M, Theorem 1.1 implies that f is a p-subharmonic function for p > 1. This is equivalent to f is a subsolution of the p-Laplace equation, i.e., $\operatorname{div}(|df|^{p-2})df \geq 0$ . In view of $f \geq 0$ , we have $f\operatorname{div}(|df|^{p-2})df \geq 0$ . It follows from Theorem 2.2 that f is constant.

Corollary 5.1. Every $C^2$ , $L^q$ convex function on a complete noncompact Riemannian manifold M is constant for any q > p - 1 > 0

Definition 2.1. For a given $q \in \mathbb{R}$ , a function or a differential form or a bundle-valued differential form f has p-finite growth (or, simply, is p-finite) if there exists $x_0 \in M$ such that <span id="page-5-4"></span>(2.1) $$\liminf_{r \to \infty} \frac{1}{r^p} \int_{B(x_0;r)} |f|^q \, dv < \infty,$$ and has p-infinite growth (or, simply, is p-infinite) otherwise. For a given $q \in \mathbb{R}$ , a function or a differential form or a bundle-valued differential form f has p-mild growth (or, simply, is p-mild) if there exist $x_0 \in M$ , and a strictly increasing sequence of $\{r_j\}_0^{\infty}$ going to infinity, such that for every $l_0 > 0$ , we have <span id="page-5-2"></span>(2.2) $$\sum_{j=\ell_0}^{\infty} \left( \frac{(r_{j+1} - r_j)^p}{\int_{B(x_0; r_{j+1}) \setminus B(x_0; r_j)} |f|^q dv} \right)^{\frac{1}{p-1}} = \infty,$$ and has p-severe growth (or, simply, is p-severe) otherwise. For a given $q \in \mathbb{R}$ , a function or a differential form or a bundle-valued differential form f has p-obtuse growth (or, simply, is p-obtuse) if there exists $x_0 \in M$ such that for every a > 0, we have <span id="page-5-3"></span>(2.3) $$\int_{a}^{\infty} \left(\frac{1}{\int_{\partial B(x_0;r)} |f|^q ds}\right)^{\frac{1}{p-1}} dr = \infty,$$ and has p-acute growth (or, simply, is p-acute) otherwise. For a given $q \in \mathbb{R}$ , a function or a differential form or a bundle-valued differential form f has p-moderate growth (or, simply, is p-moderate) if there exist $x_0 \in M$ , and $\psi(r) \in \mathcal{F}$ , such that <span id="page-5-0"></span>(2.4) $$\limsup_{r \to \infty} \frac{1}{r^p \psi^{p-1}(r)} \int_{B(x_0;r)} |f|^q \, dv < \infty,$$ and has p-immoderate growth (or, simply, is p-immoderate) otherwise, where (2.5) $$\mathcal{F} = \{ \psi : [a, \infty) \longrightarrow (0, \infty) | \int_{a}^{\infty} \frac{dr}{r\psi(r)} = \infty \text{ for some } a \ge 0 \}.$$ (Notice that the functions in $\mathcal{F}$ are not necessarily monotone.) For a given $q \in \mathbb{R}$ , a function or a differential form or a bundle-valued differential form f has p-small growth (or, simply, is p-small) if there exists $x_0 \in M$ , such that for every a > 0, we have <span id="page-5-1"></span>(2.6) $$\int_{a}^{\infty} \left( \frac{r}{\int_{B(x_0;r)} |f|^q \, dv} \right)^{\frac{1}{p-1}} dr = \infty \,,$$ and has p-large growth (or, simply, is p-large) otherwise.

Definition 2.2. For a given $q \in \mathbb{R}$ , a function or a differential form or a bundle-valued differential form f has p-balanced growth (or, simply, is p-balanced) if f has one of the following: p-finite, p-mild, p-obtuse, p-moderate, or p-small growth, and has p-imbalanced growth (or, simply, is p-imbalanced) otherwise. The above definitions of "p-balanced, p-finite, p-mild, p-obtuse, p-moderate, p-small" and their counter-parts "p-imbalanced, p-infinite, p-severe, p-acute, p-immoderate, p-large" growth depend on q, and q will be specified in the context in which the definition is used.