text:
\section{Proof of Theorem~\ref{thm:sparseMoment} (Sparse Moment Relaxation)}
\label{sec:supp-proof-sparseRelaxation}
\begin{proof}
Let us first show that the sparse moment relaxation~\eqref{eq:sparseMoment} is indeed a valid relaxation,~\ie~(a) the sparse set of monomials $\monoleq{\xextend}{2\rbasisset}$ contains all the monomials in the objective function $f(\xextend)$ of the POP~\eqref{eq:pop} (otherwise, the objective function of the relaxation~\eqref{eq:sparseMoment} is not equivalent to the objective function of the POP~\eqref{eq:pop}); and (b) the sparse set of moments $\moments_{2\rbasisset}$ contains all the moments appearing in the localizing matrices of~\eqref{eq:sparseMoment} (otherwise, the optimization contains undefined variables). To see (a), from the sparsity of the objective function $f$ (\cf~property~\ref{prop:pop-objective} of Proposition~\ref{prop:pop}), we know that $f = \sumallpoints f_i$ and each $f_i$ at most contains monomials of type $\monoleq{\vxx}{2}$, $\theta_i$ and $\theta_i \cdot \monoleq{\vxx}{2}$ (\cf~the expression of $f_i$ in eq.~\eqref{eq:TLSBinary}), all of which are included in the sparse set of monomials $\monoleq{\xextend}{2\rbasisset}$ (recall $\monoleq{\xextend}{\rbasisset} = [1,\vxx\tran,\vtheta\tran,\monoindeg{\vxx}{2}\tran,\vtheta\tran \kron \vxx\tran]\tran$). To see (b), we observe that $h^x$ and $g$ only contain monomials $\monoleq{\vxx}{2}$, and $\moments_2$ only contain monomials $\monoleq{\xextend}{2}$, so the product $\monoleq{\vxx}{2} \kron \monoleq{\xextend}{2}$ is included in $\monoleq{\xextend}{2\rbasisset}$. Hence, the moments in the localizing matrices $\MM_1\parentheses{h^x\moments_2}$ and $\MM_1\parentheses{g\moments_2}$ are included in the moment vector $\moments_{2\rbasisset}$. Similarly, $h^\theta$ only contains monomials $\monoleq{\vtheta}{2}$, and $\moments_{2\rbasisset_x}$ only contains monomials $\monoleq{\vxx}{2}$, so the product $\monoleq{\vtheta}{2} \kron \monoleq{\vxx}{2}$ is included in $\monoleq{\xextend}{2\rbasisset}$. Hence, the moments in the localizing matrices $\MM_{\rbasisset_x}\parentheses{h^\theta \moments_{2\rbasisset_x}}$ are also included in the moment vector $\moments_{2\rbasisset}$.

{\bf Lower Bound}. Then we prove that $p^\star_{\rbasisset} \leq p^\star_2$,~\ie~the optimal value of the sparse relaxation~\eqref{eq:sparseMoment} is a lower bound of the optimal value of the dense relaxation~\eqref{eq:denseMoment} at order $\rorder=2$. We prove this by showing that the feasible set of the dense relaxation is contained in the feasible set of the sparse relaxation. To see this, consider $\moments_4$ as an arbitrary point in the feasible set of the dense relaxation~\eqref{eq:denseMoment},~\ie~$\moments_4$ satisfies $\moment_\zero = 1$, $\MM_2\parentheses{\moments_4} \succeq 0$, $\MM_1\parentheses{h \moments_2} = \zero,\forall h \in \vh$, and $\MM_1\parentheses{g \moments_2} \succeq 0,\forall g \in \vg$. Then $\moments_{2\rbasisset} \subset \moments_4$, the sub-vector of $\moments_4$ corresponding to the sparse set of monomials $\monoleq{\xextend}{2\rbasisset}$, must be feasible for the sparse relaxation~\eqref{eq:sparseMoment}. This is because $\MM_\rbasisset\parentheses{\moments_{2\rbasisset}} \succeq 0$ must hold as $\MM_\rbasisset\parentheses{\moments_{2\rbasisset}}$ is a principal sub-matrix of the full moment matrix $\MM_2\parentheses{\moments_4}$; $\MM_{\rbasisset_x}\parentheses{h \moments_{2\rbasisset_x}} = \zero$ must hold as $\MM_{\rbasisset_x}\parentheses{h \moments_{2\rbasisset_x}}$ is a principal sub-matrix of the full localizing matrix $\MM_1\parentheses{h \moments_2}, \forall h \in \vh^\theta$; and $\MM_1\parentheses{h\moments_2}$ and $\MM_1\parentheses{g\moments_2}$ are the same as the localizing matrices in the dense relaxation.

{\bf Rounding and Relative Duality Gap}. Property~\ref{prop:dense-gap} of the dense relaxation still holds for the sparse relaxation. Because $p^\star_{\rbasisset} \leq p^\star_2$ and $p^\star_2 \leq f^\star$, we have $p^\star_{\rbasisset} \leq f^\star$ is also a valid lower bound for $f^\star$, the true optimal objective value of the POP. Additionally, since the sparse set of monomials $\monoleq{\xextend}{\rbasisset}$ still contains $\monoleq{\xextend}{1}$, the degree-1 monomials of $\vxx$ and $\vtheta$, one can use the same rounding method (\ie~spectral decomposition and the projection onto the feasible set $\calxextend$ in~\eqref{eq:projection}) to obtain a feasible solution $\hatxextend$, which gives a value $\hatf = f(\hatxextend)$ that satisfies $p^\star_{\rbasisset} \leq f^\star \leq \hatf$. The relative duality gap can then be calculated similar to eq.~\eqref{eq:relDualityGapDense} as:
\bea
\eta_\rbasisset = \frac{\hatf - p^\star_\rbasisset}{\hatf},
\eea
where a smaller $\eta_\rbasisset$ implies a tighter relaxation and $\eta_\rbasisset = 0$ if and only if the relaxation is tight.

{\bf Optimality Certificate}. Showing property~\ref{prop:dense-certificate} of the dense relaxation also holds for the sparse relaxation is non-trivial because Theorem~\ref{thm:sufficientTruncatedK} does not hold for an arbitrary sparse moment matrix (\ie~a moment matrix with rows and columns indexed by a sparse set of monomials $\monoleq{\xextend}{\rbasisset} \subset \monoleq{\xextend}{2}$). Therefore, we will show that the sparse moment matrix $\MM_\rbasisset\parentheses{\moments_{2\rbasisset}}$ can be \emph{extended} to a full moment matrix $\MM_2\parentheses{\moments_4}$ when $\rank{\MM_\rbasisset\parentheses{\moments_{2\rbasisset}}} = 1$. Let us first introduce the notion of a flat extension.

\begin{definition}[\bf Flat Extension]
\label{def:flatExtension}
Given two moment matrices $\MM_{\rbasisset}\parentheses{\moments_{2\rbasisset}}$ and $\MM_{\calA}\parentheses{\moments_{2\calA}}$, with $\rbasisset \subset \calA$ (recall that the rows and columns of $\MM_{\rbasisset}\parentheses{\moments_{2\rbasisset}}$ (resp. $\MM_{\calA}\parentheses{\moments_{2\calA}}$) are indexed by monomials $\monoleq{\xextend}{\rbasisset}$ (resp. $\monoleq{\xextend}{\calA}$)), then $\MM_{\calA}\parentheses{\moments_{2\calA}}$ is said to be a flat extension of $\MM_{\rbasisset}\parentheses{\moments_{2\rbasisset}}$ if $\MM_{\rbasisset}\parentheses{\moments_{2\rbasisset}}$ coincides with the sub-matrix of $\MM_{\calA}\parentheses{\moments_{2\calA}}$ indexed by $\monoleq{\xextend}{\rbasisset}$ and $\rank{\MM_{\rbasisset}\parentheses{\moments_{2\rbasisset}}} = \rank{\MM_{\calA}\parentheses{\moments_{2\calA}}}$.
\end{definition}

Our goal is to show that $\MM_{\rbasisset}\parentheses{\moments_{2\rbasisset}}$ admits a flat extension $\MM_2\parentheses{\moments_4}$ when $\rank{\MM_{\rbasisset}\parentheses{\moments_{2\rbasisset}}} = 1$, so that $\rank{\MM_2\parentheses{\moments_4}}=1$ is also true, in which case we recover the dense moment relaxation and obtain an optimality certificate. To do so, we will show that the sparse moment matrix $\MM_{\rbasisset}\parentheses{\moments_{2\rbasisset}}$ satisfies the \emph{generalized flat extension theorem} in~\cite{Laurent09AdM-generalFlatExtension}, stated below.

\begin{theorem}[\bf Generalized Flat Extension~{\cite[Theorem 1.4]{Laurent09AdM-generalFlatExtension}}]
\label{thm:generalFlatExtension}
Given a monomial basis $\monoleq{\xextend}{\calC}$, define its \emph{closure} to be the set:
\bea
\monoleq{\xextend}{\calC^+} \doteq \monoleq{\xextend}{\calC} \cup \left(  \cup_{i=1}^\dimxextend p_i \monoleq{\xextend}{\calC} \right)  \doteq \left\{ \xextend^\valpha,p_1\xextend^\valpha,\dots,p_{\dimxextend}\xextend^\valpha | \valpha \in \calC \right\}.
\eea
For example, let $\dimxextend = 3$, and $\monoleq{\xextend}{\calC} = [p_1]$, then $\monoleq{\xextend}{\calC^+} = [p_1,p_1^2,p_1p_2,p_1p_3]$. In addition, the monomial set $\monoleq{\xextend}{\calC}$ is said to be \emph{connected to 1} if $1 \in \monoleq{\xextend}{\calC}$ and every monomial $\xextend^\valpha$ can be written as $\xextend^\valpha = p_{i_1}p_{i_2}\cdots p_{i_k}$ with $p_{i_1},p_{i_1}p_{i_2},...,p_{i_1}p_{i_2}\cdots p_{i_k} \in \monoleq{\xextend}{\calC}$. For example $[1,p_1,p_1p_2]$ is connected to 1, but $[1,p_1p_2]$ is not. Then the generalized flat extension theorem states:

If $\monoleq{\xextend}{\calC}$ is connected to 1, and $\MM_{\calC^+}\parentheses{\moments_{2\calC^+}}$ is a flat extension of $\MM_{\calC}\parentheses{\moments_{2\calC}}$, then $\MM_{\calC^+}\parentheses{\moments_{2\calC^+}}$ admits a unique flat extension $\MM_\rorder\parentheses{\moments_{2\rorder}}$ for any $\rorder \geq \alpha_\max$, where $\alpha_\max \doteq \max\{|\valpha|: \valpha \in \calC^+\}$ denotes the maximum degree of the monomials in $\monoleq{\xextend}{\calC^+}$.
\end{theorem}

Using Theorem~\ref{thm:generalFlatExtension}, let $\monoleq{\xextend}{\calC} \doteq \monoleq{\xextend}{\rbasisset_x} = [1,\vxx\tran]\tran$, then obviously $\monoleq{\xextend}{\calC}$ is connected to 1. The closure of $\monoleq{\xextend}{\calC}$ is $\monoleq{\xextend}{\calC^+} = \monoleq{\xextend}{\rbasisset} = [1,\vxx\tran,\vtheta\tran,\monoindeg{\vxx}{2},\vtheta\tran \kron \vxx\tran]
\tran$. If $\rank{\MM_\rbasisset\parentheses{\moments_{2\rbasisset}}} = 1$, then $\rank{\MM_\rbasisset\parentheses{\moments_{2\rbasisset}}} = \rank{\MM_{\rbasisset_x}\parentheses{\moments_{2\rbasisset_x}}} = 1$ and $\MM_\rbasisset\parentheses{\moments_{2\rbasisset}}$ is a flat extension of $\MM_{\rbasisset_x}\parentheses{\moments_{2\rbasisset_x}}$. Therefore, $\MM_\rbasisset\parentheses{\moments_{2\rbasisset}}$ admits a flat extension $\MM_\rorder\parentheses{\moments_{2\rorder}}$ for any $\rorder \geq 2$. In particular, setting $\rorder = 2$, we recover a dense moment matrix $\MM_2\parentheses{\moments_4}$ with $\rank{\MM_2\parentheses{\moments_4}}=1$. It remains to show that the moments $\moments_4$ (obtained from the flat extension) satisfy the constraints on the localizing matrices in the dense relaxation~\eqref{eq:denseMoment}. The only different constraint between the dense relaxation~\eqref{eq:denseMoment} at $\rorder=2$ and the sparse relaxation~\eqref{eq:sparseMoment} is that, the constraint $\MM_1\parentheses{h \moments_2} = \zero$ has been relaxed to $\MM_1\parentheses{h \moments_{2\rbasisset_x}} = \zero$ for $h \in \vh^\theta$. However, we observe that the top-left entry of $\MM_1\parentheses{h \moments_{2\rbasisset_x}}$ is $\moment_1 - \moment_{\theta_i^2}$ when $h = h^{\theta_i} = 1-\theta_i^2$, and $\moment_1 - \moment_{\theta_i^2} = 0$ implies $\moment_{\theta_i^2} = \moment_{\theta_i}^2 = 1$ (due to $\rank{\MM_2\parentheses{\moments_4}} =1 $), which implies $\moment_{\theta_i} = \pm 1$ and the solution is indeed binary and supported on $\calxextend$ and must satisfy $\MM_1\parentheses{h \moments_2} = \zero$.
\end{proof}

config: arxiv


TITLE: how to show $c=0$,$f:(0,\infty)\rightarrow\mathbb{R}$ is differentiable, $\lim_{x\rightarrow\infty}f(x)=1$,
QUESTION [2 upvotes]: $f:(0,\infty)\rightarrow\mathbb{R}$ is differentiable, $\lim_{x\rightarrow\infty}f(x)=1$, $\lim_{x\rightarrow\infty}f'(x)=c$   we need to show $c=0$
well, I tried like this $|f(x)-1|<\epsilon\forall x>M$, where $M$ is very large, $|f'(x)-c|<\epsilon\forall x>M$,  what more I can say?thank you.

REPLY [3 votes]: Since $\lim_{x \to \infty} \frac{f'(x)}{1}$ exists, by L'Hospital
$$\lim_{x \to \infty} \frac{f(x)}{x}=\lim_{x \to \infty} \frac{f'(x)}{1}=c \,.$$
But the first limit is $0$.
Alternately Use the MVT on the interval $[x,2x]$. You have
$$\frac{f(2x)-f(x)}{x}=f'(c_x)$$
Now, when $x \to \infty$ we have $f'(c_x) \to f(c)$ and the LHS gos to $0$.

REPLY [0 votes]: Yet another proof:
Let $0<x$. By the MVT in $(x,x+1)$:
$$\exists \xi\in (x,x+1):f'(\xi)=\frac{f(x+1)-f(x)}{x+1-x}=f(x+1)-f(x)$$
Letting $x\to +\infty$, $\xi\to +\infty$ and so
$$\lim_{x\to +\infty}f'(x)=1-1=0$$


TITLE: Is existence of cosmic dark energy falsifiable yet?
QUESTION [0 upvotes]: Is "existence of cosmic dark energy" falsifiable yet? And what about dark matters? Let me rephrase it again(pardon me for my previous poor expression)-
Is there any condition discovered which can be used to determine the "existence of cosmic dark energy" is true or false without any ambiguity?
I'm not a physics guy. Just being curious.

REPLY [0 votes]: The word "falsifiable" is a philosophical term which just means that the theory can predict statements which may (or may not) conflict with available observations.  This question is essentially just asking whether or not there are any predictions made by the dark energy theory which could conflict with observations that have already been made, or could be made with current technology.  Thus, the answer is a simple yes, the model is currently falsifiable.  The accelerating rate of expansion determines a value for the cosmological constant and this value helps determine the curvature of the universe.  One would naturally predict, based upon the rate of accelerating expansion and other observations, that the universe should be nearly flat, and this is indeed observed.  If the universe were curved then this theory would have been falsified.
The epicycles may be interpreted either as a mathematical device or as a 'real' representation of what's going on.  Either way this theory is falsifiable if one measures the planet's positions with sufficient precision - this model cannot incorporate perturbations.  Basically, any good theory is falsifiable, and every one will get falsified eventually until the final theory is found.


text:
\begin{document}

\title[Liouville theorems]{Liouville results for fully nonlinear equations modeled on H\"ormander vector fields. I. The Heisenberg group}

\author{Martino Bardi}
\author{Alessandro Goffi}

\date{\today}
\subjclass[2010]{Primary: 35B53, 35J70, 35J60; Secondary: 49L25, 35H20.}
\keywords{Fully nonlinear equation, degenerate elliptic equation, subelliptic equation, H\"ormander condition, Liouville theorems, Heisenberg
 group.}
 \thanks{
 The authors are members of the Gruppo Nazionale per l'Analisi Matematica, la Probabilit\`a e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). The authors were partially supported by the research project ``Nonlinear Partial Differential Equations: Asymptotic Problems and Mean-Field Games" of the Fondazione CaRiPaRo.

This research was largely carried out while the second-named author was Ph.D. fellow at Gran Sasso Science Institute and the main results are part of his PhD thesis.}
\address{Department of Mathematics ``T. Levi-Civita", University of Padova, Via Trieste 63, 35121 Padova, Italy} \email{bardi@math.unipd.it}
\address{Department of Mathematics ``T. Levi-Civita", University of Padova, Via Trieste 63, 35121 Padova, Italy} \email{alessandro.goffi@math.unipd.it}

\maketitle
\begin{abstract}
This paper studies Liouville properties for viscosity sub- and supersolutions of fully nonlinear degenerate elliptic PDEs, under the main assumption that
the operator has a family of generalized subunit vector fields that satisfy the H\"ormander condition. A general set of sufficient conditions is given such that all subsolutions bounded above are constant; it includes the existence of a supersolution out of a big ball, that explodes at infinity. Therefore for a large class of operators the problem is reduced to finding such a Lyapunov-like function. This is
done here for the vector fields that generate the Heisenberg group, giving explicit conditions on the sign and size of the first and zero-th order terms in the equation.

The optimality of the conditions is shown via several examples. A sequel of this paper applies the methods to other Carnot groups and to Grushin geometries.
\end{abstract}
\tableofcontents

\section{Introduction}
\label{intro}
In this paper we study Liouville properties for viscosity sub- or supersolutions of fully nonlinear degenerate elliptic equations
\begin{equation}\label{basic}
F(x, u, Du, D^2u)=0 \quad\text{ in }\R^d ,
\end{equation}
where $F : \R^d\times\R\times\R^d\times\mathcal S_d\to \R$ is at least continuous and proper, i.e., nondecreasing in the second entry and non-increasing in the last entry (with respect to the partial order of symmetric matrices). Our main assumption is the existence of a family
 $\mathcal{X}=(X_1,...,X_m)$ of
  vector fields satisfying the H\"ormander bracket generating condition and {\em subunit} for $F$ in the following sense: for all $i=1,...,m$
\begin{equation}\label{subunitH}
\sup_{\gamma>0} F(x,0,p,I-\gamma p\otimes p)> 0 \quad \forall p\in \R^{d} \;\text{ such that } \; X_i(x)\cdot p\ne 0 .
\end{equation}
This generalizes the classical definition of subunit vectors for linear operators by Fefferman and Phong \cite{FP} and was introduced in our recent paper \cite{BG}. Typical examples are subelliptic equations of the form
\begin{equation}\label{1L-}
G(x,u,D
u,(D_{\mathcal{X}}^2u)^*)=0\ ,
\end{equation}
where $(D_{\mathcal{X}}^2 u)_{ij} =X_i (X_j u)$ is the intrinsic (or horizontal) Hessian associated to $\mathcal{X}$,   $Y^*$ is the symmetrized matrix of $Y$,

and $G$ is proper and strictly decreasing with respect to the last entry.

\smallskip
Before explaining our results, let us recall some of the many Liouville-type properties  for elliptic equations known  in the literature, the most related to our work.
The classical Liouville theorem for harmonic functions on the whole space states that the only harmonic functions in $\R^d$ bounded from above or below are constants, and it is a consequence of mean-value formulas or, more generally, of the Harnack inequality. Such result actually holds for classical solutions to more general uniformly elliptic equations, provided the zeroth order coefficient has the appropriate sign for the maximum principle, and the equation is homogeneous, see, e.g.,  the monograph \cite{GT}.

For  inhomogeneous equations the property is false,
for instance $\Delta(|x|^2)=2d$ in $\R^d$.

The Liouville property
holds also in the much larger class of merely subharmonic functions
(i.e., subsolutions of $-\Delta u=0$) bounded from above
if the space dimension is $d=2$,
 by exploiting the behavior of the fundamental solution $\log|x|$ and using the Hadamard Three-Circle Theorem (see, e.g., \cite[Theorem 2.29]{PW}, or
  Theorem \ref{liolin} below for a different proof).
  However, this result
   fails in higher dimensions $d\geq3$: for instance,
   $u_1(x):=-(1+|x|^2)^{-1/2}$ and $u_2(x):=-(1+|x|^2)^{-1}$ are nonpositive
    subharmonic functions in $\R^3$ and, respectively, in $\R^d$ with $d\geq4$.

 For
 linear degenerate elliptic equations,
 mean-value properties and Harnack-type inequalities were proved in many cases,
 typically for vector fields $\X$  that generate a stratified Lie group,  and Liouville theorems for solutions to such equations were proved, e.g., in   \cite{ICDCsub,
 BLU, KL2, KL1}, see also the references therein.

On the other hand, one does not expect the Liouville property for sub- or supersolutions to $-\Delta_\X u=0$ when $\X$
generates a Carnot group, because the intrinsic dimension of this geometry is larger than 2, which is the maximal one for subharmonic functions.

 In fact, in Section \ref{cex}
  we give simple explicit examples of bounded, non-constant, classical sub- and supersolutions of the sub-Laplace equation in any Heisenberg group ${\He^d}$.

Liouville theorems for nonlinear elliptic equations were first considered by Gidas and Spruck \cite{GS_cpde} for semilinear equations and
then widely investigated, also in the subelliptic and in the quasilinear settings, see, e.g., { \cite{BCDC, Italosurvey, ICDCsub,  AW, BFP, BMagliaro}} and the references therein.

For fully nonlinear equations as  \eqref{basic}, in the simpler form $F(x,D^2u)=0$ and uniformly elliptic, it was proved in
\cite[Section 4.3 Remark 4]{CC} that continuous viscosity solutions either bounded from above or below are constant. We recall that uniform ellipticity with parameters $\Lambda\geq\lambda>0$ can be defined by means of   Pucci's extremal operators (see their definition in Section \ref{puccieqs}) as
\begin{equation}\label{extremal}
\mathcal{M}^{-}_{\lambda,\Lambda}(M
)\leq F(x,r
,p
,M
) -F(x,r
,p
, 0)\leq \mathcal{M}^{+}_{\lambda,\Lambda}(M
)

\end{equation}
for all symmetric matrices $M$. The result is a consequence of the Harnack inequality
and comparison with Pucci's
 operators.

 Further related results
 for solutions of PDEs of the form $F(D^2u)=0$
 can be found in \cite[Section 1.7]{NV}

 and \cite[Theorem 1.7]{ArmstrongCPAM}, { and in   \cite[Theorem 1.5]{PP} for equations with $F$ depending also on $x$ and $Du$.}

 The first results for mere sub- or supersolutions of $F(x,D^2u)=0$
  are due to A. Cutr\iac{}  and F. Leoni  \cite{CLeoni}. They proved  that if $u\in C(\R^d)$ is either bounded { above}
 and satisfying

\begin{equation}\label{M+}
\mathcal{M}^{+}_{\lambda,\Lambda}(D^2u)\leq 0\text{ in }\R^d
\end{equation}

in viscosity sense, or bounded { below}
 and satisfying

\begin{equation}\label{M-}
\mathcal{M}^{-}_{\lambda,\Lambda}(D^2u)\geq 0\text{ in }\R^d
\end{equation}
in viscosity sense, then $u$ is constant provided that $d\leq \frac{\Lambda}{\lambda}+1$.

This can be seen as the fully nonlinear analogue of the Liouville theorem for subharmonic functions, since when $\lambda=\Lambda$ one gets the Laplacian (up to constants) and the constraint reads $d\leq 2$. Such conditions are known to be sharp: examples of nontrivial solutions to Pucci's extremal equations when $d>\frac{\Lambda}{\lambda}+1$ can be found in \cite[Remark 2]{CLeoni} and in Section \ref{cex} below.

This result was extended to the Heisenberg group $\He^d$ by Cutr\iac{}  and Tchou \cite[Theorem 5.2]{CTchou}
for the inequalities \eqref{M+} and  \eqref{M-} with $D^2u$ replaced by $(D^2_{\He^d} u)^*$. Here the condition $d\leq \frac{\Lambda}{\lambda}+1$ is replaced by $Q\leq \frac{\Lambda}{\lambda}+1$, $Q=2d+2$ standing for the homogeneous dimension of $\He^d$. An example of classical subsolution violating
 the Liouville property when $Q>\frac{\Lambda}{\lambda}+1$ is in Section \ref{cex}.

This is
consistent with the aforementioned failure of Liouville properties for
 subharmonic functions
 in the Heisenberg group.

 In \cite{CLeoni} the authors also prove
Liouville results for sub- and supersolutions of $F(x,D^2u)+u^p=0$ with $F$ uniformly elliptic, $F(x,0)=0$ and $p$ in a suitable range.

This was recently extended to  Carnot groups of Heisenberg type in \cite{Goffi}. See also \cite{ArmstrongSirakovSharp} and \cite{LeoniJMPA} for related results.

Liouville properties for PDEs involving gradient terms  of the form
$$F(x,D^2u)+g(|x|)|Du|+h(x)u^p=0 ,
$$
were first investigated by Capuzzo Dolcetta and  Cutr\iac{} \cite{CDC2003}.

They assume that $g$ is bounded and such that
\[
-\frac{\Lambda(d-1)}{|x|}\leq g(|x|)\leq \frac{\lambda-\Lambda(d-1)}{|x|}
\]
for $|x|$ large, and use suitable extensions of the Hadamard three-sphere theorem.
Note that this is a smallness condition at infinity  on the first order terms of the PDE.
See also \cite{Goffi} for similar recent results on the Heisenberg group.
Related papers for fully nonlinear PDEs with gradient dependence
are \cite{Rossi}, \cite{PP} and \cite{CFelmer}.

A new approach to Liouville properties for sub- and supersolutions of
Hamilton-Jacobi-Bellman elliptic equations involving operators of Ornstein-Uhlenbeck type was introduced
 in \cite{BCManca}, based on the strong maximum principle and the existence of a sort of Lyapunov function for the equation.
 It was applied in \cite{BC} to fully nonlinear uniformly elliptic equations of the form \eqref{basic} and to some quasilinear hypoelliptic equations, under

 assumptions on the sign of the coefficients of the first and zero-th order terms, and on their size. Here these terms must be large for large $|x|$, contrary to the results quoted above.  In the case of Pucci's operators the results of \cite{BC} are different from those in \cite{CLeoni} and fit better the treatment of uniformly elliptic equations via the inequalities \eqref{extremal}.
 In Section \ref{cex} we give examples showing their optimality.
  The paper \cite{MMT} treats a linear equation on the Heisenberg group in the same spirit.

\smallskip

In the present paper we study Liouville properties for viscosity sub- and supersolutions of equations of the form \eqref{basic} under the condition \eqref{subunitH} for a H\"ormander family $\mathcal{X}$.
Our main motivation are equations of the form \eqref{1L-}, uniformly subelliptic in the sense that $G$ satisfies the inequalities \eqref{extremal} with Pucci operators $\mathcal{M}^{\pm}_{\lambda,\Lambda}$ acting on $m$-dimensional instead of $d$-dimensional symmetric matrices.

In the first part we prove a general result under two additional assumptions, that for subsolutions are:
a
 subadditivity condition (\eqref{inf}
  of Section \ref{abs-res}) which for linear equations corresponds to the homogeneity, and the existence of a Lyapunov-like function $w$ such that $\lim_{|x|\rightarrow\infty}w(x)=+\infty$ and

supersolution of \eqref{basic}  out of a large ball (symmetric assumptions are made for supersolutions).
Here we adapt the approach of \cite{BC} to degenerate equations

by means of the new strong maximum and minimum principles
obtained by the authors in the recent paper \cite{BG} using
the generalized subunit vectors
for fully nonlinear equations \eqref{subunitH}.

In the second part of the paper we find more explicit sufficient conditions for the Liouville properties in the case of the Heisenberg group $\He^d$ by
taking  $w=\log \rho$ as  Lyapunov function,
where   $\rho$ is a norm 1-homogeneous with respect to the dilations of the group $\He^d$.

As must be expected from the results quoted before, these assumptions
 concern the sign and the strength of either the first or the zero-th order terms in the equation (or both). { They are related to recurrence conditions in the probabilistic literature,
  and are a form of dissipativity (cfr., e.g., \cite{PZ}).}
An example of our results, for the uniformly subelliptic equation
\begin{equation}\label{pde_he}

G(x,u,D_{{\mathbb{H}^d}}u,(D_{\mathbb{H}^d}^2u)^*)=0 ,  \quad \text{ in }\R^{2d+1} ,
\end{equation}
where $D_{{\mathbb{H}^d}}u$ and $D_{\mathbb{H}^d}^2u$ are the horizontal gradient and Hessian in $\He^d$, is the following: if
\[
G(x,r,p,X)\geq \mathcal{M}^-_{\lambda,\Lambda}(X)+\inf_{\alpha\in A}\{c^\alpha(x)r-b^\alpha(x)\cdot p\}
\]
with $c^\alpha\geq 0$, we prove the Liouville property for subsolutions under the condition
\[
\sup_{\alpha\in A}\{b^\alpha(x)\cdot \frac{\eta}{|x_H|^2}-c^\alpha(x)\frac{\rho^4}{|x_H|^2}\log\rho\}\leq \lambda-\Lambda(Q-1) \quad\text{ for } |x|\geq R ,
\]

where $x_H:=(x_1,...,x_{2d})\neq 0_{\R^{2d}}$, $\eta\in\R^{2d}$
is defined by

$\eta_i=x_i|x_H|^2+x_{i+d}x_{2d+1}$,

$\eta_{i+d}=x_i|x_H|^2-x_{i}x_{2d+1}$,

for $i=1,...,d$,
 and $Q=2d+2$ is the homogeneous dimension of $\He^d$. This condition is satisfied if either $c^\alpha > 0$ or $b^\alpha(x)\cdot {\eta} < 0$ for $x$ large, and under suitable growth conditions at infinity of the data.

In Section \ref{cex} we use again the  norm $\rho$ to discuss the sharpness of
 this condition.

In our companion paper \cite{BG_lio2} we apply the general results of Section \ref{general} to other classical families of H\"ormander vector fields,
namely, the generators of free step 2 Carnot groups,
 and Grushin-type fields, whose associated geometry has not a group structure.

It is well known that
 Liouville properties have many applications to various issues.
We are motivated in particular by their consequences in ergodic problems, large time stabilization in parabolic equations, and singular perturbation problems, as in, e.g., \cite{BC, BCManca, MMT, MMTesaim}.
For other forms of Liouville-type theorems for different equations let us also mention \cite{Biri} in the Heisenberg group, the recent paper \cite{Li} for PDEs arising in conformal geometry, and \cite{BGL} for versions of Pucci's extremal equations with
different
 degeneracies than in our work.

\par\smallskip

The paper is organised as follows. In section \ref{glimpse} we explain the approach to Liouville properties based on Lyapunov functions and strong maximum principles in the simple case of classical subsolutions of linear equations, for the reader's convenience, and discuss some related literature. Section \ref{general} presents an abstract result and its various applications to nonlinear equations with general H\"ormander vector fields.
In Section \ref{heisenberg} we study PDEs involving the generators of the Heisenberg group $\He^d$, in the form \eqref{pde_he} where only the horizontal gradient appears, as well as in the form \eqref{1L-} involving the Euclidean gradient. Section \ref{cex} makes a detailed comparison with the literature, in the cases of $\R^d$ and $\He^d$, and discusses by means of explicit examples the optimality of the sufficient conditions for Liouville properties.

\section{A glimpse on the method of proof for
 linear equations}\label{glimpse}
  Before showing our main results, we
  present the proof of a
   Liouville-type theorem for classical $C^2$ subsolutions to linear uniformly elliptic equations in the Euclidean framework, which serves as a guideline for our proof in the nonlinear and subelliptic setting.
   It uses only classical arguments such as strong maximum and comparison principles, { but not Harnack inequalities.}

\begin{thm}\label{liolin}
Assume
 the operator $Lu:=-\mathrm{Tr}(a
(x)D^2u)+b(x)\cdot Du{ +c(x)u}$ is uniformly elliptic, with $a:\R^d\to\mathcal{S}_d$, $b:\R^d\to\R^d$, { $c : \R^d\to [0,+\infty)$ locally bounded}.
Suppose
 there exists
$w\in C^2(\R^d\backslash\{0\})$ such that, for some $R>0$,
\begin{itemize}
\item[(i)] $Lw\geq0$ for $|x|>R$
\item[(ii)] $\lim_{|x|\to+\infty}w=+\infty$.
\end{itemize}
Let also $u\in C^2(\R^d)$ be such that $Lu\leq 0$
and ${ 0\leq u}(x)\leq C$ in $\R^d$. Then $u$ is constant.
\end{thm}
\begin{rem}\label{lapld=2}

This result is essentially  a special case, e.g., of \cite[Theorem 2.1]{BC} and applies
to the case of the Laplacian (i.e. $a_{ij}=\delta_{ij}$ and $b=0$) when $d\leq 2$; therefore it
 gives a different proof of the Liouville theorem for subsolutions, see e.g. \cite[Theorem 2.29]{PW}. Indeed, the function $w:=\log|x|$ fulfills the above assumptions, giving thus that every  subharmonic function bounded from above is constant. However, as pointed out in the introduction, this is not the case when $d\geq3$, where
  $w$ is no longer a classical supersolution of $-\Delta u=0$. { A more general result in the context of Riemannian manifolds
  can be found in \cite[Corollary 7.7]{Gry1} under the same sufficient conditions (i)-(ii).} It applies to subsolutions of $-\Delta u+b(x)\cdot Du=0$ in any space dimension $d$ under assumptions on the drift $b$ implying the existence of a Lyapunov-like function $w$

  (cfr. \cite{BC}).
\end{rem}
\begin{proof}

For $\zeta>0$ set
\[
v_\zeta(x):=u(x)-\zeta w(x)\text{ for }|x|\geq\bar{R}
\]
for some $\bar{R}>R>0$. Clearly, $v_\zeta\in C^2(\Omega_{\bar{R}})$, where
 $\Omega_{\bar{R}}:=\{x\in\R^d:|x|\geq\bar{R}\}$, and
\[
\text{
$Lv_\zeta=Lu-\zeta Lw\leq 0$ for every $x$ such that $|x|>\bar{R}$}\ .
\]
Define $C_\zeta:=\max_{\{|x|=\bar{R}\}}v_\zeta(x)$. Since
\[
\lim_{|x|\to+\infty}v_\zeta(x)=-\infty\ ,
\]
there exists $K_\zeta>\bar{R}$ such that $v_\zeta<C_\zeta$ for every $x$ such that $|x|\geq K_\zeta$. By the weak maximum principle (see \cite[Corollary 3.2]{GT})
 on the set $\{x\in\R^d:\bar{R}<|x|<K_\zeta\}$ we have
\[
\max_{\{x\in\R^d:\bar{R}<|x|<K_\zeta\}}v_\zeta(x)=\max_{\{x\in\R^d:|x|=\bar{R}\text{ or }|x|=K_\zeta\}}v_\zeta(x) = C_\zeta .
\]
Since $v_\zeta(x)<C_\zeta$ for every $x$ such that $|x|\geq K_\zeta$, we { get, for all $|y|\geq\bar R$},
\[
v_\zeta(y)
\leq C_\zeta
\leq  \max_{\{x\in\R^d:|x|=\bar{R}\}}u-\zeta \min_{\{x\in\R^d:|x|=\bar{R}\}}w\ .
\]
On one hand, letting $\zeta\to0$ we conclude
\[
u(y)\leq  \max_{\{x\in\R^d:|x|=\bar{R}\}}u , \quad \text{ for all $|y|\geq\bar{R}$}\ .
\]
On the other hand, owing to the weak maximum principle in the set $B(0,\bar{R})$ we obtain
\[
u(y)\leq  \max_{\{x\in\R^d:|x|=\bar{R}\}}u  , \quad\text{ for all $|y|<\bar{R}$}\ .
\]
Combining the above inequalities one concludes
\[
u(y)\leq \max_{\{x\in\R^d:|x|=\bar{R}\}}u, \quad \text{ for all }y\in\R^d\ .
\]
Hence, $u$ attains its { nonnegative} maximum
at some point of  $\partial B(0,\bar{R})$ and then the conclusion follows by the strong maximum principle for classical linear uniformly elliptic equations \cite[Theorem 3.5]{GT}.
\end{proof}
\begin{rem}
The same result remains true if $L$ is replaced by a degenerate elliptic operator $L_\X u:=-\sum_{i,j}X_iX_ju+b(x)\cdot D_\X u{ +c(x)u}$, provided the vector fields  $\X$ satisfy the H\"ormander condition and $b:\R^m\to\R^n$, $n\leq m$ is { smooth},
 the proof being exactly the same thanks to Bony strong maximum principle for subelliptic equations.

An example of such result is  \cite[Proposition 3.1]{MMT}.

Note also that the assumption $u\leq C$ can be replaced by $\limsup_{|x|\to\infty}u(x)/w(x)\leq 0$, whereas the sign condition $u\geq 0$ can be dropped if $c\equiv 0$.

We further remark that, when $b\equiv0$,
$L$ reduces to a Schr\"odinger-type operator. When $u$ is a solution of the equation $-\Delta u+c u=0$, the Liouville property is proved in \cite[Corollary 13.7]{Gry1} under the same sufficient conditions (i)-(ii) on Riemannian manifolds, and it is connected to recurrence and non-explosive properties of Brownian motions on Riemannian manifolds, see \cite[Theorem 5.1 and Section 13.2]{Gry1}. Our result is more general in that it allows $u$ to be merely a subsolution to the equation.

 We also refer to \cite{PZ} for a control theoretic interpretation of the Liouville property { for Ornstein-Uhlenbeck operators}.
\end{rem}

\section{The general case}\label{general}
\subsection{An abstract result} \label{abs-res}

In this Section we consider a general equation of the form
\begin{equation}\label{fully2}
F(x,u,Du,D^2u)=0 \quad \text{ in }\R^d .
\end{equation}

We will denote $F[u]:=F(x,u,Du,D^2u)$ and make the following assumptions
\begin{itemize}
  \item[(i)] { $F : \R^d\times\R\times\R^d\times\mathcal S_d\to \R$} is continuous, proper, satisfies
  \begin{equation}\label{inf}\tag{S1}
 F[\varphi-\psi]\leq F[\varphi]-F[\psi]\text{ for all }\varphi,\psi\in C^2(\R^d)
  \end{equation}
  and $F(x,r,0,0)\geq0$ for every $x\in\Omega$ and $r\geq0$.

  \item[(ii)] $F$ satisfies the comparison principle in any bounded open set $\Om$, namely, if $u$ and $v$ are, respectively, a viscosity sub-

  and  supersolution of \eqref{fully2} such that $u\leq v$ on $\partial\Om$, then $u\leq v$ in $\Om$.
\item[(iii)] There exists $R_o\geq 0$ and $w\in\LSC(\R^d)$ viscosity supersolution of \eqref{fully2} for $|x|>R_o$ and  satisfying $\lim_{|x|\rightarrow\infty}w(x)=+\infty$.
  \item[(iv)] $F$ satisfies the strong maximum principle, namely, any viscosity subsolution of \eqref{fully2} that attains an
   non-negative maximum must be constant.

\end{itemize}
To prove the analogous results for viscosity supersolutions we need to replace (i) and (iii)-(iv) above by
\begin{itemize}
  \item[(i')] $F$ is continuous, proper, satisfies
   \begin{equation}\label{sup}\tag{S2}
 F[\varphi-\psi]\geq F[\varphi]-F[\psi]\text{ for all }\varphi,\psi\in C^2(\R^d)
  \end{equation}
  and $F(x,r,0,0)\leq0$ for every $x\in\Omega$ and $r\leq0$.

  \item[(iii')] There exists $R_o\geq0$ and $W\in\USC(\R^d)$ viscosity subsolution to \eqref{fully2} for $|x|>R_o$ and satisfying $\lim_{|x|\rightarrow\infty}W(x)=-\infty$.
    \item[(iv')] $F$ satisfies the strong minimum principle.
\end{itemize}

The next result extends the proof
of Theorem \ref{liolin} for linear equations to the fully nonlinear degenerate setting. Its proof is essentially the same as the one done  in \cite{BC} for HJB
 equations,
 so we only outline it for the reader's convenience.
\begin{prop}\label{liogen1}
Assume {\upshape (i)-(iv)}. Let $u\in\USC(\R^d)$ be a viscosity subsolution to \eqref{fully2} satisfying
\begin{equation}\label{limsup}
\limsup_{|x|\rightarrow\infty}\frac{u(x)}{w(x)}\leq0\ .
\end{equation}
for $w$ as in {\upshape(iii)}. If $u\geq0$, then $u$ is constant.
\end{prop}

\begin{proof}

Define $u_{\zeta}(x):=u(x)-\zeta w(x)$ for $\zeta>0$. Possibly increasing $R_o$, we can assume that $u$ is not constant in $\overline{B}(0,R_o):=\{x\in\R^m:|x|\leq R_0\}$, otherwise we are done. Set
\begin{equation}
\label{Ceta}
C_{\zeta}:=\max_{|x|\leq R_o}u_{\zeta}(x)\ .
\end{equation}

Note that $F[C_{\zeta}]\geq0$ and $C_{\zeta}>0$ for all $\zeta$ sufficiently small.

In fact, if $C_{\zeta}=0$,
by letting $\zeta\rightarrow0$ we get $u(x)=0$ for every $x$ with $|x|\leq R_o$, a contradiction with
$u$ not constant in $\overline{B}(0,R_o)$.

 The growth condition \eqref{limsup} implies
\begin{equation*}
\limsup_{|x|\rightarrow\infty}\frac{u_{\zeta}(x)}{w(x)}\leq-\zeta<0 \quad \forall \zeta>0\ .
\end{equation*}
As a consequence, we have
\begin{equation}\label{limsupueta}
\lim_{|x|\rightarrow+\infty}u_{\zeta}(x)=-\infty\ .
\end{equation}
Then, for all $\zeta>0$ there exists $R_{\zeta}>R_o$ such that
\begin{equation}\label{ineqC}
u_{\zeta}(x)\leq C_{\zeta}\ \quad\text{for all }|x|\geq R_{\zeta}\ .
\end{equation}

The main step is proving that $u_{\zeta}$ is a viscosity subsolution of $F[u]=0$ in $\{x\in\R^d:|x|>R_o\}$.
Take $\bar{x}$ and $\varphi$ smooth such that $0=(u_{\zeta}-\varphi)(\bar{x})>(u_{\zeta}-\varphi)({x})$ for all $x$.
Assume by contradiction that $F[\varphi(\bar{x})]>0$.

Then for some $\delta>0$ and $0<r<|\bar{x}|-R_o$
\begin{equation}
\label{F>0}
F[\varphi-\delta]>0  \quad \text{ in }B(\bar{x},r) .
\end{equation}

Next take $0<k<\delta$ such that $u_{\zeta}-\varphi\leq-k<0$ on $\partial B(\bar{x},r)$.

We claim that $\zeta w+\varphi-k$  satisfies $F[\zeta w+\varphi-k]\geq0$ in $B(\bar{x},r)$. Indeed, take $\tilde{x}\in B(\bar{x},r)$ and $\psi$ smooth such that

$\zeta w +\varphi-k-\psi$ has a minimum at $\tilde{x}$.

Using that $w$ is a viscosity supersolution to \eqref{fully2},
$F$ proper, \eqref{inf}, and  \eqref{F>0} we get
\[
0\leq F[\psi(\tilde{x})-\varphi(\tilde{x})+k]\leq F[\psi(\tilde{x})-\varphi(\tilde{x})+\delta]\leq F[\psi(\tilde{x})]-F[\varphi(\tilde{x})-\delta]<F[\psi(\tilde{x})] .
\]

Then $\zeta w+\varphi-k$ is a
supersolution to $F[u]=0$ in $B(\bar{x},r)$ and $u\leq\zeta w+\varphi-k$ on $\partial B(\bar{x},r)$,
so  the comparison principle gives

$u\leq \zeta w+\varphi-k$ in $B(\bar{x},r)$,

in contradiction with the fact that $u(\bar{x})=\zeta w(\bar{x})+\varphi(\bar{x})$.

Now we can use the comparison principle in $\Om=\{x:R_o<|x|<R_{\zeta}\}$ and \eqref{ineqC} to get
$u_{\zeta}\leq C_{\zeta}$ in $\Om$.
 Therefore we have
\begin{equation*}
u_{\zeta}(x)\leq C_{\zeta}\  \quad \text{for all }|x|\geq R_o .
\end{equation*}
By letting $\zeta\rightarrow0^+$ we obtain
\begin{equation*}
u(x)\leq\max_{|y|\leq R_o}u(y)  \quad \text{for all } x\in \R^d ,
\end{equation*}
and hence $u$ attains its maximum $\bar{x}$ over $\R^d$. Since
 $u\geq0$ the SMP gives
  the desired conclusion.
\end{proof}
\begin{rem}
Note that if $u$ is bounded above, then \eqref{limsup} is satisfied.

\end{rem}

The next result says
 that the assumption $u\geq0$ can be dropped provided $r\mapsto F(x,r,p,X)$
is constant: this will be the case for some HJB operators we discuss in the next sections.
\begin{cor}\label{corlio1}
Assume {\upshape (i)-(iv)}. Let $u\in\USC(\R^d)$ be a viscosity subsolution to \eqref{fully2} satisfying \eqref{limsup}

for $w$ as in {\upshape(iii)}. Assume  $r\mapsto F(x,r,p,X)$ is constant for all $x,p, X$ and $F(x,r,0,0)=0$ for every $x\in\Omega$.

Then $u$ is constant.
\end{cor}
\begin{proof}
The proof goes along the same lines as Proposition \ref{liogen1}. It is sufficient to note that since $r\mapsto F(x,r,p,X)$ is constant for all $x,p, X$, $u+|u(\bar{x})|$, $\bar x$ standing for the maximum point in Proposition \ref{liogen1}, is again a subsolution,

and one concludes.
\end{proof}

A symmetric result holds for the case of supersolutions to \eqref{fully2}, { see \cite{BC} for the details of the proof.}

\begin{prop}\label{liogen2}
Assume {\upshape (i'),(ii),(iii') and (iv')}. Let $v\in\LSC(\R^d)$ be a viscosity supersolution to \eqref{fully2} satisfying
\begin{equation}\label{liminf}
{ \limsup_{|x|\rightarrow\infty}}\frac{v(x)}{W(x)}\leq0
\end{equation}
for $W$ as in {\upshape(iii')}. Assume either $v\leq0$, or $r\mapsto F(x,r,p,X)$ is constant for all $x,p, X$ and $F(x,r,0,0)=0$ for every $x\in\Omega$. Then $v$ is constant.
\end{prop}

\subsection{Equations with H\"ormander vector fields}
In this section we discuss Liouville properties for PDEs over H\"ormander vector fields. We recall that
the vector fields
{ $Z_1,..., Z_m$

satisfy the H\"ormander's rank condition if

\smallskip
\noindent (H) \emph {the vector fields are smooth
and the
 Lie algebra generated by them has full rank $d$ at each point.}
 \smallskip

 \noindent
  The classical smoothness requirement on $Z_{i}$ is $C^\infty$,} but it can be reduced to $C^k$ for a suitable $k$, and considerably more if the Lie brackets are interpreted in a generalized sense, see \cite{FR} and the references therein.

Before stating the main result for subsolutions, we recall a crucial scaling assumption for the validity of the strong maximum principle for fully nonlinear subelliptic equations together with the concept of generalized subunit vector field.
\begin{itemize}
\item[(SC)] For some $\phi : (0,1]\to (0,+\infty]$, $F$
satisfies
\begin{equation*}
F(x,\xi s,\xi p,\xi X)\geq\ \phi(\xi)F(x,s,p,X)
\end{equation*}
for all $\xi\in(0,1]$,
 $s\in[-1,0]$,
  $x\in\Omega$, $p\in\R^d\backslash\{0\}$, and $X\in\Sym_d$;
\end{itemize}

\begin{defn}\label{subunit}  $Z\in \R^{d}$ is a generalized \emph{subunit vector} (briefly, SV)

 for $F=F(x,r,p,X)$ at $x\in\Omega$ if

\begin{equation*}
\sup_{\gamma>0} F(x,0,p,I-\gamma p\otimes p)> 0 \quad \forall p\in \R^{d} \;\text{ such that } \; Z\cdot p\ne 0 ;
\end{equation*}
$Z:\Omega\rightarrow\R^{d}$ is a \emph{subunit vector field} (briefly, SVF)

if $Z(x)$ is SV for $F$ at $x$ for every $x\in\Omega$.

\end{defn}
The name is motivated by the the notion introduced  by C. Fefferman and D.H. Phong \cite {FP} for linear operators.
\begin{thm}\label{liosub}
Let $F$ be such that {\upshape (i),(ii), (iii), and (SC)} hold. Furthermore assume that $F$ admits $Z_1,...,Z_m$ generalized subunit vector fields satisfying the H\"ormander condition {\upshape (H)}.
 Let $u\in\USC(\R^d)$ be a viscosity subsolution to \eqref{fully2} satisfying \eqref{limsup} for $w$ as in {\upshape(iii)}. Assume either $u\geq0$, or $r\mapsto F(x,r,p,X)$ is constant for all $x,p, X$ and $F(x,r,0,0)=0$ for every $x\in\Omega$.
 Then $u$ is constant.
\end{thm}
\begin{proof}
The proof is a consequence of Proposition \ref{liogen1} and Corollary \ref{corlio1}, recalling that under {\upshape(i), (SC)} and the existence of subunit vector fields for $F$, the strong maximum principle holds (cf \cite[Corollary 2.6]{BG}).
\end{proof}

Similarly, in the case of supersolutions we have the following result by replacing (SC) with
\begin{itemize}
\item[(SC')] For some $\phi : (0,1]\to (0,+\infty]$, $F$
satisfies
\begin{equation*}
F(x,\xi s,\xi p,\xi X)\leq\ \phi(\xi)F(x,s,p,X)
\end{equation*}
for all $\xi\in(0,1]$,
 $s\in[-1,0]$,
  $x\in\Omega$, $p\in\R^d\backslash\{0\}$, and $X\in\Sym_d$;
\end{itemize}
and the condition in Definition \ref{subunit} is replaced with
\begin{equation*}
\inf_{\gamma>0} F(x,0,p,\gamma p\otimes p-I)> 0 \quad \forall p\in \R^{d} \;\text{ such that } \; Z\cdot p\ne 0 .
\end{equation*}
{ The proof is a simple consequence of
 Proposition \ref{liogen2},
 and the strong minimum principle Corollary 2.12 in
 \cite
 {BG}.
}
\begin{thm}\label{liosuper}
Let $F$ be such that {\upshape (i'),(ii),(iii'), and (SC')} hold. Furthermore, assume that $F$ admits $Z_1,...,Z_m$ generalized subunit vector fields satisfying the H\"ormander condition as above. Let $v\in\LSC(\R^d)$ be a viscosity supersolution to \eqref{fully2} satisfying \eqref{liminf} for $W$ as in {\upshape(iii')}. Assume either $v\leq0$, or $r\mapsto F(x,r,p,X)$ is constant for all $x,p, X$ and $F(x,r,0,0)=0$ for every $x\in\Omega$. Then $v$ is constant.
\end{thm}

{ Next we apply the last two theorems to subelliptic equations of the form

\begin{equation}
\label{1Lbis}
G(x,u,D_{\mathcal{X}}u,(D_{\mathcal{X}}^2u)^*)=0 ,  \quad \text{ in }\R^d ,
\end{equation}
where $\mathcal{X}=(X_1,...,X_m)$ are $C^{1,1}$ vector fields
on $\R^d$ satisfying the H\"ormander condition (H), $D_{\mathcal{X}}u:=(X_1 u,...,X_m u)$, $(D_{\mathcal{X}}^2 u)_{ij} :=X_i (X_j u)$, and $Y^*$ is the symmetrized matrix of $Y$.
Here { $G : \R^d\times\R\times\R^m\times\mathcal S_m\to\R$} is at least continuous, proper, satisfying \eqref{inf},  and it is \emph{elliptic for any $x$ and $p$ fixed}
 in the following sense:

\begin{equation}
\label{Gell}
\sup_{\gamma>0} G(x,0,q,X-\gamma q\otimes q)> 0 \quad \forall \, x\in \Om,\; q\in \R^{m}, \; q\ne 0 , \; X\in \Sym_m .
\end{equation}

After choosing a basis in Euclidean space we write $X_j=\sigma^j\cdot D$, with $\sigma^j:\R^d\rightarrow\R^d$, and $\sigma=\sigma(x)=[\sigma^1(x),...,\sigma^m(x)]\in\R^{d\times m}$.
Then
\begin{equation*}
D_{\mathcal{X}}u=\sigma^T Du=(\sigma^1\cdot Du,...,\sigma^m\cdot Du)
\end{equation*}
and
\begin{equation*}
X_i (X_j u)=(\sigma^T D^2 u\ \sigma)_{ij} +(D\sigma^j\ \sigma^i)\cdot Du\ .
\end{equation*}

Therefore,  for $u\in C^2$,
\begin{equation*}
(D_{\mathcal{X}}^2 u)^*=\sigma^T D^2 u\sigma+
g(x,Du) \,, \qquad (g(x,P))_{ij}:=\frac12[(D\sigma^j\ \sigma^i)\cdot p+(D\sigma^i\ \sigma^j)\cdot p] ,
\end{equation*}

and we can  rewrite the equation \eqref{1Lbis} in Euclidean coordinates, i.e., in the form $F(x,u,Du,D^2 u)=0$, by taking
\begin{equation}\label{FG}
F(x,r,p,X)=G(x,r,\sigma^T(x)p,\sigma^T(x)X\sigma(x)+g(x,Du)) .
\end{equation}
The ellipticity condition \eqref{Gell} implies that
the vector fields  $X_j=\sigma^j\cdot D$ are subunit for $F$  (cfr. \cite[Lemma 3.1]{BG}).
Moreover, if $G$ satisfies (SC) or (SC'), also $F$ does. Assume finally that $F$ given by \eqref{FG} satisfies property (ii) about the weak comparison principle. Then we have the following.

\begin{cor}
Under the assumptions listed above the equation \eqref{1Lbis} has the Liouville properties for viscosity sub- and supersolutions of Theorems \ref{liosub} and
 \ref{liosuper} .
\end{cor}
}

\subsection{Equations driven by Pucci's subelliptic operators}
\label{puccieqs}

{ Given a family of $m$ vector fields and
 the corresponding Hessian matrix, we consider the Pucci's extremal operators over such matrices instead of the classical Euclidean Hessians.

Following Caffarelli and Cabr\'e \cite{CC}, } we fix $0<\lambda\leq\Lambda$, denote with $\Sym_{m}$ the set of $m\times m$ symmetric matrices, and let
\[
\mathcal{A}_{\lambda,\Lambda}:=\{A\in\Sym_m
:\lambda|\xi|^2\leq A\xi\cdot\xi\leq\Lambda|\xi|^2\ ,\forall\xi\in\R^m
\}\ .
\]

{ For  $M\in\Sym_{m}$ the \textit{maximal} and \textit{minimal
 operator} are defined as
\begin{equation*}
\mathcal{M}^{+}_{\lambda,\Lambda}(M):=\sup_{A\in \mathcal{A}_{\lambda,\Lambda}}\mathrm{Tr}(-AM)
 , \qquad \mathcal{M}^{-}_{\lambda,\Lambda}(M):=\inf_{A\in\mathcal{A}_{\lambda,\Lambda}}\mathrm{Tr}(-AM)\ .
\end{equation*}

}

If $e_1\leq...\leq e_d$ are the ordered eigenvalues of the matrix $M$, one can check  that \cite[Section 2.2]{CC}

\begin{equation}
\label{representationM+}
\mathcal{M}^{+}_{\lambda,\Lambda}(M)=-\Lambda\sum_{e_k<0}e_k-\lambda\sum_{e_k>0}e_k , \qquad
\mathcal{M}^{-}_{\lambda,\Lambda}(M)=-\Lambda\sum_{e_k>0}e_k-\lambda\sum_{e_k<0}e_k .
\end{equation}

Now we prove the Liouville property for subsolutions of the equation
\begin{equation}\label{pucci1}
\mathcal{M}^-_{\lambda,\Lambda}((D^2_{\mathcal{X}}u)^*)+H_i(x,u,D_{\mathcal{X}}u)=0 \quad \text{ in }\R^d\ ,
\end{equation}
where
\begin{equation}\label{Hi}
H_i(x,r,p):=\inf_{\alpha\in A}\{c^\alpha(x)r-b^\alpha(x)\cdot p\}
\end{equation}
and for
 supersolutions of
\begin{equation}\label{pucci2}
\mathcal{M}^+_{\lambda,\Lambda}((D^2_{\mathcal{X}}u)^*)+H_s(x,u,D_{\mathcal{X}}u)=0 \quad \text{ in }\R^d\ ,
\end{equation}
where
\begin{equation}\label{Hs}
H_s(x,r,p):=\sup_{\alpha\in A}\{c^\alpha(x)r-b^\alpha(x)\cdot p\}\ .
\end{equation}
{ Here $A$ is a set of indices such that $H_i$ and $H_s$ are finite.} We assume that { $b^\alpha : \R^d \to \R^m$ }
 is locally Lipschitz in $x$ uniformly in $\alpha$, i.e., for all $R>0$ there exists $K_R>0$ such that
\begin{equation}\label{blip}
\sup_{|x|,|y|\leq R, \alpha\in A}|b^\alpha(x)-b^\alpha(y)|\leq K_R|x-y|
\end{equation}
and
\begin{equation}\label{cass}
c^\alpha(x)\geq0\text{ and continuous in $|x|\leq R$ uniformly in $\alpha$}.
\end{equation}

\begin{cor}\label{pucciminmax} { Assume the vector fields $\mathcal{X}$
are $C^{1,1}$
and satisfy the H\"ormander condition (H).}

\begin{itemize}
\item [(a)] Under the previous assumptions
 on $H_i$, let $u\in\USC(\R^d)$ be a viscosity subsolution to \eqref{pucci1} satisfying \eqref{limsup} for $w$ as in {\upshape(iii)} . If either $u\geq0$ or $c^\alpha(x)\equiv0$, then $u$ is constant.
\item [(b)] Under the previous assumptions
 on $H_s$, let $v\in\LSC(\R^d)$ be a viscosity supersolution to \eqref{pucci2} satisfying \eqref{liminf} for $W$ as in {\upshape(iii')}. If either $v\leq0$ or $c^\alpha(x)\equiv0$, then $v$ is constant.
\end{itemize}
\end{cor}
\begin{proof}
{ The proofs of (a) and (b) are consequences, respectively, of Theorems \ref{liosub} and \ref{liosuper}. The  operators $\mathcal{M}^-_{\lambda,\Lambda}((D^2_{\mathcal{X}}u)^*)$ and $\mathcal{M}^+_{\lambda,\Lambda}((D^2_{\mathcal{X}}u)^*)$ enjoy, respectively, the property \eqref{inf} and \eqref{sup} as a consequence of the property of duality (i.e., $\mathcal{M}^-_{\lambda,\Lambda}(M)=-\mathcal{M}^+_{\lambda,\Lambda}(-M)$
) and { the
inequalities
$$\mathcal{M}^-_{\lambda,\Lambda}(M+N)\leq \mathcal{M}^-_{\lambda,\Lambda}(M)+\mathcal{M}^+_{\lambda,\Lambda}(N), \quad \mathcal{M}^+_{\lambda,\Lambda}(M+N)\geq \mathcal{M}^-_{\lambda,\Lambda}(M)+\mathcal{M}^+_{\lambda,\Lambda}(N)$$
for every $M,N\in\Sym_d$, see \cite[Lemma 2.10
]{CC}.
}
Moreover, they are positively $1$-homogeneous, so they satisfy the scalings (SC) and (SC').}

The comparison principle (ii)
 holds for both equations in view of \cite[Example 4.6]{BG}. Finally, observe that when $c^\alpha\equiv0$, then $G(x,r,0,0)=0$ for every $x\in\Omega$, $r\in\R$, and $r\mapsto G(x,r,p,X)$ is constant for every $x,p,X$.

\end{proof}

{ Corollary \ref{pucciminmax} concerns only operators that are either convex or concave with respect to the derivatives of the solution. However, we
 will use it in the next Section \ref{fully} to study general fully nonlinear uniformly subelliptic equations.

 A different, although similar, class of extremal operators

was  introduced by C. Pucci in the seminal paper \cite{Pucci66} (for the Euclidean Hessian). Here we consider them on the Hessian associated to a the vector fields $\mathcal{X}$.

Consider}  for $\alpha>0$ the class of matrices
\begin{equation*}
\mathcal{B}_{\alpha}:=\{A\in\Sym_m:A\xi\cdot\xi\geq\alpha|\xi|^2,\mathrm{Tr}(A)=1,\forall\xi\in\R^m\} ,
\end{equation*}
and define
\begin{equation}\label{Pucci+}
\mathcal{P}^{+}_{\alpha}(M):=\sup_{A\in \mathcal{B}_{\alpha}}\mathrm{Tr}(-AM) , \qquad \mathcal{P}^{-}_{\alpha}(M)=\inf_{A\in\mathcal{B}_{\alpha}}\mathrm{Tr}(-AM)\ .
\end{equation}

As pointed out in \cite{Pucci66} (see also \cite[Chapter 17]{GT}), { these operators can be rewritten in terms of  the ordered eigenvalues of the matrix $M\in\Sym_m$ as follows }
\begin{equation}
\label{representation+}
\mathcal{P}^{+}_{\alpha}(M)=-\alpha\sum_{k=2}^{m
}e_k-[1-(m
-1)\alpha]e_1=-\alpha\mathrm{Tr}(M)-(1-m
\alpha)e_1
\,,
\end{equation}

\begin{equation}
\label{representation-}
\mathcal{P}^{-}_{\alpha}(M)=-\alpha\sum_{k=1}^{m-1}e_k-[1-(m-1)\alpha]e_m=-\alpha\mathrm{Tr}(M)-(1-m\alpha)e_m\, .
\end{equation}
{ The Liouville properties of Corollary \ref{pucciminmax} hold under the same assumptions for the equations \eqref{pucci1} and \eqref{pucci2} with the operators $\mathcal{M}^-_{\lambda,\Lambda}((D^2_{\mathcal{X}}u)^*)$ and $\mathcal{M}^+_{\lambda,\Lambda}((D^2_{\mathcal{X}}u)^*)$ replaced, respectively, by $\mathcal{P}^{-}_{\alpha}((D^2_{\mathcal{X}}u)^*)$ and $\mathcal{P}^{+}_{\alpha}((D^2_{\mathcal{X}}u)^*)$. { In
Section \ref{full_Hei}} { we will give some explicit results for extremal equations involving $\mathcal{P}^{\pm}$ on the Heisenberg group.}

}

\subsection{Fully nonlinear uniformly subelliptic equations}\label{fully}

In this section we consider the
{ general fully nonlinear
 subelliptic equation
\begin{equation}\label{2L}
G(x,u,D_{\mathcal{X}}u,(D_{\mathcal{X}}^2u)^*)=0 \quad \text{ in }\R^d
\end{equation}

when { $G : \R^d\times\R\times\R^m\times\mathcal S_m\to\R$} satisfies
the following form of uniform ellipticity}
\begin{equation}\label{US}
\mathcal{M}^-_{\lambda,\Lambda}(M-N)\leq G(x,r,p,M)-G(x,r,p,N)\leq \mathcal{M}^+_{\lambda,\Lambda}(M-N)
\end{equation}
for every $(x,r,p)\in \Omega\times\R\times\R^m$ and $M,N\in\Sym_m$ with $N\geq0$. By taking $N=0$ we get
\begin{equation*}
\mathcal{M}^-_{\lambda,\Lambda}(M)\leq G(x,r,p,M)-G(x,r,p,0)\leq \mathcal{M}^+_{\lambda,\Lambda}(M)\ ,
\end{equation*}
and, as a consequence, by setting $H(x,r,p):=G(x,r,p,0)$, one can infer Liouville results for viscosity subsolutions and supersolutions to { \eqref{2L} by comparison with the equation \eqref{pucci1} and \eqref{pucci2} of the previous section.

For this purpose we assume either}
\begin{equation}\label{G>H}
G(x,r,p,0)\geq H_i(x,r,p) \quad \forall \, x \in \R^d , r\in\R , p\in\R^m ,
\end{equation}
for a concave Hamiltonian of the form \eqref{Hi}, { or
\begin{equation}\label{G<H}
G(x,r,p,0)\leq H_s(x,r,p)
\end{equation}
for a convex $H_s$ as in \eqref{Hs}.
}

\begin{cor}
\label{fullymin}
 { Assume the vector fields $\mathcal{X}$
are $C^{1,1}$
and satisfy the H\"ormander condition (H), \eqref{blip}, \eqref{cass}, and the ellipticity condition \eqref{US}.}

\begin{itemize}
\item [(a)]

Let \eqref{G>H} hold and
 $u\in\USC(\R^d)$ be a viscosity subsolution to \eqref{2L} satisfying \eqref{limsup} for $w$ as in {\upshape(iii)}. If either $u\geq0$ or $c^\alpha(x)\equiv0$, then $u$ is constant.
\item [(b)]
Let \eqref{G<H} hold and $v\in\LSC(\R^d)$ be a viscosity supersolution to \eqref{2L} satisfying \eqref{liminf} for $W$ as in {\upshape(iii')}. If either $v\leq0$ or $c^\alpha(x)\equiv0$, then $v$ is constant.
\end{itemize}
\end{cor}
\begin{proof}
It is sufficient to observe that $u$ and $v$ satisfies the differential inequalities
\[
\mathcal{M}^-_{\lambda,\Lambda}((D^2_{\mathcal{X}}u)^*)+H_i(x,u,D_{\mathcal{X}}u)\leq0 \text{ in }\R^d \,,
\]
{ \[
\mathcal{M}^+_{\lambda,\Lambda}((D^2_{\mathcal{X}}v)^*)+H_s(x,v,D_{\mathcal{X}}v)\geq0 \text{ in }\R^d \,,
\] }
and apply Corollary \ref{pucciminmax}.
\end{proof}

{ The same kind of result holds for equations of the form
{
\begin{equation}\label{non-h-grad}

G(x,u,Du, (D_{\mathcal{X}}^2u)^*) =0\text{ in }\R^d\ ,
\end{equation}
for $G : \R^d\times\R\times\R^d\times\mathcal S_m\to\R$,} where the dependence is on the Euclidean gradient $Du$ instead of the horizontal one $D_{\mathcal{X}}u$. We assume $G$ satisfies \eqref{US} and either
\begin{equation}\label{G>Hp}

{ G(x,r,p,0)}\geq H_i(x,r,p) ,  \quad \forall \, x, p\in \R^d , r\in\R ,
\end{equation}
for a concave Hamiltonian of the form \eqref{Hi}, but with  $b^\alpha : \R^d \to \R^d$ a vector field in $\R^d$,  or
\begin{equation}\label{G<Hp}

{ G(x,r,p,0)}\leq H_s(x,r,p)     ,  \quad \forall \, x, p\in \R^d , r\in\R ,
\end{equation}
for a convex Hamiltonian of the form \eqref{Hs} with  $b^\alpha : \R^d \to \R^d$. On $b^\alpha$ and $c^\alpha$ we make the same assumptions \eqref{blip}, \eqref{cass}, and the fields $\mathcal{X}$
are $C^{1,1}$
and satisfy
(H). The arguments leading to Corollaries \ref{pucciminmax} and \ref{fullymin} give the following.
\begin{cor}
\label{eu-grad}
Under the conditions listed above a subsolution (resp., supersolution) of \eqref{non-h-grad} with assumption \eqref{G>Hp} (resp.,  \eqref{G<Hp}) verifies the Liouville property { \upshape (a)} (resp.,  { \upshape (b)})of Corollary \ref{fullymin}.
\end{cor}
}

\subsection{{Normalized
$p$-Laplacian.}}

The result of the last section encompasses degenerate equations of the form
\[
{-}|D_\X u|^{2-p}\mathrm{div}_\X(|D_\X u|^{p-2}D_\X u)=0 \quad \text{ in }\R^d ,
\]
{ where $\mathrm{div}_\X$ is the intrinsic divergence over the fields of the family $\X$.}

In fact the operator $E(D_\X u, (D^2_\X u)^*):=-|D_\X u|^{2-p}\mathrm{div}_\X(|D_\X u|^{p-2}D_\X u)$, called normalized or game-theoretic
$p$-Laplacian, can be
 rewritten as $-\mathrm{Tr}[A(D_\X u)(D^2_\X u)^*]$,

with
\[
A(D_\X u)=I_m+(p-2)\frac{D_\X u\otimes D_\X u}{|D_\X u|^2} \,.
\]
In other words,
$$E(D_\X u, (D^2_\X u)^*)=-\Delta_\X u-(p-2)|D_\X u|^{-2}\Delta_{\X,\infty}u ,$$
 where
$\Delta_{\X,\infty}$ is the  $\infty$-Laplacian operator over the fields $\X$.
 It is immediate to see that {
\[
\min\{1,p-1\}|\xi|^2\leq A(D_\X u)\xi\cdot \xi\leq\max\{1,p-1\}|\xi|^2\ ,
\]
showing that $E$ is uniformly subelliptic for $p\in(1,\infty)$.
Therefore such nonlinear operators

can be compared with Pucci's extremal operators $\mathcal{M}^{\pm}_{\lambda,\Lambda}$ with $\lambda=\min\{1,p-1\}$ and $\Lambda=\max\{1,p-1\}$ over H\"ormander vector fields. They were studied recently by several authors, see e.g.
  \cite{Pina} for the case of Carnot groups, and the references therein. The game theoretic $p$-Laplace operator on $\X$ is the sublaplacian if $p=2$, whereas for  $p=1$ it drives the evolutive equation
 describing the motion of level sets by sub-Riemannian mean curvature.}

\section{
{ The Heisenberg vector fields}}
\label{heisenberg}
The aim of this section is to specialize the results obtained in the previous one to viscosity subsolutions of \eqref{2L} fulfilling \eqref{US} over Heisenberg vector fields. We briefly recall some standard facts on the Heisenberg group. For further details we refer the reader to the monograph \cite{BLU}. The Heisenberg group $\He^d$ can be identified with $(\R^{2d+1},\circ)$, where $2d+1$ stands for the topological dimension and the group law $\circ$ is defined by
\begin{equation*}
x\circ y=\left(x_1+y_1,...,x_{2d}+y_{2d},x_{2d+1}+y_{2d+1}+2\sum_{i=1}^d(x_iy_{i+d}-x_{i+d}y_i ) \right) .
\end{equation*}
{
We denote with $x$  a point of $\R^{2d+1}$ and set
\[
x_H:=(x_1,...,x_{2d})\ .
\]
}
The $d$-dimensional Heisenberg algebra is the Lie algebra spanned by the $m=2d$ vector fields
\begin{equation*}
X_i=\partial_i+2x_{i+d}\partial_{2d+1}\ ,
\end{equation*}
\begin{equation*}
X_{i+d}=\partial_{i+d}-2x_{i}\partial_{2d+1}\ ,
\end{equation*}
for $i=1,...,d$.
Such vector fields satisfy the commutation relations
\[
[X_i,X_{i+d}]=-4\partial_{2d+1}\text{ and }[X_i,X_j]=0\text{ for all $j\neq i+d$, }i\in\{1,...,d\}\ .
\]
and are 1-homogeneous with respect to the family of (anisotropic) dilations
\[
\delta_\lambda(x)=(\lambda x_1,...,\lambda x_{2d},\lambda^2x_{2d+1}) \ ,\lambda>0\ ,
\]

Following \cite[Definition 5.1.1]{BLU}, it is useful to consider the following homogeneous norm defined via the stratification property of $\He^d$
\begin{equation}\label{homoH}
\rho(x)=\left(\left(\sum_{i=1}^{2d}(x_i)^2\right)^2+x_{2d+1}^2\right)^{\frac14}\ .
\end{equation}
which is 1-homogeneous with respect to the previous group of dilations. { We emphasize that
 this norm is easier to compute than
  the Carnot-Carath\'eodory norm.}

\subsection{Fully nonlinear PDEs on the Heisenberg group}
\label{full_Hei}
In the next result we provide sufficient conditions for the validity of the Liouville property for viscosity subsolutions to \eqref{pucci1}. We search the Lyapunov functions of property (iii) and (iii') among the radial ones (e.g. $w=\log\rho$ and $W=-\log\rho$).

Here and in the next examples we exploit a classical chain rule to compute the horizontal gradient and Hessian of a ``radial" function with respect to the homogeneous norm $\rho$. Indeed, for a sufficiently smooth radial function $f=f(\rho)$ and given a system of vector fields $\X=\{X_1,...,X_m\}$, we have
\begin{equation*}
D_{\X}f(\rho)=f'(\rho)D_{\X}\rho
\end{equation*}
and
\begin{equation*}
D^2_{\X}f(\rho)=f'(\rho)D^2_{\X}\rho+f^{''}(\rho)D_{\X}\rho\otimes D_{\X}\rho\ .
\end{equation*}
In this section we denote the Heisenberg horizontal gradient and symmetrized Hessian by $D_{\mathbb{H}^d}$ and $(D^2_{\mathbb{H}^d})^*$.
We premise the following auxiliary result taken from \cite[Lemma 3.1 and Lemma 3.2]{CTchou}.

\begin{lemma}\label{hesH}
Let $\rho$ be defined in \eqref{homoH}. Then, { for $|x_H|\neq0$,
\[
D_{\mathbb{H}^d}\rho = \frac{\eta}{\rho^3} , \qquad |D_{\mathbb{H}^d}\rho|^2=\frac{|x_H|^2}{\rho^2}\leq 1\, ,
\]
where $\eta\in\R^{2d}$
is defined by
\begin{equation}
\label{eta}
\eta_i:=x_i|x_H|^2+x_{i+d}x_{2d+1}\ , \qquad
\eta_{i+d}:=x_{i+d}|x_H|^2-x_{i}x_{2d+1} .
\end{equation}
for $i=1,...,d$. Moreover }
\[
D^2_{\mathbb{H}^d}\rho=-\frac{3}{\rho}D_{\mathbb{H}^d}\rho\otimes D_{\mathbb{H}^d}\rho+\frac{1}{\rho}|D_{\mathbb{H}^d}\rho|^2I_{2d}+\frac{2}{\rho^3}\begin{pmatrix}B & C\\-C & B\end{pmatrix}\ ,
\]
where the matrices $B=(b_{ij})$ and $C=(c_{ij})$ are defined as follows
\[
b_{ij}:=x_ix_j+x_{d+i}x_{d+j}\ ,\qquad c_{ij}:=x_ix_{d+j}-x_jx_{d+i}
\]
for $i,j=1,...,d$ (in particular $B=B^T$ and $C=-C^T$).

 In addition, for a radial function $f=f(\rho)$ we have
\[
D^2_{\mathbb{H}^d}f(\rho)=\frac{f'(\rho)|D_{\mathbb{H}^d}\rho|^2}{\rho}I_{2d}+2\frac{f'(\rho)}{\rho^3}\begin{pmatrix}B & C\\-C & B\end{pmatrix}+\left(f''(\rho)-3\frac{f'(\rho)}{\rho}\right)D_{\mathbb{H}^d}\rho\otimes D_{\mathbb{H}^d}\rho\ .
\]
{ and its eigenvalues are $f''(\rho)|D_{\mathbb{H}^d}\rho|^2, 3f'(\rho)\frac{|D_{\mathbb{H}^d}\rho|^2}{\rho}$, which are simple, and $f'(\rho)\frac{|D_{\mathbb{H}^d}\rho|^2}{\rho}$ with multiplicity $2d-2$}.
\end{lemma}

\begin{thm}\label{Cor1H}
Let $\mathcal{X}=\{X_1,....,{ X_{2d}}
\}$ be the system of vector fields generating the Heisenberg group $\He^d$. Assume that \eqref{blip} and \eqref{cass} are in force and
\begin{equation}\label{condcor1}
\sup_{\alpha\in A}\{b^\alpha(x)\cdot \frac{\eta}{|x_H|^2}-c^\alpha(x)\frac{\rho^4}{|x_H|^2}\log\rho\}\leq \lambda-\Lambda(Q-1)
\end{equation}
for $\rho$ sufficiently large and $|x_H|\neq0$, where $Q=2d+2$ is the homogeneous dimension of $\mathbb{H}^d$, $b^\alpha(x)$ takes values in $\R^{2d}$, and $\eta=(\eta_i,\eta_{i+d})$ is defined { by \eqref{eta}}.

\begin{itemize}
\item[(A)] Let $u\in\USC(\R^{2d+1})$ be a viscosity subsolution of \eqref{pucci1} such that

\begin{equation}
\label{grow+}
\limsup_{|x|\rightarrow\infty}\frac{u(x)}{\log\rho(x)}\leq0\ .
\end{equation}
If either $c^\alpha(x)\equiv0$ or $u\geq0$, then $u$ is a constant.
\item[(B)] Let $v\in\LSC(\R^{2d+1})$ be a viscosity supersolution of \eqref{pucci2} such that
\begin{equation}
\label{grow-}
\liminf_{|x|\rightarrow\infty}\frac{v(x)}{\log\rho(x)}\geq0\ .
\end{equation}
If either $c^\alpha(x)\equiv0$ or $v\leq0$, then $v$ is a constant.
\end{itemize}
\end{thm}
\begin{rem}
When $b\equiv c\equiv 0$ and $\lambda=\Lambda=1$ (i.e. \eqref{pucci1} becomes $-\Delta_{\He^d}u=0$) condition \eqref{condcor1} gives $\lambda-\Lambda(Q-1)=(2-Q)\geq0$, which is not satisfied in  the Heisenberg group because $Q\geq4$. This is consistent with the failure of the Liouville property for sub- and supersolutions of the Heisenberg sub-Laplacian {  that we prove in Section \ref{cex} below.}
\end{rem}
\begin{proof}
We only have to check property {\upshape(iii)} { for
 the Lyapunov function
 $w(x)
 =\log\rho(x)$. Note} that $\lim_{|x|\to\infty}w
 (x)=\infty$ because
 $\rho\to\infty$ as $|x|\to\infty$.

{ By Lemma \ref{hesH}  applied to the radial function $w$}
the eigenvalues of $(D^2_{\He^d}w)^*$ are
\[
-\frac{|x_H|^2}{\rho^4} \,\text{ and }\, 3\frac{|x_H|^2}{\rho^4},\quad \text{which are simple} ,
\]
and
\[
\frac{|x_H|^2}{\rho^4}\text{ with multiplicity $2d-2$ }
\]
when $|D_{\He^d}\rho|\neq0$.
 { Otherwise all the eigenvalues vanish identically and  $w
$ is trivially a supersolution to \eqref{pucci1} because $\mathcal{M}^-_{\lambda,\Lambda}((D^2_{\He^d}w)^*)=0$
 and $c^\alpha u\geq0$.}

 Hence, we are able to compute the Pucci's minimal operator at points where $|x_H|\neq0$ owing to Lemma \ref{hesH} as
\begin{equation*}
\mathcal{M}^-_{\lambda,\Lambda}((D^2_{\He^d}w)^*)=\{-\Lambda(2d+1)+\lambda\}\frac{|x_H|^2}{\rho^4}\ .
\end{equation*}
Thus, $w$ is a supersolution at all points where
\begin{equation*}
\{-\Lambda(2d+1)+\lambda\}\frac{|x_H|^2}{\rho^4}+\inf_{\alpha\in A}\{c^\alpha(x)\log\rho-b^\alpha(x)\cdot\frac{\eta}{\rho^4}\}\geq0\ ,
\end{equation*}
{ because $D_{\He^d}w= \eta/\rho^4$ by Lemma \ref{hesH}. }
In particular, this inequality holds when $\rho$ is sufficiently large under condition \eqref{condcor1} by recalling that $Q=2d+2$. Similarly, one can check that \eqref{condcor1} implies that the function $W(\rho)=-\log\rho$ is a subsolution to \eqref{pucci2} for $|x|$ sufficiently large at points where $|D_{\He^d}\rho|\neq0$. Therefore Corollary \ref{pucciminmax} gives the conclusion.
\end{proof}

\begin{rem}
Condition \eqref{condcor1} is comparable to that obtained in \cite[condition (2.17)]{BC}, but here typical quantities of Carnot groups appear. One may think that the ratio
\[
\frac{\rho^4}{|x_H|^2}=\frac{|x_H|^4+|x_V|^2}{|x_H|^2}
\]
plays exactly the same role as $|x|^2$ in \cite[condition (2.17)]{BC}, while the dimension $d$ of the Euclidean setting is precisely replaced by its sub-Riemannian counterpart $Q$, as expected.
\end{rem}
\begin{rem}
A simple condition that implies \eqref{condcor1}, and therefore the Liouville property,  is
\begin{equation*}
\limsup_{|x|\rightarrow\infty}\sup_{\alpha\in A} b^\alpha(x)\cdot \frac{\eta}{|x_H|^2} <\lambda-\Lambda(Q-1)\ ,
\end{equation*}
since $c\geq0$. Compare the above condition to that in \cite[Remark 2.4]{BC}: $Q$ replaces the dimension $d$ of the Euclidean case and $x\in\R^d$ is replaced by the vector $\eta/|x_H|^2\in\R^{2d}$, where $\eta$ is defined by \eqref{eta}.

\end{rem}
{ We can now give explicit conditions for  the Liouville properties
for the general subelliptic equation \eqref{2L} on the Heisenberg group.}

\begin{cor}
Assume that the operator $G$ { satisfies \eqref{US},
$\X=\{X_1,...,X_{2d}\}$}  are the Heisenberg vector fields,

and \eqref{blip}, \eqref{cass}, and \eqref{condcor1} are satisfied.

\noindent {\upshape(A)} Assume
 \eqref{G>H} and $u\in\USC(\R^{2d+1})$ is a subsolution of \eqref{2L} satisfying  { \eqref{grow+}.}

 If either $c^\alpha(x)\equiv0$ or $u\geq0$, then $u$ is constant.

\noindent {\upshape(B)} Assume
 \eqref{G<H} and $v\in\LSC(\R^{2d+1})$ is a supersolution of \eqref{2L} satisfying  { \eqref{grow-}.}

 If either $c^\alpha(x)\equiv0$ or $v\leq0$, then $v$ is constant.

\end{cor}
\begin{proof}
It is enough to exploit that $u$ { (resp., $v$)} is a subsolution to \eqref{pucci1} (resp., a supersolution to \eqref{pucci2}) over the Heisenberg group and then apply Theorem \ref{Cor1H}-(A) (resp., Theorem \ref{Cor1H}-(B)).
\end{proof}

We specialize the last corollaries to a class of examples in order to compare with those in \cite{BC}. Consider again
the fully nonlinear uniformly subelliptic PDE \eqref{2L}
and assume that either
\begin{equation}
\label{hypH}
G(x,r,p,0)\geq-\bar{b}(x)\cdot p-g(x)|p|+\bar{c}(x)r\,,
\end{equation}
{ or
\begin{equation}
\label{hypH2}

{G}(x,r,p,0)\leq-\bar{b}(x)\cdot p+g(x)|p|+\bar{c}(x)r \,,
\end{equation}
where $\bar{b}:\R^{2d+1}\rightarrow\R^{2d}$ and $g:\R^{2d+1}\rightarrow\R$ are locally Lipschitz, $\bar{c}$ is continuous,} $g\geq0$, and $\bar{c}\geq0$.

\begin{cor}
\label{corollbcg}
Assume that the operator $G$ in \eqref{2L} satisfies \eqref{US} and
\begin{equation}\label{condcor1bis}
\bar{b}(x)\cdot \frac{\eta}{|x_H|^2}+g(x)\frac{|\eta|}{|x_H|^2}\leq\bar{c}(x)\frac{\rho^4}{|x_H|^2}\log\rho+\lambda-\Lambda(Q-1) \,,
\end{equation}
{ for $\rho$ sufficiently large and $|x_H|\neq0$, where $\eta$ is defined by \eqref{eta} and $Q=2d+2$.}

\noindent {\upshape(A)} Suppose  that \eqref{hypH} holds and
$u\in\USC(\R^{2d+1})$ is a viscosity subsolution of \eqref{2L} satisfying \eqref{grow+}. If

either $c^\alpha(x)\equiv0$ or $u\geq0$, then $u$ is a constant.

{ \noindent {\upshape(B)} Suppose  that \eqref{hypH2} holds and
 $v\in\LSC(\R^{2d+1})$ is a viscosity supersolution of \eqref{2L} satisfying \eqref{grow-}}. If
either $v\leq0$ or $c^\alpha(x)\equiv0$, then $v$ is constant.
\end{cor}
\begin{proof} { As in the proof of Theorem \ref{Cor1H} we must only check that $w=\log\rho$ is a supersolution.
Observe that $-|D_{\He^d}w|=-|\sigma^TDw|=\min_{|\alpha|=1}\{-\alpha\cdot \sigma^TDw\}$. Hence we can write the right-hand side of the inequality \eqref{hypH} with $p=D_{\He^d}w$ as
\begin{equation*}
\inf_{\alpha\in A}\{\bar{c}w-(\bar{b}+g\alpha)\cdot\sigma^TDw\}\ ,
\end{equation*}
}
where $A=\{\alpha\in\R^{2d}:|\alpha|=1\}$. Moreover
{ $D_{\He^d}w=\frac{1}{\rho^4}\eta$ by Lemma \ref{hesH}.

Then $w$ is a supersolution where
\begin{equation*}
\{-\Lambda(2d+1)+\lambda\}\frac{|x_H|^2}{\rho^4}+ \bar{c}\log\rho+\inf_{\alpha\in A}\{-(\bar{b}+g\alpha)\cdot\frac{\eta}{\rho^4}\}\geq0 \,,
\end{equation*}
and this inequality is satisfied for $\rho$  large enough if \eqref{condcor1bis} holds.

Arguing in a similar manner we prove (B) by showing that $W=-\log\rho$ is a subsolution.}
\end{proof}
{ \begin{ex} [Schr\"odinger-type equations] \label{schro} For nonnegative subsolutions of the equation
\[
\mathcal{M}^-_{\lambda,\Lambda}((D^2_{\He^d}u)^*) + \bar{c}(x)u =0 \quad \text{ in }\R^{2d+1}
\]
the Liouville property holds if
\[
\liminf_{
|x|\to\infty} \bar{c}(x)\frac{\rho^4(x)}{|x_H|^2}\log\rho(x)>{\Lambda(Q-1)-\lambda}\ ,
\]
a results that appears to be new even in the linear case $\lambda=\Lambda$.
\end{ex} }
{ \begin{ex} [A horizontal Ornstein-Uhlenbeck equation] Consider  subsolutions of the equation
\[
\mathcal{M}^-_{\lambda,\Lambda}((D^2_{\He^d}u)^*) - \gamma(x) \eta(x) \cdot D_{\He^d}u =0 \quad \text{ in }\R^{2d+1}\
\]
where $\gamma(x) >0$ and $\eta$ is defined by \eqref{eta}, i.e.,
\[
\eta(x)=x_H|x_H|^2+ x_{2d+1}x_H^\perp \,,\qquad x_H^\perp:=(x_{d+1},...,x_{2d}, -x_1,...,-x_d) \,.
\]
Since $\eta\cdot\eta
=|x_H|^2 \rho^4$, the condition \eqref{condcor1bis} becomes
\[
\liminf_{|x|\to\infty} \gamma(x)\rho^4(x)>{\Lambda(Q-1)-\lambda}
\]
and then the Liouville property holds.

\end{ex} }

We end this subsection with a result on the following equations driven by the Pucci's extremal operators $\mathcal{P}^\pm_\lambda$ defined by \eqref{Pucci+} (here $\lambda=\alpha$)

\begin{equation}\label{pucci1bis}
\mathcal{P}^-_{\lambda}((D^2_{\mathcal{X}}u)^*)+H_i(x,u,D_{\mathcal{X}}u)=0\quad \text{ in }\R^{2d+1}\ ,
\end{equation}

\begin{equation}\label{pucci2bis}
\mathcal{P}^+_{\lambda}((D^2_{\mathcal{X}}u)^*)+H_s(x,u,D_{\mathcal{X}}u)=0\quad \text{ in }\R^{2d+1}\ .
\end{equation}
Sufficient conditions for the Liouville property can be
 obtained by comparing
  $\mathcal{P}^\pm$ with $\mathcal{M}^\pm$ as follows
\begin{equation*}
\mathcal{P}^+_{\lambda}(M)\leq\mathcal{M}^{+}_{\lambda,\lambda+(1-d\lambda)}(M) \,,\quad \mathcal{P}^-_{\lambda}(M)\geq\mathcal{M}^{-}_{\lambda,\lambda+(1-d\lambda)}(M)\,, \quad \forall\, M\in\Sym_{2d} \,.
\end{equation*}

However,
by exploiting the representation formulas \eqref{representation+} and \eqref{representation-}
for $\mathcal{P}^\pm$ we can get
 optimal sufficient conditions.
\begin{cor}\label{Cor1P}
Let $\mathcal{X}=\{X_1,....,X_{2d}
\}$ be the system of vector fields generating the Heisenberg group $\He^d$. Assume
 \eqref{blip}, \eqref{cass},
  and
\begin{equation}\label{condcor1p}
\sup_{\alpha\in A}\{b^\alpha(x)\cdot \frac{\eta}{|x_H|^2}-c^\alpha(x)\frac{\rho^4}{|x_H|^2}\log\rho\}\leq 4d\lambda -3

\end{equation}
for $\rho$ sufficiently large and
$|D_{\He^d}\rho|\neq0$.

{ Then the same conclusions as in Theorem \ref{Cor1H}  hold for subsolutions of \eqref{pucci1bis}

 and supersolutions of \eqref{pucci2bis}. }

\end{cor}
\begin{proof}
The proof is
 the same as Theorem \ref{Cor1H} using the Lyapunov function $w(\rho)=\log\rho$ and the
  formulas
   \eqref{representation+} and \eqref{representation-}.

By the expression of the eigenvalues of $(D^2_{\He^d}w)^*$ in the proof of Theorem \ref{Cor1H} one finds
\[
\mathcal{P}^-_\lambda((D^2_{\He^d}w)^*)=(4d\lambda-3)\frac{|x_H|^2}{\rho^4}\ .
\]
Similarly, one uses $W=-\log\rho$ as Lyapunov function for the maximal operator $\mathcal{P}^+_\lambda$.
\end{proof}
\begin{rem}

Condition \eqref{condcor1p} is better than \eqref{condcor1} with $\Lambda=\lambda+(1-2d\lambda)$ and $\lambda<\frac{1}{2d}$, since
\begin{equation*}
-2d\lambda-(1-2d\lambda)(2d+1)<-3+4d\lambda\ .
\end{equation*}
\end{rem}

\subsection{Comparison with the literature and sharpness of the conditions}\label{cex}
In this section we make a comparison with the results in the literature, showing the sharpness of our conditions and those of \cite{BC, CLeoni, CTchou} via several counterexamples.

\subsubsection{{ The Euclidean case}}

{ Corollary 2.4 of \cite{BC}
 states a Liouville-type result
 that in the case without lower order terms holds } { for the inequalities
\[
\mathcal{M}^{-}_{\lambda,\Lambda}(D^2u)\leq 0\text{ in }\R^d , \qquad \mathcal{M}^{+}_{\lambda,\Lambda}(D^2u)\geq 0\text{ in }\R^d ,
\]
the former for viscosity subsolutions bounded above, and the second for supersolutions bounded from below,

  when $d\leq \frac{\lambda}{\Lambda}+1$.
  This complements the result  of \cite{CLeoni} on \eqref{M+} and \eqref{M-} recalled in the Introduction,

  but with a more restrictive condidion on $d$, which is, however, still sharp for the Lapalcian (${\lambda}={\Lambda}$).

  The next  counterexample shows that the
 inequalities can have nonconstant solutions when $d>\frac{\lambda}{\Lambda}+1$.}

\begin{cex}

For $d\geq2$ { set

$\beta:=\frac{\Lambda}{\lambda}(d-1)+1
$
and consider }the function
\[
u_2(x)=\begin{cases}
\frac18[\beta(\beta-2)|x|^4-2(\beta^2-4)|x|^2+\beta(\beta+2)]&\text{ if }|x|<1\ ,\\
\frac{1}{|x|^{\beta-2}}&\text{ if }|x|\geq1\ .
\end{cases}
\]
{ Since it
is radial,
 the eigenvalues of the Hessian matrix can be
 computed
 by \cite[Lemma 3.1]{CLeoni} and one checks that it
is } a
 classical solution to $\mathcal{M}^+_{\lambda,\Lambda}(D^2u_2)\geq0$ in $\R^d$.
 { Moreover it is bounded and not constant if $\beta>2$, which is equivalent to $d>\lambda/\Lambda+1$, so the Liouville property for supersolutions to $\mathcal{M}^+(D^2u)=0$ is false in this case}. Similarly, $v_2=-u_2$ gives a counterexample for solutions to $\mathcal{M}^-_{\lambda,\Lambda}(D^2v_2)\leq0$ in $\R^d$.

\end{cex}

{ \begin{rem}
The paper \cite{CLeoni} studies a similar but different problem with respect to \cite{BC}, namely,
the Liouville property
for viscosity supersolutions to $\mathcal{M}^-_{\lambda,\Lambda}(D^2u)=0$ in $\R^d$ and for
subsolutions to $\mathcal{M}^+_{\lambda,\Lambda}(D^2u)=0$.
They prove it under the  less restrictive condition $d\leq \frac{\Lambda}{\lambda}+1$ \cite[Theorem 3.2]{CLeoni}. Note, however, that their theorem cannot be applied to general uniformly elliptic operators via the inequalities \eqref{US}, whereas the results in \cite{BC} allow such application.

 The next example shows that also the condition in \cite{CLeoni} is optimal.
 \end {rem}
}

 \begin{cex} [From
 \cite{CLeoni}] { Set $\alpha:=\frac{\lambda}{\Lambda}(d-1)+1$ and consider  the function}

 \[
u_3(x)=\begin{cases}
-\frac18[\alpha(\alpha-2)|x|^4-2(\alpha^2-4)|x|^2+\alpha(\alpha+2)]&\text{ if }|x|<1\ ,\\
-\frac{1}{|x|^{\alpha-2}}&\text{ if }|x|\geq1\ ,
\end{cases}
\]
which is a
 classical solution to $\mathcal{M}^+_{\lambda,\Lambda}(D^2u_3)\leq0$ in $\R^{d}$. { Moreover it is bounded and not constant if $\alpha>2$, which is equivalent to $d>\Lambda/\lambda+1$}. Similarly, $v_3=-u_3$ yields a counterexample for the corresponding property for the minimal operator.
 \end{cex}

\subsubsection{{ The Heisenberg case: sublaplacians.}}

Liouville's theorem for classical harmonic functions on the Heisenberg group is a consequence of the Harnack inequality, see, e.g., \cite[Theorem 8.5.1]{BLU}, { or mean-value formulas \cite[Theorem 3.1]{ICDCsub}}. However, the Liouville property for classical subsolutions (resp., supersolutions) bounded from above (resp., below) of
\[
-\Delta_{\He^d}u=0\quad \text{ in }{ \mathbb{H}^d\simeq} \R^{2d+1}
\]
is false { for all dimensions $d$, as the next example shows.
}
 We recall that  $Q:=2d+2$ is the homogeneous dimension of $\mathbb{H}^d$ and $\rho(x)$ is the homogeneous norm defined in \eqref{homoH}.
\begin{cex}
The function
\[
\tilde u(x)=\begin{cases}
\frac18[Q(Q-2)\rho^4-2(Q^2-4)\rho^2+Q(Q+2)]&\text{ if }\rho\leq1\ ,\\
\frac{1}{\rho^{Q-2}}&\text{ if }\rho\geq1\ ,
\end{cases}
\]

is a bounded classical supersolution to $-\Delta_{\He^d}u=0$ in $\R^{2d+1}$. Indeed, when $\rho\leq 1$ one  applies Lemma \ref{hesH} to  the radial function
\[
\tilde u=\tilde f(\rho)=\frac18[Q(Q-2)\rho^4-2(Q^2-4)\rho^2+Q(Q+2)],
\]
and gets,
at points where $|D_{\mathbb{H}^d}\rho|\neq0$,
\begin{multline*}
-\Delta_{\mathbb{H}^d} \tilde u=-\mathrm{Tr}(D^2_{\mathbb{H}^d} \tilde f(\rho))\\
=-\frac{Q-2}{2\rho^2}|x_H|^2\left\{[3Q\rho^2-(Q+2)]+3[Q\rho^2-(Q+2)]+(2d-2)[Q\rho^2-(Q+2)]\right\}\\
=-\frac{Q-2}{2\rho^2}|x_H|^2Q(\rho^2-1)(Q+2)\geq 0 \,,
\end{multline*}
due to the fact that $\rho^2\leq1$ and $Q\geq4$. At points where $|x_H|=0$ all the eigenvalues of $D^2_{\mathbb{H}^d} \tilde f(\rho)$ vanish and hence $\tilde u$ is a solution of the sub-Laplace equation. When $\rho\geq1$, instead, one observes that $\tilde u$ is  the fundamental solution of the sub-Laplace equations on the Heisenberg group found by G.B. Folland \cite{Folland}.
Similarly, $v=-\tilde u$ gives a bounded subsolution to $-\Delta_{\He^d}u=0$ in $\R^{2d+1}$.
\end{cex}

\subsubsection{{ The Heisenberg case: fully nonlinear operators.}}

{ The Liouville property in this context was first studied by Cutr\`i and Tchou  \cite
{CTchou}}

for viscosity supersolutions bounded from below of  $\mathcal{M}^-_{\lambda,\Lambda}((D^2_{\He^d}u)^*)$ $=0$ in $\R^{2d+1}$ and for
subsolutions bounded from above to $\mathcal{M}^+_{\lambda,\Lambda}((D^2_{\He^d}u)^*)=0$.
{ Their  Theorem 5.2 states that such functions are constant provided that $Q\leq \frac{\Lambda}{\lambda}+1$. The next is a new example showing that this condition is sharp.}

 \begin{cex}\label{cexalphatilde}
Set $\tilde \alpha:=\frac{\lambda}{\Lambda}(Q-1)+1$.
We show that for $\tilde\alpha>2$, { i.e., $Q> \frac{\Lambda}{\lambda}+1$,}
 \[
u_4(x)=\begin{cases}
-\frac18[\tilde \alpha(\tilde \alpha-2)\rho^4-2(\tilde \alpha^2-4)\rho^2+\tilde \alpha(\tilde \alpha+2)]&\text{ if }\rho<1\ ,\\
-\frac{1}{\rho^{\tilde \alpha-2}}&\text{ if }\rho\geq1\ ,
\end{cases}
\]
is a bounded from above classical solution to $\mathcal{M}^+_{\lambda,\Lambda}((D^2_{\He^d}u_4)^*)\leq0$ in $\R^{2d+1}$ and it is not constant. Indeed, denote by $u_4(x)=f_4(\rho)$. For $\rho<1$ we have

\[
f'_4(\rho)=-\frac{\tilde \alpha-2}{2}\rho[\tilde \alpha\rho^2-(\tilde \alpha+2)]\ ,
\]
and
\[
f''_4(\rho)=-\frac{\tilde \alpha-2}{2}[3\rho^2\tilde\alpha-(\tilde\alpha+2)]
\]
Recalling that $|D_{\He^d}\rho|^2=|x_H|^2/\rho^2$, by Lemma \ref{hesH} the eigenvalues of the radial function $f_4(\rho)$
\[
e_1=|D_{\He^d}\rho|^2f''_4(\rho)=-\frac{\tilde \alpha-2}{2\rho^2}|x_H|^2[3\rho^2\tilde\alpha-(\tilde \alpha+2)] ,
\]
\[
e_2=3|D_{\He^d}\rho|^2\frac{f'_4(\rho)}{\rho}=-3\frac{\tilde \alpha-2}{2\rho^2}|x_H|^2[\tilde \alpha\rho^2-(\tilde \alpha+2)] ,
\]
which are both simple, and
\[
e_3=|D_{\He^d}\rho|^2\frac{f'_4(\rho)}{\rho}=-\frac{\tilde \alpha-2}{2\rho^2}|x_H|^2[\tilde \alpha\rho^2-(\tilde \alpha+2)] ,
\]
which has multiplicity $2d-2$.
Observe that, when $\rho<1$ and $\tilde \alpha>2$, the eigenvalues $e_2,e_3$ are always positive. Moreover,
for $\rho^2\leq\frac{\tilde \alpha+2}{3\tilde\alpha}<1$, also $e_1$ is positive and hence $\mathcal{M}^+_{\lambda,\Lambda}((D^2_{\He^d}u_4)^*)\leq0$. When $1>\rho^2>\frac{\tilde \alpha+2}{3\tilde\alpha}$, $e_1<0$, and hence by Lemma \ref{hesH}
\begin{multline*}
\mathcal{M}^+_{\lambda,\Lambda}((D^2_{\He^d}u_4)^*)=\Lambda\frac{\tilde \alpha-2}{2\rho^2}|x_H|^2[3\rho^2\tilde\alpha-(\tilde \alpha+2)]\\+\lambda\left\{\frac{\tilde \alpha-2}{2\rho^2}|x_H|^2[\tilde \alpha\rho^2-(\tilde \alpha+2)](2d-2)+3\frac{\tilde \alpha-2}{2\rho^2}|x_H|^2[\tilde \alpha\rho^2-(\tilde \alpha+2)]\right\}\\
=\frac{\tilde \alpha-2}{2\rho^2}|x_H|^2\left\{\lambda[\tilde \alpha\rho^2-(\tilde \alpha+2)](2d-2)+3\tilde \alpha\rho^2-3(\tilde \alpha+2)]+\Lambda[3\rho^2\tilde\alpha-(\tilde \alpha+2)]\right\}\\
=\frac{\tilde \alpha-2}{2\rho^2}|x_H|^2\left\{\tilde \alpha\rho^2[(2d+1)\lambda+3\Lambda] -\lambda(2d+1)(\tilde \alpha+2)-\Lambda(\tilde\alpha+2)\right\}\\
=\frac{\tilde \alpha-2}{2\rho^2}|x_H|^2\left\{\tilde \alpha\rho^2[(Q-1)\lambda+3\Lambda] -[\lambda(Q-1)+\Lambda](\tilde \alpha+2)\right\}\\
=\frac{\tilde \alpha-2}{2\rho^2}|x_H|^2\left\{[\lambda(Q-1)+\Lambda](-\tilde\alpha-2+\tilde\alpha\rho^2)+2\Lambda\tilde\alpha\rho^2\right\}\\
\leq \frac{\tilde \alpha-2}{2\rho^2}|x_H|^2\left\{-2[\lambda(Q-1)+\Lambda]+2\Lambda\tilde\alpha\right\}
=\frac{\tilde \alpha-2}{2\rho^2}|x_H|^2\left\{-2\lambda(Q-1)+2\Lambda(\tilde\alpha-1)\right\}=0\ ,
\end{multline*}
where the last equality is true in view of $\tilde{\alpha}-1=\frac{\lambda}{\Lambda}(Q-1)$. When $\rho>1$ we have
\[
f'_4(\rho)=-(2-\tilde\alpha)\rho^{1-\tilde{\alpha}} ,
\]
\[
f''_4(\rho)=-(2-\tilde\alpha)(1-\tilde\alpha)\rho^{-\tilde \alpha} ,
\]
and the eigenvalues are
\[
e_4=|D_{\He^d}\rho|^2f''_4(\rho)=-\frac{|x_H|^2(2-\tilde \alpha)(1-\tilde \alpha)}{\rho^{\tilde \alpha+2}} ,
\]
\[
e_5=3|D_{\He^d}\rho|^2\frac{f'_4(\rho)}{\rho}=-3\frac{|x_H|^2(2-\tilde \alpha)}{\rho^{\tilde \alpha+2}} ,
\]
and
\[
e_6=|D_{\He^d}\rho|^2\frac{f'_4(\rho)}{\rho}=-\frac{|x_H|^2(2-\tilde \alpha)}{\rho^{\tilde \alpha+2}}
\]
with multiplicity $2d-2$. Therefore, for $\rho\geq1$, we have
\[
\mathcal{M}^+_{\lambda,\Lambda}((D^2_{\He^d}u_4)^*)=\frac{|x_H|^2(2-\tilde \alpha)}{\rho^{\tilde \alpha+2}}\left[\Lambda(1-\tilde\alpha)+\lambda(Q-1)\right]=0\ ,
\]
Similarly, $v_4=-u_4$ yields a counterexample for the corresponding property of the miminal operator.
\end{cex}
\smallskip
{ Next we discuss the optimality of our Theorem \ref{Cor1H} in the case without lower order terms, i.e., $H_i=H_s=0$. Then the condition \eqref{condcor1} becomes $Q\leq\frac{\lambda}{\Lambda}+1$, which is not satisfied in  the Heisenberg group because $Q\geq4$. This  is consistent with the failure of the Liouville property for sub- and supersolutions of the Heisenberg Laplacian observed before.

The next
example shows that the Liouville property fails also for supersolutions bounded from below of $\mathcal{M}^+_{\lambda,\Lambda}((D^2_{\He^d}u)^*)=0$ and subsolutions bounded from above of  $\mathcal{M}^-_{\lambda,\Lambda}((D^2_{\He^d}u)^*)=0$, for all $\lambda, \Lambda$, and $d$. Therefore, we conclude that the presence of suitable lower order terms in Theorem \ref{Cor1H} is necessary
for the Liouville property.

}

\begin{cex} { Set $\tilde \beta:=\frac{\Lambda}{\lambda}(Q-1)+1$. Note that $\tilde \beta > 2$ because $Q\geq 4>\frac{\lambda}{\Lambda}+1$.}
In the same way as in Counterexample \ref{cexalphatilde}, one can verify that the function
\[
u_5(x)=\begin{cases}
\frac18[\tilde \beta(\tilde \beta-2)\rho^4-2(\tilde \beta^2-4)\rho^2+\tilde \beta(\tilde \beta+2)]&\text{ if }\rho<1\ ,\\
\frac{1}{\rho^{\tilde \beta-2}}&\text{ if }\rho\geq1\ .
\end{cases}
\]
is a bounded, nonconstant,  classical supersolution to $\mathcal{M}^+_{\lambda,\Lambda}((D^2_{\He^d}u_5)^*)=0$.

\end{cex}

{
\subsection{Equations with Heisenberg Hessian and Euclidean gradient.}\label{sec;euc-grad}

Here we consider equations of the form \eqref{1L-}, i.e., \eqref{non-h-grad}, namely,
\begin{equation}\label{non-h-grad-2}
G(x,u, Du, (D_{
\He^d}^2u)^*)=0\text{ in }\R^d\ ,
\end{equation}
with $G : \R^{2d+1}\times\R\times\R^{2d+1}\times\mathcal S_{2d} \to\R$, so they involve the Heisenberg Hessian $D_{
\He^d}^2u$ and the
 Euclidean gradient $Du$. As at the end of Section \ref{fully} we assume $G$ is uniformly subelliptic and its first order part is bounded from below by a concave Hamiltonian $H_i$ or from above by a convex one $H_s$.
Then Corollary \ref{eu-grad} gives one of the Liouville properties if we find a suitable super- or subsolution out of a big ball. The next result gives an explicit sufficient condition
saying that the vector fields $b^\al$ in the drift part of $H_i$ and $H_s$ point toward the origin for $|x|$ large enough, as in the Ornstein-Uhlenbeck operators. It involves  the homogeneous norm $\rho$ of the Heisenberg group defined by \eqref{homoH}.
\begin{cor}\label{cor_euc1}
Assume that the operator $G$ { satisfies \eqref{US},
where
$\X=\{X_1,...,X_{2d}\}$}  are the Heisenberg vector fields,

and \eqref{blip} and  \eqref{cass} hold.
Suppose there exist $\g_1,\dots, \g_{2d+1}\in\R$ with $\min_i\g_i=\g_o>0$ and such that
\begin{equation}\label{OUtype}
\sup_\al b^\al(x)\cdot D\rho(x) \leq -\sum_{i=1}^{2d+1}\g_ix_i\partial_i\rho + o\left(\frac1{\rho^3} \right)\qquad\text{ as }\rho\to \infty.
\end{equation}
\noindent {\upshape(A)} Assume
 \eqref{G>Hp} and $u\in\USC(\R^{2d+1})$ is a subsolution of \eqref{non-h-grad-2} satisfying  { \eqref{grow+}.}

 If either $c^\alpha(x)\equiv0$ or $u\geq0$, then $u$ is constant.

\noindent {\upshape(B)} Assume
 \eqref{G<Hp} and $v\in\LSC(\R^{2d+1})$ is a supersolution of \eqref{non-h-grad-2} satisfying  { \eqref{grow-}.}

 If either $c^\alpha(x)\equiv0$ or $v\leq0$, then $v$ is constant.

\end{cor}
\begin{proof}
We check that $w=\log\rho$ is a supersolution. Let $C_1:= \Lambda(2d+1)-\lambda > 0$. As in the proof of Theorem \ref{Cor1H}, $w$ is a supersolution at all points where
\begin{equation}\label{tobecheck}
-C_1\frac{|x_H|^2}{\rho^4}+\inf_{\alpha\in A}\left\{c^\alpha(x)\log\rho-b^\alpha(x)\cdot\frac{D\rho}{\rho}\right\}\geq0\ .
\end{equation}
Since $D\rho = (2|x_H|^2 x_H, x_{2d+1})/(2\rho^3)$, we get  from \eqref{OUtype}  that the left hand side is larger than
\begin{multline*}
-C_1\frac{|x_H|^2}{\rho^4}+\frac 1{2\rho^4}\left( 2\sum_{i=1}^{2d}\g_ix_i^2|x_H|^2+ \g_{2d+1}x_{2d+1}^2 + o(1)\right)\geq
\\
\frac 1{\rho^4}\left( |x_H|^2 (\g_o|x_H|^2 - C_1) +\frac{\g_o}2 x_{2d+1}^2+ o(1)\right) \geq 0 \,,
\end{multline*}
for $\rho$ large enough, by taking either $|x_H|^2 > C_1/\g_o$, or $|x_H|^2 \leq C_1/\g_o$ and $x_{2d+1}^2>2C_1^2/\g_o^2$.
\end{proof}

The last result is based on a condition of positivity of the coefficients $c^\al$ at infinity similar to
 Example \ref{schro}.

\begin{cor}\label{cor_euc2}
In the assumptions of Corollary \ref{cor_euc1} replace \eqref{OUtype} with
\begin{equation}\label{c_a}
\liminf_{|x|\to\infty} \inf_{\alpha\in A}c^\alpha(x)
\log\rho(x)>0
 \,,
\end{equation}
and either
\begin{equation}\label{sign}
\limsup_{|x|\to\infty} \sup_{\alpha\in A}b^\alpha(x)\cdot D\rho(x)\leq 0 \,,
\end{equation}
or
\begin{equation}\label{order}
 \sup_{\alpha\in A}|b^\alpha(x)|  = o(\rho) \qquad\text{ as }\rho\to \infty.
\end{equation}
Then the  conclusions of Corollary \ref{cor_euc1} hold true.
\end{cor}
\begin{proof}
We check again the inequality \eqref{tobecheck}. Condition \eqref{sign} implies that
$-b^\alpha(x)\cdot
{D\rho}/{\rho}\geq o(1)$ as $\rho\to \infty$ uniformly in $\al$, and the same occurs under \eqref{order} because $D\rho = O(1)$. Also ${|x_H|^2}/{\rho^4}\leq 1/\rho^2 =o(1)$. Then condition \eqref{c_a} implies  \eqref{tobecheck} for $|x|$ large enough.
\end{proof}
\begin{rem}
Corollary \ref{cor_euc1} generalizes to fully nonlinear equations the Liouville properties for linear Ornstein-Uhlenbeck operators with Heisenberg sub-laplacian proved in \cite{MMT}.

The condition \eqref{c_a} in Corollary \ref{cor_euc2} obviously holds if $c^\alpha(x)\geq c_o>0$ for $|x|$ large enough, and in such case the condition \eqref{order} can be weakened to $\sup_{\alpha\in A}|b^\alpha(x)|  = o(\rho\log \rho)$.
\end{rem}
}


TITLE: Solving Problems In Consecutive Days
QUESTION [4 upvotes]: During a common year a student solves at least $1$ problem in combinatorics, each day. But, so he would not get too tired, he solves at most $12$ problems per week. Prove that there exist several consecutive days during which student has solved, exactly, $20$ problems.
What I have tried is dividing $365$ days in $19$ $``boxes"$. Hence $365 > 19*19$ there must be a $``box"$ with $20$ consecutive days. Also, hence $19 = 2*7+5$ he can't have solved more than $2*12+5$ problems.
But from this I cannot infer how I get there is no $21$ days in the $``box"$.
Is my approach even good? Thanks in advance

REPLY [1 votes]: $a_i$ being the number of problems solved until the $i^{th}$ day, we have for any consecutive $11$ weeks
$$1\leq a_1<a_2<\ldots<a_{77}\leq 132$$ and
$$21\leq a_1+20<a_2+20<\ldots<a_{77}+20\leq 152$$
Then we have a total of $154$ positive integers all of which are less than or equal to $152$.Then by pigeon hole principle there exist $i$ and $j$ such that $a_i=a_j+20$. Then $a_i-a_j=20$. Then the problem solver has solved on $(j+1)^{th}$ day,$(j+2)^{th}$ day $\ldots$ $i^{th}$ day a total of exactly $20$  problems.


TITLE: Infinite square well separation of variables issue
QUESTION [1 upvotes]: When solving the time independent Schrodinger equation for the infinite square well in 2 or 3 dimensions (I'll use 2 dimensions for brevity), we use separation of variables, first assuming that our solution is of the form $\psi(x,y)=f(x)g(y)$ and we get to the point where we have that
$$\frac{1}{f}\frac{\partial^2f}{\partial x^2}+\frac{1}{g}\frac{\partial^2g}{\partial y^2}=\frac{-2m}{\hbar}E$$
My problems begin at this point. Clearly the two terms on the LHS of the above equation must be equal to a constant. Thus we have that
$$\frac{1}{f}\frac{\partial^2f}{\partial x^2}=C_x \,\,\,\,\,\,\,\frac{1}{g}\frac{\partial^2g}{\partial y^2}=C_y$$
But this is never how the problem is actually solved. Instead, it is always assumed that we actually have
$$\frac{1}{f}\frac{\partial^2f}{\partial x^2}=-k^2_x \,\,\,\,\,\,\,\frac{1}{g}\frac{\partial^2g}{\partial y^2}=-k^2_y$$
My problem is that this does not allow either of the terms of the form $\frac{1}{f}\frac{\partial^2f}{\partial x^2}$ to equal a positive constant. Yet one of them equalling a positive constant should still be a valid possibility. I am aware that in this example, $E>0$ because $E$ must always be greater than the value of the potential within the well. But even if we make the assumption that $E>0$, we can still have either one of $C_x$ or $C_y$ being positive because the sum of a positive $C_x$ and negative $C_y$ can still overall be negative provided $C_x<-C_y$. Thus the constants in my second equation should be able to be both positive or negative and still satisfy the Schrodinger equation. It seems to me that by assuming that both the terms $\frac{1}{f}\frac{\partial^2f}{\partial x^2}$ and $\frac{1}{f}\frac{\partial^2g}{\partial y^2}$ be equal to negative constants, we should lose out on many valid solutions to the schrodinger equation (all the cases where $ C_x<-C_y $ for example).
Any help on this issue would be most appreciated!

REPLY [2 votes]: The problem of having $$\frac{1}{f}\frac{d^2f}{dx^2}=C\tag{1}$$ with $C>0$ is that its solution is $$F(x)=Ae^{\sqrt{C}x}+Be^{-\sqrt{C}
x},\tag{2}$$
but if you impose the boundary condition $f(0)=f(L)=0$, you will get $A=B=0.$