# On weighted contractivity and $L^{2}$-Hardy inequalities

Yaozhong Qiu  Department of Mathematics, Imperial College London, 180 Queen’s
Gate, London, SW7 2AZ, United Kingdom<EMAIL_ADDRESS>
###### Abstract.

We show a symmetric Markov diffusion semigroup satisfies a weighted
contractivity condition if and only if a $L^{2}$-Hardy inequality holds, and
we give a sufficient condition for the former. We then give some applications.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Main results
    1. 3.1 Multiplicative subcommutation
    2. 3.2 The abstract Hardy inequality
    3. 3.3 The logarithmic $Q=2$ case
  4. 4 Applications
    1. 4.1 Stratified Lie groups
      1. 4.1.1 Weighted inequalities
      2. 4.1.2 Half-space inequalities
      3. 4.1.3 Radial inequalities
      4. 4.1.4 One-sided laplacian comparisons
      5. 4.1.5 Hörmander and degenerate operators in general
    2. 4.2 Hyperbolic space
    3. 4.3 Rellich inequalities
    4. 4.4 Weak Hardy inequalities

##  1\. Introduction

In this paper, we prove $L^{2}$-Hardy inequalities from the viewpoint of
symmetric Markov semigroups $(P_{t})_{t\geq 0}$ generated by a diffusion $L$
with carré du champ $\Gamma$ and reversible measure $\mu$. Our starting point
is the general theory of Markov semigroups which associates a property (or
properties) of the semigroup with a functional inequality, for instance the
hypercontractivity and exponentially fast entropic convergence to equilibrium
of the semigroup with the logarithmic Sobolev inequality. Here we shall
associate the titular weighted contractivity of the semigroup with the Hardy
inequality, and give a sufficient condition for the former following the ideas
of [RZ21] where the authors studied the generalised Bakry-Émery calculus for a
carré du champ with an additional multiplicative term.

The classical Bakry-Émery calculus is typically associated with the study of
semigroups generated by elliptic diffusions, for instance when
$L=\Delta-\nabla U\cdot\nabla$ is a (euclidean) Kolmogorov type operator on
$\mathbb{R}^{m}$ where $\nabla$ and $\Delta$ are respectively the euclidean
gradient and laplacian and $U:\mathbb{R}^{m}\rightarrow\mathbb{R}$ is a
suitable real-valued function. When $L$ is nonelliptic however, for instance
when $\nabla$ and $\Delta$ are respectively replaced with the subgradient
$\nabla_{\mathbb{H}^{m}}$ and sublaplacian $\Delta_{\mathbb{H}^{m}}$ on the
Heisenberg group $\mathbb{H}^{m}$, the classical curvature(-dimension)
condition is violated (see, for instance, [Bak+08, pp. 3]).

In the nonelliptic setting, the theory of generalised Bakry-Émery calculi has
bore fruit. Roughly speaking, the classical Bakry-Émery calculus studies the
“gamma calculus” or “$\Gamma_{2}$-calculus” of the carré du champ operator
$\Gamma$, whereas generalised Bakry-Émery calculi study the gamma calculi of
other (perhaps similar) operators. For instance, [BB12, BG16] developed a
generalised curvature-dimension condition adapted to the subelliptic (or
subriemannian) setting and proved analogues of the Poincaré and logarithmic
Sobolev inequalities by studying a carré du champ with an additional gradient
term containing derivatives in the so-called “vertical” direction. [Bau17]
used a similar idea to study the kinetic Fokker-Planck equation and recovered
hypocoercive type estimates for convergence to equilibrium which appeared in
[Vil06]. We refer the reader to [Mon19] for more details on generalised Bakry-
Émery calculi and their applications. Our analysis falls under the scope of
the framework of [Mon19] and, in the language of that paper, corresponds to
the study of $\Phi^{W}(f)=W^{2}f^{2}$. Note $\Phi^{W}$ contains none of the
traditional gradient fields, but it can still be regarded as a generalised
carré du champ associated to a differential operator of order zero.

The contents of this paper are organised as follows. In §3, we study the
generalised Bakry-Émery calculus, denoted by $\Gamma^{W}$, for $\Phi^{W}$, and
prove a correspondence between the curvature condition
$\Gamma^{W}(f)\geq\gamma\Phi^{W}(f)$ and the pointwise subcommutation
$\Phi^{W}(P_{t}f)\leq e^{-2\gamma t}P_{t}\Phi^{W}(f)$. We then prove a
correspondence between the integrated subcommutation

$\int\Phi^{W}(P_{t}f)d\mu\leq e^{-2\gamma t}\int\Phi^{W}(f)d\mu$

and the functional inequality

$\int L(W^{2})f^{2}d\mu\leq 2\int W^{2}\Gamma(f)d\mu-2\gamma\int
W^{2}f^{2}d\mu,$

that is, for $\gamma=0$, a correspondence between the weighted contractivity
of the semigroup and the Hardy inequality. Following this, we restate a
sufficient condition first appearing in [RZ21] for the curvature condition,
which sequentially implies the pointwise subcommutation, the integrated
subcommutation, and finally the Hardy inequality. In §4, we give some
applications of our results in various settings. We refer the reader to
[Dav99, BEL15] for a review of Hardy inequalities and their applications.

##  2\. Preliminaries

An abstract definition of a diffusion can be given in the absence of a
differentiable structure together with an abstract notion of regularity, but
we shall restrict our attention here to $\mathbb{R}^{m}$ where a diffusion $L$
is defined as a second order differential operator of the form

(2.1) $L=\sum_{i,j=1}^{m}a_{ij}\partial_{ij}+\sum_{i=1}^{m}b_{i}\partial_{i}$

where the real-valued coefficients $a_{ij},b_{i}$ are smooth for all
$i,j=1,\cdots,m$ and the matrix $a(x)=(a_{ij}(x))_{i,j=1}^{m}$ is a symmetric
positive semidefinite quadratic form for all $x\in\mathbb{R}^{m}$. If $a$ is
in fact (uniformly) positive definite, then we say $L$ is (uniformly)
elliptic.

In the Bakry-Émery language, intrinsic to $L$ is its carré du champ

(2.2) $\Gamma(f,g)=\frac{1}{2}(L(fg)-fLg-gLf)$

which, at least in the setting of diffusions, encodes the second order part of
$L$, in that if $L$ takes the form (2.1), then

$\Gamma(f,g)=\sum_{i,j=1}^{m}a_{ij}\partial_{i}f\partial_{j}g=a(\nabla_{\mathbb{R}^{m}}f,\nabla_{\mathbb{R}^{m}}g).$

Note since $a$ is positive semidefinite the carré du champ is nonnegative,
which, in particular, implies the carré du champ is a symmetric bilinear
positive semidefinite form and therefore satisfies the Cauchy-Schwarz
inequality

(2.3) $\Gamma(f,g)^{2}\leq\Gamma(f,f)\Gamma(g,g).$

We record some properties of the carré du champ that appear in the sequel. The
first is that the carré du champ allows us to succinctly express the fact $L$
satisfies the chain rule

(2.4) $L\phi(f)=\phi^{\prime}(f)Lf+\phi^{\prime\prime}(f)\Gamma(f)$

for all smooth functions $\phi:\mathbb{R}\rightarrow\mathbb{R}$. Consequently,
the carré du champ, which is a derivation in both arguments, also satisfies
the chain rule

(2.5) $\Gamma(\phi(f),g)=\phi^{\prime}(f)\Gamma(f,g).$

The second is that the carré du champ is related to $L$ through an integration
by parts formula. For this we assume the existence of a measure $\mu$ on
$\mathbb{R}^{m}$ which is invariant and reversible for $L$, meaning
respectively

(2.6) $\int Lfd\mu=0$

and

(2.7) $\int fLgd\mu=\int gLfd\mu.$

The integration by parts formula now follows from directly integrating the
definition (2.2) of the carré du champ:

(2.8) $\int fLgd\mu=-\int\Gamma(f,g)d\mu.$

In fact, this identity says the pair $(\Gamma,\mu)$ characterises $L$ entirely
provided $\mu$ is invariant and reversible for $L$, and otherwise it
characterises the symmetric part $\frac{1}{2}(L+L^{*})$ of $L$ for $L^{*}$ the
adjoint of $L$ in $L^{2}(\mu)$. For this reason the data of a symmetric Markov
semigroup can be characterised by a Markov triple consisting of $\Gamma,\mu$,
and the underlying space.

The reader may notice we have made no mention of the function space on which
the previous identities should hold, and, indeed, we recognise the previous
identities may a priori not make any sense without further qualifications on
the function space to which $f$ and $g$ belong. For instance, the definition
of the carré du champ ostensibly requires that the domain of $L$ be at least
an algebra, while the existence and uniqueness of the invariant and reversible
measure apparently depends on the function space on which we would like (2.6)
and (2.7) to hold. As a standard example, if $L=\Delta$ is the euclidean
laplacian, then it holds by Fourier inversion that

$\int\Delta fdx=0,\quad\forall f\in L^{2}(\mathbb{R}^{m}),$

but this obviously does not hold for every $f$ of arbitrary regularity. It
turns out that the space of smooth compactly supported functions is “nice
enough” for our purposes and will allow us to rather cleanly sidestep much of
the subtlety in the interpretation of the previous identities (see [BGL+14,
Chapter 3]).

We omit the functional analytic proofs required to construct the semigroup,
but a sketch is the following. The diffusion $L$, called the generator of the
semigroup, is an essentially selfadjoint nonnegative operator on
$C_{c}^{\infty}(\mathbb{R}^{m})\subset L^{2}(\mu)$ and admits a unique
selfadjoint Friedrichs extension to $L^{2}(\mu)$ which we continue to denote
by $L$. One then defines the semigroup $(P_{t})_{t\geq
0}\vcentcolon=(e^{tL})_{t\geq 0}$ by the spectral theorem. The “Markov”
descriptor asserts $P_{t}$ preserves the space of nonnegative functions and
preserves the unit function in the sense $P_{t}1=1$. More precisely,
$(P_{t})_{t\geq 0}$ is a priori only sub-Markov, meaning $P_{t}f\leq 1$ for
all $f\leq 1$, since the unit function may not belong to $L^{2}(\mu)$, for
instance in the generic case where $\mu$ is Lebesgue measure. Nonetheless we
can still understand the Markov property at least formally in the sense
$L1=0\implies P_{t}1=1$. The “symmetric” descriptor asserts $P_{t}$ is
selfadjoint in $L^{2}(\mu)$, a property it inherits from the fact $L$ is
selfadjoint in $L^{2}(\mu)$ by the reversibility (2.7) of $\mu$. Moreover,
$P_{t}$ is mass-preserving, that is

(2.9) $\int P_{t}fd\mu=\int fd\mu,$

by the invariance (2.6) of $\mu$. It can also be shown $(P_{t})_{t\geq 0}$ is
a (strongly continuous) contraction semigroup over $L^{2}(\mu)$, meaning
$\lVert P_{t}f\rVert_{2}\leq\lVert f\rVert_{2}$ for all $f\in L^{2}(\mu)$. By
Riesz-Thorin interpolation, it extends to a contraction semigroup over
$L^{p}(\mu)$ for all $1\leq p\leq\infty$ consistent, at least for
$1<p<\infty$, with the view of $P_{t}f$ solving the abstract heat equation
$\partial_{t}u(x,t)=Lu(x,t)$ with Cauchy data $u(x,0)=f(x)$. In particular, we
can understand the Markov property when $(P_{t})_{t\geq 0}$ is regarded as a
semigroup over $L^{\infty}(\mu)$, even if in most applications the sub-Markov
property typically suffices. Note the strong continuity of $(P_{t})_{t\geq 0}$
is not important here, but is important for reasons beyond the scope of this
review.

Before we close this chapter, we remark that although we will be interested in
applying semigroup theoretic techniques with the goal being to prove Hardy
inequalities, the study of (symmetric) Markov semigroups often intersects
other fields of research. For instance, under some mild conditions, a Markov
semigroup expresses the law of a Markov process which itself is the solution
of a stochastic differential equation, so that properties in one setting
transfer to another. There is also a one-to-one correspondence between a
semigroup and its generator and a mechanism to pass from one to the other, so
that properties of the generator can be expressed in terms of properties of
the semigroup and vice versa. More relevant to this paper is the connection
between a semigroup and a functional inequality.

The discussion in this chapter has been distilled from the following. We refer
the reader to [Led00, Bak04, Wan06, Bau14, BGL+14] for more details on Markov
semigroups, and to [Dav89, ENB00] for more functional analytic aspects.

##  3\. Main results

###  3.1. Multiplicative subcommutation

In the sequel, we shall assume we are given a symmetric Markov semigroup
$(P_{t})_{t\geq 0}$ generated by a diffusion $L$ with carré du champ $\Gamma$
and invariant and reversible measure $\mu$. The underlying space $X$ is a
smooth manifold, $W\in C^{\infty}(X)$ is smooth (but typically not compactly
supported), and $f\in C_{c}^{\infty}(X)$ is smooth and compactly supported.
Note since $f\in C_{c}^{\infty}(X)$ that invariance of $\mu$ follows from
reversibility, as seen by taking in the reversibility identity (2.7) a
function $g$ such that $g\equiv 1$ on $\operatorname{supp}f$. Thus it suffices
to say $L$ has reversible measure $\mu$.

######  Remark 1.

In fact, that invariance is weaker than reversibility holds more generally. If
(2.7) holds for $f,g\in L^{2}(\mu)$ and $\mu$ is $\sigma$-finite, then it
suffices to take a sequence $(g_{n})_{n\geq 0}\subset L^{2}(\mu)$ of
nonnegative functions such that $g_{n}\nearrow 1$.

In what follows, we assume all integration happens over $X$ unless otherwise
stated. We adopt the notation $\Phi^{W}(f,g)=W^{2}fg$ and denote by
$\Gamma^{W}$ the gamma calculus of $\Phi^{W}$, in a slight departure from the
language of [Mon19, RZ21], and we adopt the traditional shorthands
$\Gamma(f)\vcentcolon=\Gamma(f,f)$ and
$\Gamma^{W}(f)\vcentcolon=\Gamma^{W}(f,f)$. In doing so, $\Gamma^{W}$ can be
regarded as a $W^{2}$-weighted carré du champ, in that if $W\equiv 1$ then
$\Gamma^{1}\equiv\Gamma$ recovers the usual carré du champ. Note our results
will not require a $\Gamma_{2}^{W}$-calculus, that is a gamma calculus for
$\Gamma^{W}$.

Our first objective is a description of the Bakry-Émery type curvature
condition for the pointwise $\Phi^{W}$-subcommutation in analogue with the
classical curvature condition $\Gamma_{2}\geq\rho\Gamma$ for the pointwise
$\Gamma$-subcommutation $\Gamma(P_{t}f)\leq e^{-2\rho t}P_{t}\Gamma(f)$. The
following result describes the curvature condition and provides a sufficient
condition.

######  Theorem 3.1 ([RZ21, Theorem 3]).

The pointwise subcommutation

(3.1) $\Phi^{W}(P_{t}f)=\left\lvert WP_{t}f\right\rvert^{2}\leq e^{-2\gamma
t}P_{t}\left\lvert Wf\right\rvert^{2}=e^{-2\gamma t}P_{t}\Phi^{W}(f)$

holds for some constant $\gamma\in\mathbb{R}$ and $t\geq 0$ if and only if the
curvature condition

(3.2)
$\displaystyle\Gamma^{W}(f)=\frac{1}{2}(L(W^{2})f^{2}+2\Gamma(W^{2},f^{2})+2W^{2}\Gamma(f))\geq\gamma\Phi^{W}(f)$

holds, and the curvature condition (3.2) holds if

(3.3) $\inf_{W\neq 0}\frac{LW}{W}-3\frac{\Gamma(W)}{W^{2}}\geq\gamma>-\infty.$

The proof of the curvature condition and the sufficient condition was
originally done for (euclidean) Kolmogorov type operators in [RZ21], but
generalising to arbitrary diffusions is straightforward. The main ingredients
of the proof are the respective chain rules (2.4) and (2.5) for $L$ and
$\Gamma$, and the Cauchy-Schwarz inequality (2.3) for $\Gamma$.

Our next objective is the correspondence between the integrated version of the
pointwise subcommutation (3.1), that is the integrated
$\Phi^{W}$-subcommutation, and the functional inequality appearing in the
introduction. The main point is that, as usual in semigroup theory,
differentiation at time $t=0$ yields an equivalence with a functional
inequality.

######  Theorem 3.2.

The integrated subcommutation

(3.4) $\int\left\lvert WP_{t}f\right\rvert^{2}d\mu\leq e^{-2\gamma
t}\int\left\lvert Wf\right\rvert^{2}d\mu$

holds if and only if

(3.5) $\int L(W^{2})f^{2}d\mu\leq 2\int W^{2}\Gamma(f)d\mu-2\gamma\int
W^{2}f^{2}d\mu.$

That is, the functional inequality (3.5) is equivalent to the weighted
contractivity condition

(3.6) $\lVert P_{t}f\rVert_{L^{2}(W^{2}d\mu)}=\lVert
WP_{t}f\rVert_{L^{2}(\mu)}\leq e^{-\gamma t}\lVert
Wf\rVert_{L^{2}(\mu)}=e^{-\gamma t}\lVert f\rVert_{L^{2}(W^{2}d\mu)}.$

######  Remark 2.

If $\gamma\neq 0$ then this property can be regarded as a weighted contraction
property for the (nondiffusive) semigroup generated by $L+\gamma$.

###### Proof.

Define $I(t)=\int W^{2}(P_{t}f)^{2}d\mu$. With $g=P_{t}f$,

$\displaystyle I^{\prime}(t)$ $\displaystyle=2\int W^{2}gLgd\mu$
$\displaystyle=\int W^{2}L(g^{2})d\mu-2\int W^{2}\Gamma(g)d\mu$
$\displaystyle=\int L(W^{2})g^{2}d\mu-2\int W^{2}\Gamma(g)d\mu$

by the chain rule (2.4) and reversibility (2.7). If (3.4) holds then
$I^{\prime}(t)\leq-2\gamma I(t)$ which is (3.5) at $t=0$, and conversely if
(3.5) holds then $I^{\prime}(t)\leq-2\gamma I(t)$ yields by Gronwall’s lemma
$I(t)\leq e^{-2\gamma t}I(0)$ which is (3.4). ∎

The connection between (3.1) and (3.4) is that (3.1) is clearly sufficient,
but perhaps not necessary, for (3.4). Thus as mentioned earlier in the
introduction, the sufficient condition (3.3) sequentially verifies the
curvature condition (3.2), the pointwise subcommutation (3.1), the integrated
subcommutation (3.4), and finally the functional inequality (3.5).

Before we show how to recover the Hardy inequality from (3.5), let us
emphasise that unlike in the usual Bakry-Émery framework, our functions $f$ do
not belong to $L^{2}(\mu)$ but instead to $C_{c}^{\infty}(X)\subset
L^{2}(\mu)$. For most choices of $W$ in the sequel it turns out that $W\notin
L^{2}(\mu)$, but nonetheless the actions of $L$ and $P_{t}$ make sense for
smooth $W$ growing slow enough at infinity. That $W\notin L^{2}(\mu)$ is
important however since it throws into question the validity of the
reversibility identity (2.7) and integration by parts formula (2.8). But since
integration by parts is valid whenever one factor is weakly differentiable and
the other is smooth and compactly supported, if $f\in C_{c}^{\infty}(X)$ then
we have access to the full suite of formulae (2.2-8).

###  3.2. The abstract Hardy inequality

We shall now specialise to the case $\gamma=0$ and show how the functional
inequality (3.5) gives rise to a Hardy inequality. Note the form of (3.5)
already has the “correct” order in the sense $L(W^{2})$ is two orders lower
than $W^{2}$. We start with two lemmata. The first is a simple transformation
of (3.5) and the second is a sufficient geometric condition to verify the
first.

######  Lemma 3.1.

Let $W$ be any function verifying (3.1) with constant $\gamma=0$. Then

(3.7) $\int\left(\frac{LW}{W}-\frac{\Gamma(W)}{W^{2}}\right)f^{2}d\mu\leq
2\int\Gamma(f)d\mu$

for all $f\in C_{c}^{\infty}(X\setminus\\{W=0\\})$.

###### Proof.

If $W$ verifies (3.1), then it verifies (3.4) with the same constant
$\gamma=0$ and therefore

$\int L(W^{2})f^{2}d\mu\leq 2\int W^{2}\Gamma(f)d\mu$

by Theorem 3.2. Replacing $f$ with $f/W$, we find

$\displaystyle
2\int\left(\frac{LW}{W}+\frac{\Gamma(W)}{W^{2}}\right)f^{2}d\mu$
$\displaystyle=\int L(W^{2})\frac{f^{2}}{W^{2}}d\mu$ $\displaystyle\leq 2\int
W^{2}\Gamma\left(\frac{f}{W}\right)d\mu$ $\displaystyle\leq
4\int\Gamma(f)d\mu+4\int\frac{\Gamma(W)}{W^{2}}f^{2}d\mu$

since

$\Gamma\left(\frac{f}{W}\right)\leq
2\frac{\Gamma(f)}{W^{2}}+2\frac{\Gamma(W)}{W^{4}}f^{2}$

by Young’s inequality and since $\Gamma$ is a derivation. ∎

######  Lemma 3.2.

Let $\psi\geq 0$ verify

(3.8) $L\psi=\frac{Q-1}{\psi}\Gamma(\psi)$

in the distributional sense, for some $Q\in\mathbb{R}_{\neq 2}$. Then
$W=\psi^{p}$ verifies (3.1) with constant $\gamma=0$ whenever $p$ is between
$0$ and $\frac{Q-2}{2}$.

###### Proof.

It suffices to show $W=\psi^{p}$ verifies (3.3) by Theorem 3.1. By (2.4) and
(2.5),

$\frac{LW}{W}-3\frac{\Gamma(W)}{W^{2}}=(p(Q-1)+p(p-1)-3p^{2})\frac{\Gamma(\psi)}{\psi^{2}}$

and clearly $p(p+Q-2)-3p^{2}$ is nonnegative whenever $p$ is between $0$ and
$\frac{Q-2}{2}$. ∎

######  Remark 3.

If $Q=2$ then clearly $p(p+Q-2)-3p^{2}=-2p^{2}$ is nonnegative if and only if
$p=0$, but of course this recovers the trivial multiplier $W\equiv 1$. It may
well be the case $\Gamma(\psi)\psi^{-2}$ is bounded, in which case $W$
verifies (3.3) with constant $\gamma=-2p^{2}\sup\Gamma(\psi)\psi^{-2}$, but in
this paper we are typically interested in the case $\gamma=0$ and with choices
of $\psi$ for which it will typically be the case $\Gamma(\psi)\psi^{-2}$ is
unbounded.

Our next result concerns the Hardy inequality in both its usual high-
dimensional form and its critical $2$-dimensional form with logarithmic
correction.

######  Theorem 3.3.

Let $\psi\geq 0$ verify (3.8) for some $Q\neq 2$. Then the Hardy inequality

(3.9)
$\int\frac{\Gamma(\psi)}{\psi^{2}}f^{2}d\mu\leq\left(\frac{2}{Q-2}\right)^{2}\int\Gamma(f)d\mu$

holds for all $f\in C_{c}^{\infty}(X\setminus\\{\psi=0\\})$. If $Q=2$ then

(3.10) $\int\frac{\Gamma(\psi)}{\psi^{2}\log^{2}\psi}f^{2}d\mu\leq
4\int\Gamma(f)d\mu$

holds for all $f\in C_{c}^{\infty}(X\setminus\\{\psi\in\\{0,1\\}\\})$.

The requirement that $f$ be supported away from the singularities of the
weight on the left hand side can be removed, for instance if
$\Gamma(\psi)\psi^{-2}$ is locally integrable near its singularities. However,
since we are interested in Hardy inequalities with various weights on either
side, to simplify the exposition we give $f$ enough regularity to sidestep
such considerations. Note as usual the inequalities above hold for $f$ in a
larger space by standard density arguments.

To motivate the expression (3.9), the prototypical example is that of
$L=\Delta$ the euclidean laplacian on $\mathbb{R}^{m}$ whose carré du champ
$\Gamma(f)=\left\lvert\nabla f\right\rvert^{2}$ is the square of the euclidean
gradient and reversible measure $d\mu$ is Lebesgue measure $dx$ on
$\mathbb{R}^{m}$. If $\psi=\left\lvert x\right\rvert$ is the euclidean norm
and $m\geq 3$, then $\psi$ verifies (3.8) with constant $Q=m$, and (3.9) is
then precisely the sharp Hardy inequality on $\mathbb{R}^{m}$ while (3.10) is
the sharp Hardy inequality with logarithmic correction on $\mathbb{R}^{2}$. In
general, $Q$ represents a “homogeneous dimension” of sorts which we elucidate
in the examples to follow. Note although the constant $(\frac{2}{Q-2})^{2}$ is
not optimal in general, it turns out that it is optimal in a number of generic
cases. (This may be somewhat surprising given the starting point is a
sufficient condition for weighted contractivity.)

###### Proof.

First suppose $Q\neq 2$. Since $\psi$ verifies (3.8) with constant $Q\neq 2$,
by Lemma 3.2 the multiplier $W=\psi^{p}$ for $p=\frac{Q-2}{2}$ verifies (3.1)
with constant $\gamma=0$, and by Lemma 3.1 it holds

$\frac{(Q-2)^{2}}{2}\int\frac{\Gamma(\psi)}{\psi^{2}}f^{2}d\mu\leq
2\int\Gamma(f)d\mu$

since
$\frac{LW}{W}-\frac{\Gamma(W)}{W^{2}}=p(p+Q-2)-p^{2}=\frac{(Q-2)^{2}}{2}\Gamma(\psi)\psi^{-2}$.
Rearranging, we find

$\int\frac{\Gamma(\psi)}{\psi^{2}}f^{2}d\mu\leq\left(\frac{2}{Q-2}\right)^{2}\int\Gamma(f)d\mu$

which is (3.9).

If $Q=2$ define $\Phi=\log\psi$. Then

$L\Phi=\frac{1}{\psi}L\psi-\frac{\Gamma(\psi)}{\psi^{2}}=0,$

i.e. $\Phi$ verifies (3.8) with constant $Q=1$. Noting
$\Gamma(\Phi)=\Gamma(\psi)\psi^{-2}$ we see that (3.10) follows from (3.9). ∎

######  Remark 4.

The reader may notice $\Phi<0$ in the region $\\{0<\psi<1\\}$. The case $Q=2$
is more delicate than it appears here, and we leave this discussion to the end
of the chapter.

One advantage of rewriting the Hardy inequality in the Bakry-Émery language is
that it allows for weights of the form $\psi^{\alpha}$ to appear somewhat
easily. This type of weighted Hardy inequality is also sometimes called the
Caffarelli-Kohn-Nirenberg inequality.

######  Corollary 3.1.

Let $\psi\geq 0$ verify (3.8) for some $Q\in\mathbb{R}$. Then the Hardy
inequality

(3.11)
$\int\psi^{\alpha}\frac{\Gamma(\psi)}{\psi^{2}}f^{2}d\mu\leq\left(\frac{2}{Q+\alpha-2}\right)^{2}\int\psi^{\alpha}\Gamma(f)d\mu$

holds for all $f\in C_{c}^{\infty}(X\setminus\\{\psi=0\\})$ and $\alpha$ such
that $Q+\alpha\neq 2$. If $Q+\alpha=2$ then

(3.12)
$\int\psi^{\alpha}\frac{\Gamma(\psi)}{\psi^{2}\log^{2}\psi}f^{2}d\mu\leq
4\int\psi^{\alpha}\Gamma(f)d\mu$

holds for all $f\in C_{c}^{\infty}(X\setminus\\{\psi\in\\{0,1\\}\\})$.

###### Proof.

If $C^{1}\ni\omega\geq 0$ then the operator defined by

$L_{\omega}f=\omega Lf+\Gamma(\omega,f)$

is a diffusion with the same reversible measure $\mu_{\omega}=\mu$ as $L$ but
with a weighted carré du champ $\Gamma_{\omega}(f,g)=\omega\Gamma(f,g)$. Thus
assuming $\psi$ verifies (3.8) with constant $Q\in\mathbb{R}$ for $L$, by
taking $\omega=\psi^{\alpha}$ we find

$\displaystyle L_{\omega}\psi=\psi^{\alpha}L\psi+\Gamma(\psi^{\alpha},\psi)$
$\displaystyle=\psi^{\alpha}\frac{Q-1}{\psi}\Gamma(\psi)+\alpha\psi^{\alpha-1}\Gamma(\psi)$
$\displaystyle=\frac{Q+\alpha-1}{\psi}\psi^{\alpha}\Gamma(\psi)$
$\displaystyle=\frac{Q+\alpha-1}{\psi}\Gamma_{\omega}(\psi,\psi),$

i.e. $\psi$ verifies (3.8) with constant $Q+\alpha$ for $L_{\omega}$ and the
result follows from Theorem 3.3. ∎

######  Remark 5.

Alternatively, the proof can be done so that the factor $\psi^{\alpha}$ is
carried by $\mu$ instead of $\Gamma$. If $\sigma\in C^{1}$ then the operator
defined by $L_{\sigma}f=Lf+\Gamma(\sigma,f)$ is a diffusion with the same
carré du champ $\Gamma_{\sigma}=\Gamma$ as $L$ but with a weighted reversible
measure $d\mu_{\sigma}=e^{\sigma}d\mu$. By taking
$\sigma=\log\psi^{\alpha}=\alpha\log\psi$ we find

$L_{\sigma}\psi=L\psi+\Gamma(\sigma,\psi)=\frac{Q-1}{\psi}\Gamma(\psi)+\frac{\alpha}{\psi}\Gamma(\psi)=\frac{Q+\alpha-1}{\psi}\Gamma_{\sigma}(\psi),$

i.e. $\psi$ verifies (3.8) with constant $Q+\alpha$ for $L_{\sigma}$ as
expected.

A second advantage of proving the Hardy inequality in this way is that the
arguments are easily adapted to the case where only a one-sided laplacian
comparison holds, that is when only an upper bound or lower bound on $L\psi$
is given. In particular, the former occurs naturally in the setting of
riemannian manifolds $(M,g)$ with nonnegative Ricci curvature as a consequence
of the celebrated laplacian comparison theorem.

######  Corollary 3.2.

Let $\psi\geq 0$ verify (3.8) from below, i.e. $\psi$ verifies

(3.13) $L\psi\geq\frac{Q-1}{\psi}\Gamma(\psi)$

for some $Q\in\mathbb{R}$. Then (3.11) holds for any $\alpha$ such that
$Q+\alpha>2$.

######  Corollary 3.3.

Let $\psi\geq 0$ verify (3.8) from above, i.e. $\psi$ verifies

(3.14) $L\psi\leq\frac{Q-1}{\psi}\Gamma(\psi)$

for some $Q\in\mathbb{R}$. Then (3.11) holds for any $\alpha$ such that
$Q+\alpha<2$.

Note the above corollaries are silent on the case $Q+\alpha=2$. In fact, the
logarithmic Hardy inequality (3.10) can be recovered in this case, but we
defer its proof to the next section.

###### Proof.

We assume we are given a lower bound, the upper bound being similar. In
verifying (3.3), note a lower bound (3.13) on $L\psi$ furnishes a lower bound
on $L\psi^{p}$ whenever $p>0$. But since $p=\frac{Q-2}{2}$, this implies
$Q>2$. In fact, $p$ must be between $0$ and $\frac{Q-2}{2}$ as far as
replicating the proof of Theorem 3.3 is concerned, so $Q>2$ is necessary. Thus
if $\psi$ verifies (3.13) with constant $Q$ for $L$, then it verifies (3.13)
with constant $Q+\alpha$ for $L_{\omega}$ and $\omega=\psi^{\alpha}$ by the
proof of Corollary 3.1, so $Q+\alpha>2$ is necessary. ∎

Returning to the case of equality, let us emphasise that the more important
fact from the viewpoint of semigroup theory is that the Hardy inequalities
here follow from the weighted contraction property (3.6). Reframed to
emphasise the connection with semigroups, the theorems given in this section
may be reformulated as follows.

######  Theorem 3.4.

Let $\psi\geq 0$ verify (3.8) with constant $Q\neq 2$, and let $f\in
C_{c}^{\infty}(X\setminus\\{\psi=0\\})$. Then $L$ satisfies the weighted
contraction property

$\lVert\psi^{(Q-2)/2}P_{t}f\rVert_{L^{2}(\mu)}\leq\lVert\psi^{(Q-2)/2}f\rVert_{L^{2}(\mu)}$

if and only if

$\int\psi^{Q-2}\frac{\Gamma(\psi)}{\psi^{2}}f^{2}d\mu\leq\frac{1}{(Q-2)^{2}}\int\psi^{Q-2}\Gamma(f)d\mu.$

More generally, for any $\alpha$ such that $Q+\alpha\neq 2$ and with
$\omega=e^{\sigma}=\psi^{\alpha}$, the operators $L_{\omega}=\omega
L+\Gamma(\omega,-)$ and $L_{\sigma}=L+\Gamma(\sigma-)$ respectively generating
$(P_{t}^{\omega})_{t\geq 0}$ and $(P_{t}^{\sigma})_{t\geq 0}$ satisfy the
weighted contraction property

$\lVert\psi^{(Q+\alpha-2)/2}\widetilde{P}_{t}f\rVert_{L^{2}(\mu)}\leq\lVert\psi^{(Q+\alpha-2)/2}f\rVert_{L^{2}(\mu)}$

for $\widetilde{P}_{t}$ either $P_{t}^{\omega}$ or $P_{t}^{\sigma}$ if and
only if

$\int\psi^{Q+2\alpha-2}\frac{\Gamma(\psi)}{\psi^{2}}f^{2}d\mu\leq\frac{1}{(Q+\alpha-2)^{2}}\int\psi^{Q+2\alpha-2}\Gamma(f)d\mu.$

In particular, the Hardy inequality (3.11) is equivalent to the weighted
contractivity of both $(P_{t}^{\omega})_{t\geq 0}$ and
$(P_{t}^{\sigma})_{t\geq 0}$ where $\omega=e^{\sigma}=\psi^{\beta}$ and
$\beta=\frac{\alpha-(Q-2)}{2}$ with respect to the multiplier $W=\psi^{p}$ for
$p=\frac{\alpha+Q-2}{4}$.

###  3.3. The logarithmic $Q=2$ case

As mentioned earlier, the case $Q=2$ is more delicate than the proof of
Theorem 3.3 would make it seem, since $\Phi=\log\psi$ is negative if
$0<\psi<1$. Since the function $\psi$ is replaced with $\Phi$ and if $\Phi$
verifies (3.8) with constant $Q=1$, it makes no sense to speak of
$W=\Phi^{(Q-2)/2}=(\log\psi)^{-1/2}$.

What makes the theory consistent in this case is the observation that (3.8) is
invariant with respect to nonzero scalar multiplication, that is it holds for
$\Phi$ if and only if it holds for $\lambda\Phi$ for any $\lambda\neq 0$. We
proceed as follows. There are two cases: either $f$ is compactly supported in
$\\{0<\psi<1\\}$, or $\\{\psi>1\\}$. If the former, we replace $\psi$ with
$-\Phi$, and if the latter, we replace $\psi$ with $\Phi$. The Hardy
inequality in both cases reads exactly the same so this distinction makes no
difference to the logarithmic Hardy inequality (3.12), but nonetheless this
distinction is important since, for instance, it affects the statements of
Corollaries 3.2, 3.3 and Theorem 3.4 in the logarithmic case.

######  Theorem 3.5.

Let $\psi\geq 0$ verify (3.8) with constant $Q=2$, and either let $f\in
C_{c}^{\infty}(\\{0<\psi<1\\})$ and $\Phi=-\log\psi$ or let $f\in
C_{c}^{\infty}(\\{\psi>1\\})$ and $\Phi=\log\psi$. Then $L$ satisfies the
weighted contraction property

$\lVert\Phi^{-1/2}P_{t}f\rVert_{L^{2}(\mu)}\leq\lVert\Phi^{-1/2}f\rVert_{L^{2}(\mu)}$

if and only if

$\int\frac{\Gamma(\psi)}{\psi^{2}\Phi^{3}}f^{2}d\mu\leq\int\frac{1}{\Phi}\Gamma(f)d\mu.$

More generally, for $\alpha=2-Q$ and with $\omega=e^{\sigma}=\psi^{\alpha}$,
the operators $L_{\omega}=\omega L+\Gamma(\omega,-)$ and
$L_{\sigma}=L+\Gamma(\sigma,-)$ respectively generating
$(P_{t}^{\omega})_{t\geq 0}$ and $(P_{t}^{\sigma})_{t\geq 0}$ satisfy the
weighted contraction property

$\lVert\Phi^{-1/2}\widetilde{P}_{t}f\rVert_{L^{2}(\mu)}\leq\lVert\Phi^{-1/2}f\rVert_{L^{2}(\mu)}$

for $\widetilde{P}_{t}$ either $P_{t}^{\omega}$ or $P_{t}^{\sigma}$ if and
only if

$\int\psi^{\alpha}\frac{\Gamma(\psi)}{\psi^{2}\Phi^{3}}f^{2}d\mu\leq\int\psi^{\alpha}\frac{1}{\Phi}\Gamma(f)d\mu.$

######  Remark 6.

The substitution $f\mapsto f/W=f\Phi^{1/2}$ of Lemma 3.1 recovers (3.12).

######  Corollary 3.4.

Let $\psi\geq 0$ verify (3.8) from below, i.e. $\psi$ verifies (3.13) for some
$Q\in\mathbb{R}$. Then (3.12) holds for $\alpha=2-Q$ provided $f\in
C_{c}^{\infty}(\\{0<\psi<1\\})$.

######  Corollary 3.5.

Let $\psi\geq 0$ verify (3.8) from above, i.e. $\psi$ verifies (3.14) for some
$Q\in\mathbb{R}$. Then (3.12) holds for $\alpha=2-Q$ provided $f\in
C_{c}^{\infty}(\\{\psi>1\\})$.

###### Proof.

We assume we are given a lower bound, the upper bound being similar. If $\psi$
verifies the laplacian lower bound (3.13) with constant $Q=2$ then $L\Phi\geq
0$, that is $\Phi$ is subharmonic and verifies (3.13) with constant $Q=1$. But
for a one-sided laplacian comparison with constant $Q=1<2$ to yield a Hardy
inequality in this setup, we need a laplacian upper bound (3.14). Replacing
$\Phi$ with $-\Phi$, the result now follows from Corollary 3.3 since the
choice $-\Phi$ corresponds to the region $\\{0<\psi<1\\}$. ∎

######  Remark 7.

This conclusion is consistent with the fact if $\psi$ verifies (3.8), that is
from both below and above, then (3.12) holds for $f$ supported in either
region.

The reader may notice we have sidestepped an important subtlety, namely the
underlying space on which the semigroups $(P_{t})_{t\geq 0}$ in this chapter
act. Indeed, the problem is that even if $f$ is compactly supported, $P_{t}f$
is generally not, as is readily checked by considering the heat semigroup on
$\mathbb{R}^{m}$. Nonetheless the arguments presented in this chapter hold if
$L$ is regarded as an operator acting on $C_{c}^{\infty}(\Omega)\subset
H_{0}^{1}(\Omega)$ for a suitable domain $\Omega\subset\mathbb{R}^{m}$, for
instance either $\\{0<\psi<1\\}$ or $\\{\psi>1\\}$ in the logarithmic case.

##  4\. Applications

There is a large literature of Hardy inequalities in various settings. Here we
document a few Hardy inequalities, some appearing in the existing literature,
and with a particular focus on the subelliptic case. In the sequel, we shall
typically denote by $m$ and $n$ euclidean dimension, and we shall denote by
$d$ and $N$ two special functions on a stratified Lie group to be defined
later. We shall also assume in the logarithmic case that $f$ is supported away
from $\\{\psi=1\\}$ for the appropriate $\psi$ verifying (3.8).

###  4.1. Stratified Lie groups

A Lie group $\mathbb{G}=(\mathbb{R}^{m},\circ)$ is said to be stratified of
step $r\in\mathbb{N}$ if its Lie algebra $\mathfrak{g}$ of left invariant
vector fields admits the decomposition

$\mathfrak{g}=\bigoplus_{i=0}^{r-1}\mathfrak{g}_{i}$

where $\mathfrak{g}_{i}$ are linear subspaces of $\mathfrak{g}$ such that
$\mathfrak{g}_{i}=[\mathfrak{g}_{0},\mathfrak{g}_{i-1}]$ for $1\leq i\leq
r-1$, that is the subalgebra $\mathfrak{g}_{0}$ generates $\mathfrak{g}$ in
the Lie sense. While the topological dimension of $\mathbb{G}$ is $m$, it is
the homogeneous dimension, defined as
$Q(\mathbb{G})\vcentcolon=\sum_{i=0}^{r-1}(i+1)\dim(\mathfrak{g}_{i})$ where
$\dim(\mathfrak{g}_{i})$ is the topological dimension of $\mathfrak{g}_{i}$ as
a $\mathbb{R}$-vector space, which encodes the stratification.

The prototypical example of a nontrivial stratified Lie group is the
Heisenberg group $\mathbb{H}^{1}$, equal to $\mathbb{R}^{3}$ in the sense of
sets. Here the euclidean gradient
$\nabla=(\partial_{1},\partial_{2},\partial_{3})$ is replaced with the
subgradient $\nabla_{\mathbb{H}^{1}}=(X_{1},X_{2})$, also called the
horizontal gradient, and where $X_{1},X_{2}$ are the canonical vector fields
generating the Lie algebra $\mathfrak{h}_{1}$ of $\mathbb{H}^{1}$. Similarly,
the euclidean laplacian $\Delta=\nabla\cdot\nabla$ is replaced with the
sublaplacian
$\Delta_{\mathbb{H}^{1}}=\nabla_{\mathbb{H}^{1}}\cdot\nabla_{\mathbb{H}^{1}}=X_{1}^{2}+X_{2}^{2}$.
This is a subelliptic (and in particular hypoelliptic) Hörmander type operator
for which the classical Bakry-Émery calculus fails, although some Bakry-Émery
type gradient subcommutation bounds can be recovered (see, for instance,
[DM05, Bak+08]). We refer the reader to [BLU07] for more details on the
construction of the subgradient $\nabla_{\mathbb{G}}$ and thus the
sublaplacian $\Delta_{\mathbb{G}}=\nabla_{\mathbb{G}}\cdot\nabla_{\mathbb{G}}$
on a general stratified Lie group $\mathbb{G}$, and to [RS19] for a review of
Hardy inequalities in the specific setting of stratified Lie groups.

The first result in this setting is a “horizontal” Hardy inequality. The
stratification of $\mathfrak{g}=\oplus_{i=0}^{r-1}\ \mathfrak{g}_{i}$ induces
a coordinate system on
$\mathbb{G}\cong\mathfrak{g}\cong\mathbb{R}^{m}=\otimes_{i=0}^{r-1}\
\mathbb{R}_{i}$ where
$\mathbb{R}_{i}\cong\mathbb{R}^{\dim(\mathfrak{g}_{i})}$, and the first,
generating, stratum $\mathfrak{g}_{0}$ is typically called the horizontal
stratum. The coordinates $x\in\mathbb{R}_{0}$ associated to $\mathfrak{g}_{0}$
are called the horizontal coordinates and their euclidean norm $\left\lvert
x\right\rvert$ is called the horizontal norm. Let us emphasise again that the
diffusion $L$ is the sublaplacian
$\Delta_{\mathbb{G}}=\nabla_{\mathbb{G}}\cdot\nabla_{\mathbb{G}}$, its carré
du champ $\Gamma(f)=\left\lvert\nabla_{\mathbb{G}}f\right\rvert^{2}$ is the
square of the subgradient, and its reversible measure $d\mu$ is Lebesgue
measure $d\xi$ on $\mathbb{R}^{m}$, which by [BLU07, Corollary 1.3.21]
coincides with the left (and right) Haar measure on $\mathbb{G}$. In the
sequel, we denote points in $\mathbb{G}$ by $\xi$ and specify explicitly the
stratum of particular coordinates if necessary, with $x$ typically denoting
coordinates in the first stratum.

######  Corollary 4.1.

Let $\mathbb{G}$ be a stratified Lie group, $x^{\prime}\subset x$ a
subcoordinate system of dimension $n\geq 2$ consisting solely of coordinates
in the first stratum, and let $f\in
C_{c}^{\infty}(\mathbb{G}\setminus\\{\left\lvert
x^{\prime}\right\rvert=0\\})$. Then

$\int_{\mathbb{G}}\left\lvert
x^{\prime}\right\rvert^{\alpha}\frac{f^{2}}{\left\lvert
x^{\prime}\right\rvert^{2}}d\xi\leq\left(\frac{2}{n+\alpha-2}\right)^{2}\int_{\mathbb{G}}\left\lvert
x^{\prime}\right\rvert^{\alpha}\left\lvert\nabla_{\mathbb{G}}f\right\rvert^{2}d\xi$

holds for any $\alpha$ such that $n+\alpha\neq 2$, and

$\int_{\mathbb{G}}\left\lvert
x^{\prime}\right\rvert^{2-n}\frac{f^{2}}{\left\lvert
x^{\prime}\right\rvert^{2}\log^{2}\left\lvert x^{\prime}\right\rvert}d\xi\leq
4\int_{\mathbb{G}}\left\lvert
x^{\prime}\right\rvert^{2-n}\left\lvert\nabla_{\mathbb{G}}f\right\rvert^{2}d\xi.$

This was first proven by [D’A04, Theorem 3.3] on the Heisenberg group
$\mathbb{H}^{m}$, and later generalised to a stratified Lie group $\mathbb{G}$
by [RS17, Corollary 5.7].

###### Proof.

The subgradient coincides with the euclidean gradient for a function depending
only on horizontal coordinates, so $\left\lvert x^{\prime}\right\rvert$
verifies (3.8) with constant $Q=n$ for $\Delta_{\mathbb{G}}$. ∎

The second result is a “homogeneous” Hardy inequality. On any stratified Lie
group of homogeneous dimension $Q(\mathbb{G})\geq 3$, which we assume here and
in the sequel, there exists by [Fol75, Theorem 2.1] a fundamental solution $u$
of $\Delta_{\mathbb{G}}$. We call the function $N=u^{1/(2-Q(\mathbb{G}))}$ the
Korányi-Folland gauge, although on a $H$-type group (see [BLU07, Chapter 18])
we also call $N$ the Kaplan norm. In the latter setting, $N$ has an explicit
form. Namely, if $\mathbb{G}\cong\mathbb{R}^{m}_{x}\times\mathbb{R}^{k}_{z}$
is a $H$-type group, then the Kaplan norm is

$N(\xi)=N(x,z)=(\left\lvert x\right\rvert^{4}+\left\lvert
z\right\rvert^{2})^{1/4}.$

In particular, if
$\mathbb{G}=\mathbb{R}_{x}^{m}\times\mathbb{R}_{y}^{m}\times\mathbb{R}_{z}^{1}$
is the Heisenberg group $\mathbb{H}^{m}$, then $N(\xi)=N(x,y,z)=((\left\lvert
x\right\rvert^{2}+\left\lvert y\right\rvert^{2})^{2}+\left\lvert
z\right\rvert^{2})^{1/4}$ since, despite the suggestive notation, the
coordinates $(x,y)\in\mathbb{R}_{x}^{m}\times\mathbb{R}_{y}^{m}$ on
$\mathbb{H}^{m}$ belong to the first stratum while $z\in\mathbb{R}_{z}^{1}$
belongs to the second stratum.

######  Corollary 4.2.

Let $\mathbb{G}$ be a stratified Lie group and let $f\in
C_{c}^{\infty}(\mathbb{G}\setminus\\{0\\})$. Then

$\int_{\mathbb{G}}N^{\alpha}\frac{\left\lvert\nabla_{\mathbb{G}}N\right\rvert^{2}}{N^{2}}f^{2}d\xi\leq\left(\frac{2}{Q(\mathbb{G})-2}\right)^{2}\int_{\mathbb{G}}N^{\alpha}\left\lvert\nabla_{\mathbb{G}}f\right\rvert^{2}d\xi$

holds for any $\alpha$ such that $Q(\mathbb{G})+\alpha\neq 2$, and

$\int_{\mathbb{G}}N^{2-Q(\mathbb{G})}\frac{\left\lvert\nabla_{\mathbb{G}}N\right\rvert^{2}}{N^{2}\log^{2}N}f^{2}d\xi\leq
4\int_{\mathbb{G}}N^{2-Q(\mathbb{G})}\left\lvert\nabla_{\mathbb{G}}f\right\rvert^{2}d\xi.$

This was first proven by [GL90, Corollary 2.1] on $\mathbb{H}^{m}$, and later
generalised to a stratified Lie group independently by [D’A05, Corollary 3.9]
and [Kom05, Theorem 4.1].

###### Proof.

If $N=u^{1/(2-Q(\mathbb{G}))}$ then

$\displaystyle\Delta_{\mathbb{G}}N$
$\displaystyle=(Q(\mathbb{G})-1)u^{1/(2-Q(\mathbb{G}))-2}N^{2(1-Q(\mathbb{G}))}\left\lvert\nabla_{\mathbb{G}}N\right\rvert^{2}$
$\displaystyle=(Q(\mathbb{G})-1)u^{1/(Q(\mathbb{G})-2)}\left\lvert\nabla_{\mathbb{G}}N\right\rvert^{2}$
$\displaystyle=\frac{Q(\mathbb{G})-1}{N}\left\lvert\nabla_{\mathbb{G}}N\right\rvert^{2},$

i.e. $N$ verifies (3.8) with constant $Q=Q(\mathbb{G})$ for
$\Delta_{\mathbb{G}}$. ∎

######  Remark 8.

When $\mathbb{G}=(\mathbb{R}^{m},+)$ is trivial and hence when
$\nabla_{\mathbb{G}}$ and $\Delta_{\mathbb{G}}$ are respectively the euclidean
gradient and laplacian, the horizontal and homogeneous Hardy inequalities
coincide. Indeed, the horizontal norm and Korányi-Folland gauge coincide, as
do the topological, horizontal, and homogeneous dimensions.

####  4.1.1. Weighted inequalities

In proving the weighted Hardy inequalities (3.11) and (3.12) we showed it
suffices to study the quantity
$\smash{\frac{\psi\Gamma(\omega,\psi)}{\omega\Gamma(\psi)}}$ and that for
$\omega=\psi^{\alpha}$ this quantity is exactly equal to
$\alpha\in\mathbb{R}$, which led to the Caffarelli-Kohn-Nirenberg type
inequalities. It turns out $\omega$ need not always be a power of $\psi$. For
instance, on a $H$-type group $\mathbb{G}$ one always has

$\nabla_{\mathbb{G}}\left\lvert
x\right\rvert\cdot\nabla_{\mathbb{G}}N=\frac{\left\lvert
x\right\rvert^{3}}{N^{3}}\text{ and
}\left\lvert\nabla_{\mathbb{G}}N\right\rvert^{2}=\frac{\left\lvert
x\right\rvert^{2}}{N^{2}}$

(see [Ing10, pp. 51, 83]) from which it follows with $\psi=N$ and
$\omega=\left\lvert x\right\rvert^{\alpha}$ that

$\frac{\psi\Gamma(\omega,\psi)}{\omega\Gamma(\psi)}=\frac{N\nabla_{\mathbb{G}}\left\lvert
x\right\rvert^{\alpha}\cdot\nabla_{\mathbb{G}}N}{\left\lvert
x\right\rvert^{\alpha}\left\lvert\nabla_{\mathbb{G}}N\right\rvert^{2}}=\alpha$

Since $N$ verifies (3.8) with constant $Q=Q(\mathbb{G})$ for
$\Delta_{\mathbb{G}}$, it follows $N$ verifies (3.8) with constant
$Q=Q(\mathbb{G})+\alpha$ for $L_{\omega}$ and, in fact, since $\Gamma$ is a
derivation, this can be generalised to the case $\omega=\left\lvert
x\right\rvert^{\alpha}N^{\beta}$.

######  Corollary 4.3.

Let $\mathbb{G}$ be a $H$-type group and let $f\in
C_{c}^{\infty}(\mathbb{G}\setminus\\{0\\})$. Then

$\int_{\mathbb{G}}\left\lvert
x\right\rvert^{\alpha}N^{\beta}\frac{\left\lvert\nabla_{\mathbb{G}}N\right\rvert^{2}}{N^{2}}f^{2}d\xi\leq\left(\frac{2}{Q(\mathbb{G})+\alpha+\beta-2}\right)^{2}\int_{\mathbb{G}}\left\lvert
x\right\rvert^{\alpha}N^{\beta}\left\lvert\nabla_{\mathbb{G}}f\right\rvert^{2}d\xi$

holds for any $\alpha,\beta$ such that $Q(\mathbb{G})+\alpha+\beta\neq 2$, and

$\int_{\mathbb{G}}\left\lvert
x\right\rvert^{\alpha}N^{\beta}\frac{\left\lvert\nabla_{\mathbb{G}}N\right\rvert^{2}}{N^{2}\log^{2}N}f^{2}d\xi\leq
4\int_{\mathbb{G}}\left\lvert
x\right\rvert^{\alpha}N^{\beta}\left\lvert\nabla_{\mathbb{G}}f\right\rvert^{2}d\xi$

if $Q(\mathbb{G})+\alpha+\beta=2$.

######  Remark 9.

If $\alpha<0$ then $f$ must be supported away from $\\{\left\lvert
x\right\rvert=0\\}$.

One can also find inequalities where the roles of $\left\lvert x\right\rvert$
and $N$ are reversed, but the inequalities then fall under the scope of
Corollaries 3.2, 3.3, 3.4, 3.5, since

$\frac{\left\lvert x\right\rvert\Gamma(N^{\beta},\left\lvert
x\right\rvert)}{N^{\beta}\Gamma(\left\lvert x\right\rvert,\left\lvert
x\right\rvert)}=\frac{\left\lvert
x\right\rvert\nabla_{\mathbb{G}}N^{\beta}\cdot\nabla_{\mathbb{G}}\left\lvert
x\right\rvert}{N^{\beta}\left\lvert\nabla_{\mathbb{G}}\left\lvert
x\right\rvert\right\rvert^{2}}=\beta\frac{\left\lvert
x\right\rvert^{4}}{N^{4}}$

furnishes either a laplacian lower or upper bound depending on the sign of
$\beta$.

######  Corollary 4.4.

Let $\mathbb{G}$ be a $H$-type group and let $f\in
C_{c}^{\infty}(\mathbb{G}\setminus\\{\left\lvert x\right\rvert=0\\})$. Then

$\int_{\mathbb{G}}\left\lvert
x\right\rvert^{\alpha}N^{\beta}\frac{f^{2}}{\left\lvert
x\right\rvert^{2}}d\xi\leq\left(\frac{2}{n+\alpha-2}\right)^{2}\int_{\mathbb{G}}\left\lvert
x\right\rvert^{\alpha}N^{\beta}\left\lvert\nabla_{\mathbb{G}}f\right\rvert^{2}d\xi$

holds for any $\alpha$ such that $n+\alpha>2$ if $\beta\geq 0$, for any
$\alpha$ such that $n+\alpha<2$ if $\beta\leq 0$, and

$\int_{\mathbb{G}}\left\lvert
x\right\rvert^{2-n}N^{\beta}\frac{f^{2}}{\left\lvert
x\right\rvert^{2}\log^{2}\left\lvert x\right\rvert}d\xi\leq
4\int_{\mathbb{G}}\left\lvert
x\right\rvert^{2-n}N^{\beta}\left\lvert\nabla_{\mathbb{G}}f\right\rvert^{2}d\xi.$

######  Remark 10.

Note in the logarithmic case we implicitly assume $f$ is supported in the
correct region depending on the sign of $\beta$, that is depending on which of
Corollaries 3.4, 3.5 apply. Note as well the statement of the corollary is
incomplete for brevity. If $\beta\geq 0$ then in fact $\left\lvert
x\right\rvert$ also verifies a laplacian upper bound (3.14) with constant
$Q=n+\alpha+\beta$ for $L_{\omega}$. The case $n+\alpha+\beta<2$ is disjoint
from the case $n+\alpha>2$, but both imply the same Hardy inequality, constant
aside.

There are of course other inequalities that can be recovered by varying the
choice of weight $\omega$. For instance, the Hardy-Poincaré inequality

$\int_{\mathbb{R}^{m}}\frac{f^{2}}{1+\left\lvert
x\right\rvert^{2}}dx\leq\left(\frac{2}{m-2}\right)^{2}\int_{\mathbb{R}^{m}}\frac{\left\lvert
x\right\rvert^{2}}{1+\left\lvert x\right\rvert^{2}}\left\lvert\nabla
f\right\rvert^{2}dx$

on $\mathbb{R}^{m}$ for $m\geq 3$ can be generalised to the setting of
stratified Lie groups, and it all boils down to a simple calculation with
$\omega=\psi^{2+\alpha}(1+\psi^{2})^{-1}$.

####  4.1.2. Half-space inequalities

The reader may notice that in the horizontal Hardy inequality the dimension
$n$ satisfied $n\geq 2$. In fact, the case $n=1$ is perfectly acceptable in
the theory of the previous chapter, but is somewhat “special” since it
corresponds to the class of half-space inequalities. Indeed, the functions
$\psi(\xi)=\left\lvert z\right\rvert$ where $z$ is any coordinate in any
stratum in a stratified Lie group $\mathbb{G}$ verify (3.8) with constant
$Q=1$, since $\Delta_{\mathbb{G}}$ has no zeroth order (multiplicative)
components and so annihilates $\psi$. That the Hardy inequality holds on a
half-space is just the regularity condition that $f$ be compactly supported
away from the zero locus of $\psi$, since to be compactly supported away from
the hyperplane $\\{\left\lvert z\right\rvert=0\\}$ amounts to being compactly
supported on either side of it. The following Hardy inequalities were first
proven, respectively, by [Har20, Equation 4], [LY08, Theorem 1.1], and [Lar16,
Theorem 2.1].

######  Corollary 4.5.

Let $f\in C_{c}^{\infty}((0,\infty))$. Then

$\int_{0}^{\infty}\frac{f^{2}}{x^{2}}dx\leq
4\int_{0}^{\infty}(f^{\prime})^{2}dx.$

######  Corollary 4.6.

Let $f\in C_{c}^{\infty}(\mathbb{H}^{m}\cap\\{z>0\\})$. Then

$\int_{\mathbb{H}^{m}_{z>0}}\frac{\left\lvert x\right\rvert^{2}+\left\lvert
y\right\rvert^{2}}{z^{2}}f^{2}d\xi\leq\int_{\mathbb{H}^{m}_{z>0}}\left\lvert\nabla_{\mathbb{H}^{m}}f\right\rvert^{2}d\xi.$

######  Corollary 4.7.

Let $f\in C_{c}^{\infty}(\mathbb{H}^{m}\cap\\{x_{1}>0\\})$. Then

$\int_{\mathbb{H}^{m}_{x_{1}>0}}\frac{f^{2}}{x_{1}^{2}}d\xi\leq\int_{\mathbb{H}^{m}_{x_{1}>0}}\left\lvert\nabla_{\mathbb{H}^{m}}f\right\rvert^{2}d\xi.$

######  Remark 11.

In the third corollary the coordinate $x_{1}$ can be substituted with any
horizontal half-space $\\{x_{i}>0\\}$ or $\\{y_{i}>0\\}$ of $\mathbb{H}^{m}$.

###### Proof.

In each of the three cases, $\psi$ is respectively $\left\lvert
x\right\rvert$, $\left\lvert z\right\rvert$, and $\left\lvert
x_{1}\right\rvert$, and $\psi$ respectively verifies (3.8) with constant $Q=1$
for $\Delta$, $\Delta_{\mathbb{H}^{m}}$, and $\Delta_{\mathbb{H}^{m}}$. The
reversible measure is Lebesgue measure in each case and the carré du champ is
respectively

  1. (1)

$\left\lvert\nabla\left\lvert x\right\rvert\right\rvert^{2}\equiv 1$,

  2. (2)

$\left\lvert\nabla_{\mathbb{H}^{m}}\left\lvert
z\right\rvert\right\rvert^{2}=4(\left\lvert x\right\rvert^{2}+\left\lvert
y\right\rvert^{2})$, and

  3. (3)

$\left\lvert\nabla_{\mathbb{H}^{m}}\left\lvert
x_{1}\right\rvert\right\rvert^{2}\equiv 1$. ∎

######  Theorem 4.1.

Let $\mathbb{G}$ be a stratified Lie group, $x$ any coordinate in any stratum
in $\mathbb{G}$, and let $f\in C_{c}^{\infty}(\mathbb{G}\cap\\{x>0\\})$. Then

$\int_{\mathbb{G}\cap\\{x>0\\}}x^{\alpha}\frac{\left\lvert\nabla_{\mathbb{G}}x\right\rvert^{2}}{x^{2}}f^{2}d\xi\leq\left(\frac{2}{\alpha-1}\right)^{2}\int_{\mathbb{G}\cap\\{x>0\\}}x^{\alpha}\left\lvert\nabla_{\mathbb{G}}f\right\rvert^{2}d\xi$

holds for any $\alpha$ such that $\alpha\neq 1$, and

$\int_{\mathbb{G}\cap\\{x>0\\}}x\frac{\left\lvert\nabla_{\mathbb{G}}x\right\rvert^{2}}{x^{2}\log^{2}x}f^{2}d\xi\leq
4\int_{\mathbb{G}\cap\\{x>0\\}}x\left\lvert\nabla_{\mathbb{G}}f\right\rvert^{2}d\xi.$

####  4.1.3. Radial inequalities

It turns out that the euclidean Hardy inequality continues to hold even if
$\left\lvert\nabla f\right\rvert^{2}$ is replaced with
$\left\lvert\nabla\left\lvert x\right\rvert\cdot\nabla
f\right\rvert^{2}\leq\left\lvert\nabla f\right\rvert^{2}$, that is

$\int_{\mathbb{R}^{m}}\frac{f^{2}}{\left\lvert
x\right\rvert^{2}}dx\leq\left(\frac{2}{m-2}\right)^{2}\int_{\mathbb{R}^{m}}\left\lvert\nabla\left\lvert
x\right\rvert\cdot\nabla
f\right\rvert^{2}dx=\vcentcolon\left(\frac{2}{m-2}\right)^{2}\int_{\mathbb{R}^{m}}\left\lvert\partial_{r}f\right\rvert^{2}dx.$

This inequality is called a radial Hardy inequality in the literature (see,
for instance, [BE11, Ges+19]), and $\partial_{r}=\nabla\left\lvert
x\right\rvert\cdot\nabla$ is typically called the radial derivative. The study
and proof of such inequalities can be replicated in our setting with
additional work. The basic idea relies on the fact the map
$f\mapsto\nabla\left\lvert x\right\rvert\cdot\nabla f=\Gamma(\left\lvert
x\right\rvert,f)$ for $\Gamma$ the carré du champ of the euclidean laplacian
is a derivation and so its square is a diffusion.

######  Theorem 4.2.

Let $\psi\geq 0$ verify (3.8) with constant $Q\in\mathbb{R}$ for $L$, assume
$\Gamma(\psi,\Gamma(\psi))=0$, and let $f\in
C_{c}^{\infty}(X\setminus\\{\psi=0\\})$. Then

(4.1)
$\int\psi^{\alpha}\frac{\Gamma(\psi)^{2}}{\psi^{2}}f^{2}d\mu\leq\left(\frac{2}{Q+\alpha-2}\right)^{2}\int\psi^{\alpha}\Gamma(\psi,f)^{2}d\mu$

holds for any $\alpha$ such that $Q+\alpha\neq 2$, and

(4.2)
$\int\psi^{2-Q}\frac{\Gamma(\psi)^{2}}{\psi^{2}\log^{2}\psi}f^{2}d\mu\leq
4\int\psi^{2-Q}\Gamma(\psi,f)^{2}d\mu.$

###### Proof.

The starting point is the operator $L_{\psi}=-Z_{\psi}^{*}Z_{\psi}$ where
$Z_{\psi}=\Gamma(\psi,-)$ and $Z_{\psi}^{*}$ is the adjoint of $Z_{\psi}$ with
respect to $\mu$. Direct computation gives

$\int g(Z_{\psi}f)d\mu=\int g\Gamma(\psi,f)d\mu=-\int f\Gamma(\psi,g)d\mu-\int
fg(L\psi)d\mu=\vcentcolon\int f(Z_{\psi}^{*}g)d\mu$

for $Z_{\psi}^{*}=-Z_{\psi}-L\psi$ from which it follows
$L_{\psi}=Z_{\psi}^{2}+(L\psi)Z_{\psi}$. By construction, $L_{\psi}$ has carré
du champ $\Gamma_{\psi}(f)=(Z_{\psi}f)^{2}$ and reversible measure $\mu$.

Note

$\frac{\psi
L_{\psi}\psi}{\Gamma_{\psi}(\psi)}=\frac{\psi(Z_{\psi}^{2}\psi+(L\psi)Z_{\psi}\psi)}{(Z_{\psi}\psi)^{2}}=\frac{\psi\Gamma(\psi,\Gamma(\psi))}{\Gamma(\psi)^{2}}+\frac{\psi
L\psi\Gamma(\psi)}{\Gamma(\psi)^{2}}=Q-1$

from which it follows $\psi$ verifies (3.8) with constant $Q\in\mathbb{R}$ for
$L_{\psi}$. The desired conclusion follows from Corollary 3.1. ∎

We can easily recover the radial Hardy inequality in the euclidean setting,
and more generally in the subelliptic setting with the horizontal norm
$\left\lvert x\right\rvert$ and the Korányi-Folland gauge $N$ playing the role
of $\psi$. The latter however requires the analysis to occur on a $H$-type
group so that the secondary condition $\Gamma(\psi,\Gamma(\psi))=0$ can be
explicitly verified.

######  Corollary 4.8.

Let $\mathbb{G}$ be a stratified Lie group of horizontal dimension $n\geq 3$
and let $f\in C_{c}^{\infty}(\mathbb{G}\setminus\\{\left\lvert
x\right\rvert=0\\})$. Then

$\int_{\mathbb{G}}\frac{f^{2}}{\left\lvert
x\right\rvert^{2}}d\xi\leq\left(\frac{2}{n-2}\right)^{2}\int_{\mathbb{G}}\left\lvert\nabla_{\mathbb{G}}\left\lvert
x\right\rvert\cdot\nabla_{\mathbb{G}}f\right\rvert^{2}d\xi.$

###### Proof.

The follows from taking $\alpha=0$ in (4.1) and with $L=\Delta_{\mathbb{G}}$
since $\Gamma(\left\lvert x\right\rvert)\equiv 1$. ∎

######  Corollary 4.9.

Let $\mathbb{G}$ be a $H$-type group and let $f\in
C_{c}^{\infty}(\mathbb{G}\setminus\\{0\\})$. Then

$\int_{\mathbb{G}}\frac{\left\lvert\nabla_{\mathbb{G}}N\right\rvert^{4}}{N^{2}}f^{2}d\xi\leq\left(\frac{2}{Q(\mathbb{G})-2}\right)^{2}\int_{\mathbb{G}}\left\lvert\nabla_{\mathbb{G}}N\cdot\nabla_{\mathbb{G}}f\right\rvert^{2}d\xi.$

###### Proof.

With $L,\Gamma,\mu$ as before, we know $\psi=N$ verifies (3.8) with constant
$Q=Q(\mathbb{G})$ for $L$, so it suffices to show $\Gamma(N,\Gamma(N))=0$. On
a $H$-type group we find

$\displaystyle\Gamma(N,\Gamma(N))=\nabla_{\mathbb{G}}N\cdot\nabla_{\mathbb{G}}\left\lvert\nabla_{\mathbb{G}}N\right\rvert^{2}$
$\displaystyle=\nabla_{\mathbb{G}}N\cdot\nabla_{\mathbb{G}}\frac{\left\lvert
x\right\rvert^{2}}{N^{2}}\vphantom{-2\left\lvert
x\right\rvert^{2}N^{-3}\frac{\left\lvert
x\right\rvert^{2}}{N^{2}}+2N^{-2}\left\lvert x\right\rvert\frac{\left\lvert
x\right\rvert^{3}}{N^{3}}}$ $\displaystyle=\left\lvert
x\right\rvert^{2}\nabla_{\mathbb{G}}N\cdot\nabla_{\mathbb{G}}N^{-2}+N^{-2}\nabla_{\mathbb{G}}N\cdot\nabla_{\mathbb{G}}\left\lvert
x\right\rvert^{2}\vphantom{-2\left\lvert
x\right\rvert^{2}N^{-3}\frac{\left\lvert
x\right\rvert^{2}}{N^{2}}+2N^{-2}\left\lvert x\right\rvert\frac{\left\lvert
x\right\rvert^{3}}{N^{3}}}$ $\displaystyle=0.\vphantom{-2\left\lvert
x\right\rvert^{2}N^{-3}\frac{\left\lvert
x\right\rvert^{2}}{N^{2}}+2N^{-2}\left\lvert x\right\rvert\frac{\left\lvert
x\right\rvert^{3}}{N^{3}}}\qed$

######  Remark 12.

The condition $\Gamma(\psi,\Gamma(\psi))$ is a somewhat generic one in that a
stratified Lie group $\mathbb{G}$ for which its Korányi-Folland gauge $N$
satisfies $\Gamma(N,\Gamma(N))=0$ is called polarisable [BT02, Definition
2.12]. It should be noted in fact our condition is slightly stronger as a
group is polarisable if and only if $N$ is $\infty$-harmonic, but this follows
from the fact $\Gamma(N,\Gamma(N))$ vanishes. Polarisable groups are somewhat
far and few between; the $H$-type groups were the first and only examples
[BT02, Corollary 5.6] until the recent work of [Bie19, Theorem 5.2].
Nonetheless they make appearances in, for instance, the study of the
$p$-sublaplacian.

An alternate version of the radial Hardy inequality can be obtained by a
different but very similar approach. Recalling §4.1, suppose the Lie algebra
$\mathfrak{g}$ of $\mathbb{G}$ admits the stratification
$\mathfrak{g}=\oplus_{i=0}^{r-1}\ \mathfrak{g}_{i}$ and induces a coordinate
system on $\mathbb{G}\cong\otimes_{i=0}^{r-1}\ \mathbb{R}_{i}$ where
$\mathbb{R}_{i}\cong\mathbb{R}^{\dim(\mathfrak{g}_{i})}$. With
$\nabla_{\mathbb{R}_{i}}$ denoting the euclidean gradient along the $i$-th
stratum, we define the dilation operator on $\mathbb{G}$ as

$D_{\mathbb{G}}f(\xi)\vcentcolon=\sum_{i=0}^{r-1}ix_{i}\cdot\nabla_{\mathbb{R}^{i}}f(\xi).$

For instance, if $\mathbb{G}=\mathbb{R}^{m}$ is the trivial euclidean group,
then $D_{\mathbb{G}}=x\cdot\nabla$, and if
$\mathbb{G}\cong\mathbb{R}^{m}_{x}\times\mathbb{R}^{m}_{y}\times\mathbb{R}^{1}_{z}$
is the Heisenberg group $\mathbb{H}^{m}$, then
$D_{\mathbb{G}}=\sum_{i=1}^{m}x_{i}\partial_{x_{i}}+y_{i}\partial_{y_{i}}+2z\partial_{z}$.
That is, $D_{\mathbb{G}}$ is a homogeneous analogue of $x\cdot\nabla$ where
each component $x_{i}\cdot\partial_{x_{i}}$ is weighted according to the
stratum to which $x_{i}$ belongs.

The following result was first proven by [RS19, Theorem 2.1.1] where the
square of the dilation operator is called Euler operator. To make sense of
this result, we first introduce the notion of a homogeneous norm, and to this
end we define a family $(\delta_{\lambda})_{\lambda>0}$ of dilations

$\delta_{\lambda}(\xi)=\delta_{\lambda}(x_{0},\cdots,x_{r-1})=(\lambda
x_{0},\cdots,\lambda^{r}x_{r-1})$

associated to the stratification of $\mathfrak{g}$. A homogeneous quasinorm is
a smooth (almost everywhere) function
$\psi:\mathbb{G}\rightarrow\mathbb{R}_{\geq 0}$ symmetric about the origin
$\psi(-\xi)=\psi(\xi)$ and $1$-homogeneous
$\psi\circ\delta_{\lambda}=\lambda\psi$ with respect to
$(\delta_{\lambda})_{\lambda>0}$. If $\psi$ is also positive definite, i.e.
$\psi(\xi)=0$ if and only if $\xi=0$, then $\psi$ is called a homogeneous norm
(see [BLU07, Chapter 5]).

######  Theorem 4.3.

Let $\mathbb{G}$ be a stratified Lie group, $\psi\geq 0$ a homogeneous
quasinorm on $\mathbb{G}$, and let $f\in
C_{c}^{\infty}(\mathbb{G}\setminus\\{\psi=0\\})$. Then

(4.3)
$\int_{\mathbb{G}}\psi^{\alpha}f^{2}d\xi\leq\left(\frac{2}{Q(\mathbb{G})+\alpha}\right)^{2}\int_{\mathbb{G}}\psi^{\alpha}(D_{\mathbb{G}}f)^{2}d\xi$

holds for any $\alpha$ such that $Q(\mathbb{G})+\alpha\neq 0$, and

(4.4)
$\int_{\mathbb{G}}\psi^{-Q(\mathbb{G})}\frac{f^{2}}{\log^{2}\psi}d\xi\leq
4\int\psi^{-Q(\mathbb{G})}(D_{\mathbb{G}}f)^{2}d\xi.$

###### Proof.

As previously, the starting point is the operator
$L_{\mathbb{G}}=-D_{\mathbb{G}}^{*}D_{\mathbb{G}}$ where $D_{\mathbb{G}}$ is
the dilation operator and $D_{\mathbb{G}}^{*}$ is its adjoint with respect to
the Lebesgue measure $d\xi$ on $\mathbb{G}$. To compute the adjoint, note
$D_{\mathbb{G}}$ can be rewritten in the following way: disregard the
stratification
$\xi=(x_{0},\cdots,x_{r-1})\in\mathbb{R}_{0}\times\cdots\mathbb{R}_{r-1}\cong\mathbb{G}$
of the coordinates $\xi\in\mathbb{G}$ and instead write
$\xi=(y_{1},\cdots,y_{m})\in\mathbb{R}^{m}\cong\mathbb{G}$ for
$m=\sum_{i=0}^{r-1}\dim(\mathbb{R}_{i})=\dim(\mathbb{G})$. Then

$D_{\mathbb{G}}=\sum_{j=1}^{m}c_{j}D_{j}$

where $D_{j}=y_{j}\partial_{j}$ and $c_{j}=k\in\\{1,\cdots,r\\}$ if and only
if $y_{j}$ is a coordinate belonging to the $k$-th stratum. Since $D_{j}$ has
adjoint $-D_{j}-1$ with respect to Lebesgue measure, it follows

$D_{\mathbb{G}}^{*}=-\sum_{j=1}^{m}c_{j}(D_{j}+1)=-D_{\mathbb{G}}-Q(\mathbb{G})$

and therefore $L_{\mathbb{G}}=D_{\mathbb{G}}^{2}+Q(\mathbb{G})D_{\mathbb{G}}$.
By construction, $L_{\mathbb{G}}$ has carré du champ
$\Gamma_{\mathbb{G}}(f)=(D_{\mathbb{G}}f)^{2}$ and reversible measure $d\xi$,
and so now we play the same game with $\psi$ and $L_{\mathbb{G}}$. To this end
note if $\psi$ is a homogeneous quasinorm, then $D_{\mathbb{G}}\psi=\psi$ by
[RS19, Proposition 1.3.1]. Thus

$\frac{\psi L_{\mathbb{G}}\psi}{\Gamma_{\mathbb{G}}(\psi)}=\frac{\psi
D_{\mathbb{G}}^{2}\psi}{(D_{\mathbb{G}}\psi)^{2}}+\frac{\psi
Q(\mathbb{G})D_{\mathbb{G}}\psi}{(D_{\mathbb{G}}\psi)^{2}}=\frac{\psi^{2}}{\psi^{2}}+Q(\mathbb{G})\frac{\psi^{2}}{\psi^{2}}=Q(\mathbb{G})+1,$

i.e. $\psi$ verifies (3.8) with constant $Q(\mathbb{G})+2$ for $L$. The
desired conclusion follows from Corollary 3.1. ∎

######  Remark 13.

Note in the euclidean setting $D=x\cdot\nabla$ recovers the results of Theorem
4.2; indeed, $(D_{\mathbb{R}^{m}}f)^{2}=\left\lvert
x\right\rvert^{2}\left\lvert\nabla\left\lvert x\right\rvert\cdot\nabla
f\right\rvert^{2}$, so taking $\psi=\left\lvert x\right\rvert$ and replacing
$\alpha$ with $\alpha+2$ we see that (4.1) and (4.2) coincide respectively
with (4.3) and (4.4).

######  Remark 14.

The dilation operator $D_{\mathbb{G}}$ plays an important role in the analysis
of the sublaplacian $\Delta_{\mathbb{G}}$. For instance, $D_{\mathbb{G}}$
satisfies the commutation relation
$[\Delta_{\mathbb{G}},D_{\mathbb{G}}]=\Delta_{\mathbb{G}}$, and, moreover, the
invariant measure of the operator $\Delta_{\mathbb{G}}-D_{\mathbb{G}}$
coincides with the heat kernel measure $p_{1}(0,dy)$ of $\Delta_{\mathbb{G}}$
at the origin and at time one [Bak+08, p. 8]. Note in the general subelliptic
setting the Hardy inequality for the dilation operator and the radial Hardy
inequalities of Corollaries 4.8, 4.9 are different, since the “gradient” of
one is the subgradient $\nabla_{\mathbb{G}}$ whereas the gradient of the other
is the homogeneous analogue of $x\cdot\nabla$. Note as well the Hardy constant
$(\frac{2}{Q(\mathbb{G})+\alpha})^{2}$ depends only on the homogeneous
dimension $Q(\mathbb{G})$ of $\mathbb{G}$. In particular, it does not (unlike
the previous radial Hardy inequality) depend on whether or not $\psi$ verifies
(3.8) and therefore it holds for the Carnot-Carathéodory distance, the
horizontal norm, the Korányi-Folland gauge, and indeed any other homogeneous
quasinorm.

####  4.1.4. One-sided laplacian comparisons

The last class of Hardy inequalities we study in this chapter are those for
which only a one-sided laplacian comparison, that is either (3.13) or (3.14),
is satisfied. We have already seen an example in §4.1, but we prove some Hardy
inequalities for choices of $\psi$ for which a one-sided laplacian comparison
arises “naturally”.

One setting in which this is the case is the riemannian setting. The laplacian
comparison theorem asserts if $(M,g)$ is a complete riemannian manifold of
dimension $m$ with nonnegative Ricci curvature, then
$\Delta_{g}d\leq\frac{m-1}{d}$ where $\Delta_{g}$ is the Laplace-Beltrami
operator and $d$ is the riemannian distance to any basepoint. Another is the
subriemannian setting of a stratified Lie group $\mathbb{G}$ together its
sublaplacian $\Delta_{\mathbb{G}}$ and geodesic distance

$d(x)\vcentcolon=\sup_{f\text{ s.t. }\Gamma(f)\leq 1}f(x)-f(0),$

called the Carnot-Carathéodory distance. It was shown in [HZ09, Theorem 3]
that there exists a constant $c\in\mathbb{R}$ for which
$\Delta_{\mathbb{G}}d\leq\frac{c}{d}$ when $\mathbb{G}$ is a $H$-type group,
although an estimate on the precise value of $c$ is not known. Still, at least
in the weighted setting this is sufficient for the following Hardy inequality
to hold.

######  Corollary 4.10.

Let $\mathbb{G}$ be a $H$-type group, $c=c(\mathbb{G})\in\mathbb{R}$ the
smallest $c$ for which it holds $\Delta_{\mathbb{G}}d\leq\frac{c}{d}$, and let
$f\in C_{c}^{\infty}(\mathbb{G}\setminus\\{0\\})$. Then

$\int_{\mathbb{G}}d^{\alpha}\frac{f^{2}}{d^{2}}d\xi\leq\left(\frac{2}{c+\alpha-1}\right)^{2}\int_{\mathbb{G}}d^{\alpha}\left\lvert\nabla_{\mathbb{G}}f\right\rvert^{2}d\xi$

holds for any $\alpha$ such that $c+\alpha<1$.

###### Proof.

$d$ verifies a laplacian upper bound with constant $Q=c+1$ for
$\Delta_{\mathbb{G}}$ and so the result follows from Corollary 3.3 since
$\left\lvert\nabla_{\mathbb{G}}d\right\rvert=1$. ∎

The case $\alpha=0$ is interesting in that [Leh17, Theorem 1.1] proved the
unweighted Hardy inequality holds, but without an estimate on the optimal
constant. Later, [FP21, Theorem 1] proved the optimal constant on the
Heisenberg group $\mathbb{H}^{m}$ is strictly at least
$\frac{1}{m^{2}}=(\frac{2}{Q(\mathbb{H}^{m})-2})^{2}$ where
$Q(\mathbb{H}^{m})=2m+2$. Since $\mathbb{H}^{m}$ is indeed a $H$-type group, a
laplacian upper bound of the form $\Delta_{\mathbb{H}^{m}}d\leq\frac{c}{d}$
exists, but without an estimate on $c$ and without knowing $c<1$ it would
appear that our Hardy inequality is silent on whether the unweighted Hardy
inequality holds. That being said, [FP21, Theorem 1] does imply that
$c>-2m+1$, otherwise a Hardy inequality would hold with constant better than
$\frac{1}{m^{2}}$. Recall as well that the answer is not as simple as using
the horizontal Hardy inequality and using the fact the horizontal norm is
controlled by the Carnot-Carathed́ory distance, since the function space on
which the horizontal Hardy inequality holds is ostensibly smaller than
$C_{c}^{\infty}(\mathbb{G}\setminus\\{0\\})$.

Nonetheless we can still recover the unweighted Hardy inequality with the
correct function space but without the optimal constant by first proving it
for a homogeneous norm $\psi$ whose subgradient does not vanish (in contrast
to the subgradient of the Korányi-Folland gauge) and then appealing to
equivalence. The idea is simple: we wish to take the horizontal norm
$\left\lvert x\right\rvert$ in place of $d$, but since $\left\lvert
x\right\rvert$ is not a norm but a quasinorm, we add an extra $\varepsilon d$
for $0<\varepsilon\ll 1$ small. However, since we are given a laplacian upper
bound for $d$, we need to take $\varepsilon d+\left\lvert x_{1}\right\rvert$
where $x_{1}$ is the first (or rather any) horizontal coordinate, since a
laplacian upper bound requires $Q<2$ to recover an unweighted Hardy inequality
according to Corollary 3.3. This unfortunately makes the constant independent
of the horizontal dimension. If instead we were given a laplacian _lower_
bound for $d$, then taking the full horizontal norm in horizontal dimension
$n\geq 3$ would lead to a better constant. This observation suggests the
“correct” homogeneous norm to add is the Korányi-Folland gauge $N$.

######  Theorem 4.4.

Let $\mathbb{G}$ be a stratified Lie group and let $\varrho$ be a homogeneous
norm on $\mathbb{G}$ in the sense of [BLU07, Chapter 5]. Define
$\kappa(\varrho)=\inf\\{\tau>0\mid\varrho\leq\tau N\\}$ and let $f\in
C_{c}^{\infty}(\mathbb{G}\setminus\\{0\\})$. Then

$\int_{\mathbb{G}}\frac{f^{2}}{\varrho^{2}}d\xi\leq\min\left(4,\left(\frac{2}{n-2}\right)\right)^{2}\kappa^{2}(\left\lvert
x\right\rvert)\kappa^{2}(\varrho)\int_{\mathbb{G}}\left\lvert\nabla_{\mathbb{G}}f\right\rvert^{2}d\xi.$

###### Proof.

All homogeneous norms are equivalent by [BLU07, Corollary 5.1.4], so $\kappa$
is well defined. Note although $\left\lvert x\right\rvert$ is not a
homogeneous norm but a quasinorm, $\kappa(\left\lvert x\right\rvert)$ still
makes sense since $\left\lvert x\right\rvert$ is controlled by $N$. On a
$H$-type group the explicit form of $N$ makes this obvious, and on a
stratified Lie group there exists a “euclidean-esque” homogeneous norm [BLU07,
p. 229] which controls $\left\lvert x\right\rvert$.

We prove an unweighted Hardy inequality for
$\psi=\psi_{\varepsilon}\vcentcolon=\varepsilon N+\left\lvert x\right\rvert$
with $0<\varepsilon\ll 1$. Since $\psi$ is a homogeneous norm, $\psi$ is
equivalent to $\varrho$ and, if a Hardy inequality were to hold, it would be
for $f$ supported away from the zero locus $\\{\psi=0\\}=\\{0\\}$. First
suppose the horizontal dimension $n$ satisfies $n\geq 3$. Since
$\left\lvert\nabla_{\mathbb{G}}N\right\rvert^{2}$ is bounded, say without loss
of generality by $1$, we find

$\left\lvert\nabla_{\mathbb{G}}\psi\right\rvert^{2}=\left\lvert\nabla_{\mathbb{G}}\varepsilon
N\right\rvert^{2}+2\nabla_{\mathbb{G}}(\varepsilon
N)\nabla_{\mathbb{G}}\left\lvert
x\right\rvert+\left\lvert\nabla_{\mathbb{G}}\left\lvert
x\right\rvert\right\rvert^{2}\leq 1+2\varepsilon+\varepsilon^{2}$

and

$\Delta_{\mathbb{G}}\psi=\frac{\varepsilon(Q-1)}{N}\left\lvert\nabla_{\mathbb{G}}N\right\rvert^{2}+\frac{n-1}{\left\lvert
x\right\rvert}$

so that

$\displaystyle\frac{\psi\Delta_{\mathbb{G}}\psi}{\left\lvert\nabla_{\mathbb{G}}\psi\right\rvert^{2}}$
$\displaystyle\geq\frac{\varepsilon^{2}(Q-1)\left\lvert\nabla_{\mathbb{G}}N\right\rvert^{2}+\varepsilon(n-1)N\left\lvert
x\right\rvert^{-1}+\varepsilon(Q-1)\left\lvert\nabla_{\mathbb{G}}N\right\rvert^{2}\left\lvert
x\right\rvert N^{-1}+n-1}{1+2\varepsilon+\varepsilon^{2}}$
$\displaystyle\geq\frac{n-1}{1+2\varepsilon+\varepsilon^{2}},$

i.e. $\psi$ verifies a laplacian lower bound with constant
$Q=Q_{\varepsilon}=n+o(\varepsilon)$ for $\Delta_{\mathbb{G}}$. Since
$\left\lvert\nabla_{\mathbb{G}}\psi\right\rvert^{2}\geq
1-2\varepsilon+\varepsilon^{2}$ and $\psi\leq(\kappa(\left\lvert
x\right\rvert)+\varepsilon)N$, it follows by Corollary 3.2 that

$\displaystyle\int_{\mathbb{G}}\frac{f^{2}}{N^{2}}d\xi$
$\displaystyle\leq(\kappa(\left\lvert
x\right\rvert)+\varepsilon)^{2}\int\frac{f^{2}}{\psi^{2}}d\xi\vphantom{\left(\frac{2}{Q-2}\right)^{2}\left(\frac{1+\varepsilon}{1-2\varepsilon+\varepsilon^{2}}\right)^{2}\int_{\mathbb{G}}\left\lvert\nabla_{\mathbb{G}}f\right\rvert^{2}d\xi}$
$\displaystyle\leq\frac{(\kappa(\left\lvert
x\right\rvert)+\varepsilon)^{2}}{1-2\varepsilon+\varepsilon}\int_{\mathbb{G}}\frac{\left\lvert\nabla_{\mathbb{G}}\psi\right\rvert^{2}}{\psi^{2}}f^{2}d\xi\vphantom{\left(\frac{2}{Q-2}\right)^{2}\left(\frac{\kappa(\left\lvert
x\right\rvert)+\varepsilon}{1-2\varepsilon+\varepsilon^{2}}\right)^{2}\int_{\mathbb{G}}\left\lvert\nabla_{\mathbb{G}}f\right\rvert^{2}d\xi}$
$\displaystyle\leq\left(\frac{2}{Q-2}\right)^{2}\left(\frac{\kappa(\left\lvert
x\right\rvert)+\varepsilon}{1-2\varepsilon+\varepsilon^{2}}\right)^{2}\int_{\mathbb{G}}\left\lvert\nabla_{\mathbb{G}}f\right\rvert^{2}d\xi.$

The desired conclusion follows by taking $\varepsilon\rightarrow 0^{+}$.

If $n\leq 2$ then we can repeat the proof with $\left\lvert x_{1}\right\rvert$
replacing $\left\lvert x\right\rvert$, where $x_{1}$ is the first horizontal
coordinate. In this case we may prove in exactly the same way that
$\psi=\psi_{\varepsilon}=\varepsilon N+\left\lvert x_{1}\right\rvert$ verifies
a laplacian upper bound with constant $Q=Q_{\varepsilon}=1+o(\varepsilon)$ for
$\Delta_{\mathbb{G}}$, so the result follows from Corollary 3.3 since
$\left\lvert x_{1}\right\rvert\leq\left\lvert
x\right\rvert\leq\kappa(\left\lvert x\right\rvert)N$. ∎

######  Remark 15.

The previous proof can be generalised to the weighted setting but it requires
$\alpha<2$ so that the upper bound $\psi\leq(\kappa(\left\lvert
x\right\rvert+\varepsilon)N$ furnishes a lower bound on $\psi^{\alpha-2}$.

Note the constant is generally suboptimal. For instance, in horizontal
dimension $n\leq 3$ one can sidestep the equivalence $d\leq\kappa(d)N$ by
working with $d$ directly, since in low horizontal dimensions the proof
requires a laplacian upper bound. It is also entirely possible that $\varrho$
has a subgradient which is bounded away from zero, in which case an unweighted
Hardy inequality for $\varrho$ with
$\left\lvert\nabla_{\mathbb{G}}\varrho\right\rvert^{2}$ obtained via Theorem
3.3 automatically has no degeneracy. Nonetheless the very general approach
here yields unweighted Hardy inequalities for every homogeneous norm, and in
particular for $d$, irrespective of their regularity.

One can continue the analysis on $H$-type groups where the explicit formulas
for the horizontal norm and Korányi-Folland gauge can be used to study some
other homogeneous (quasi)norms. Here are two such examples which to our best
knowledge are new in the realm of $H$-type groups.

######  Corollary 4.11.

Let $\mathbb{G}$ be a $H$-type group of horizontal dimension $n$ and let
$N_{LN}$ be the Lee-Naor norm [LN06, p. 5], defined as
$N_{LN}=(N^{2}+\left\lvert x\right\rvert^{2})^{1/2}$ where $N$ and
$\left\lvert x\right\rvert$ are respectively the Korányi-Folland gauge and
horizontal norm. Let $q=\frac{Q(\mathbb{G})+n}{2}$ and let $f\in
C_{c}^{\infty}(\mathbb{G}\setminus\\{0\\})$. Then

$\int_{\mathbb{G}}N_{LN}^{\alpha}\frac{\left\lvert\nabla_{\mathbb{G}}N_{LN}\right\rvert^{2}}{N_{LN}^{2}}f^{2}d\xi\leq\left(\frac{2}{q+\alpha-2}\right)^{2}\int_{\mathbb{G}}N_{LN}^{\alpha}\left\lvert\nabla_{\mathbb{G}}f\right\rvert^{2}d\xi$

holds for any $\alpha$ such that $q+\alpha>2$.

###### Proof.

By direct computation, it can be shown

$\left\lvert\nabla_{\mathbb{G}}N_{LN}\right\rvert^{2}=2\frac{\left\lvert
x\right\rvert^{2}}{N^{2}}$

and

$\Delta_{\mathbb{G}}N_{LN}=\frac{1}{N_{LN}}\left(Q(\mathbb{G})\frac{\left\lvert
x\right\rvert^{2}}{N^{2}}+n\right)-\frac{1}{N_{LN}}\left\lvert\nabla_{\mathbb{G}}N_{LN}\right\rvert^{2}$

from which it follows

$\frac{N_{LN}\Delta_{\mathbb{G}}N_{LN}}{\left\lvert\nabla_{\mathbb{G}}N_{LN}\right\rvert^{2}}=\frac{Q(\mathbb{G})}{2}+\frac{n}{2}\frac{N^{2}}{\left\lvert
x\right\rvert^{2}}-1\geq\frac{Q(\mathbb{G})+n}{2}-1,$

i.e. $N_{LN}$ verifies a laplacian lower bound with constant
$q=\frac{Q(\mathbb{G})+n}{2}$ for $\Delta_{\mathbb{G}}$. ∎

######  Corollary 4.12.

Let $\mathbb{G}$ be a $H$-type group of horizontal dimension $n$ and let
$\psi=\left\lvert x\right\rvert^{1/2}N^{1/2}$ be the geometric mean of the
Korányi-Folland gauge and the horizontal norm. Let
$q_{\min}=\min(\frac{n+Q(\mathbb{G})-2}{2},2n-3)+1$ and
$q_{\max}=\max(\frac{n+Q(\mathbb{G})-2}{2},2n-3)+1$, and let $f\in
C_{c}^{\infty}(\mathbb{G}\setminus\\{\left\lvert x\right\rvert=0\\})$. Then

$\int_{\mathbb{G}}\psi^{\alpha}\frac{\left\lvert\nabla_{\mathbb{G}}\psi\right\rvert^{2}}{\psi^{2}}f^{2}d\xi\leq\left(\frac{2}{q_{\min}+\alpha-2}\right)^{2}\int_{\mathbb{G}}\psi^{\alpha}\left\lvert\nabla_{\mathbb{G}}f\right\rvert^{2}d\xi$

holds for any $\alpha$ such that $q_{\min}+\alpha>2$ and

$\int_{\mathbb{G}}\psi^{\alpha}\frac{\left\lvert\nabla_{\mathbb{G}}\psi\right\rvert^{2}}{\psi^{2}}f^{2}d\xi\leq\left(\frac{2}{q_{\max}+\alpha-2}\right)^{2}\int_{\mathbb{G}}\psi^{\alpha}\left\lvert\nabla_{\mathbb{G}}f\right\rvert^{2}d\xi$

holds for any $\alpha$ such that $q_{\max}+\alpha<2$.

Note for this homogeneous quasinorm its subgradient is unbounded. In
particular, it blows up along the first stratum $\\{\left\lvert
x\right\rvert=0\\}$.

###### Proof.

Again, by direct computation, it can be shown

$\displaystyle
q_{\min}-1\leq\frac{\psi\Delta_{\mathbb{G}}\psi}{\left\lvert\nabla_{\mathbb{G}}\psi\right\rvert^{2}}$
$\displaystyle=\frac{(2n-3)N^{4}+(2Q(\mathbb{G})-1)\left\lvert
x\right\rvert^{4}}{N^{4}+3\left\lvert x\right\rvert^{4}}$
$\displaystyle=\frac{(2n+2Q(\mathbb{G})-4)\left\lvert
x\right\rvert^{4}+(2n-3)(N^{4}-\left\lvert x\right\rvert^{4})}{4\left\lvert
x\right\rvert^{4}+(N^{4}-\left\lvert x\right\rvert^{4})}\leq q_{\max}-1,$

i.e. $\psi$ verifies both a laplacian lower and upper bound respectively with
constant $Q=q_{\min}$ and $Q=q_{\max}$ for $\Delta_{\mathbb{G}}$. ∎

####  4.1.5. Hörmander and degenerate operators in general

Before we close this section, we remark the homogeneous Hardy inequality falls
under the purview of a more general result for hypoelliptic Hörmander
operators. This follows from [BB17, Theorem 1.4] which asserts if
$L=\sum_{i=1}^{\ell}X_{i}^{2}$ is a hypoelliptic Hörmander operator acting on
$\mathbb{R}^{m}$ which is $2$-homogeneous with respect to a family
$(\delta_{\lambda})_{\lambda>0}$ of dilations, then $L$ admits a fundamental
solution $u$. The proof uses and generalises the existence result for
stratified Lie groups by lifting $L$ to a sublaplacian $\Delta_{\mathbb{G}}$
on a stratified Lie group $\mathbb{G}$ and, as a result, this gives an
explicit formula for $u$ in terms of the fundamental solution for
$\Delta_{\mathbb{G}}$.

For $L=\Delta_{\mathbb{G}}$ the sublaplacian on a stratified Lie group and
with the notation developed in this chapter, the corresponding dilations are
given by (as seen in §4.1.3)

$\delta_{\lambda}(x_{0},\cdots,x_{r-1})=(\lambda
x_{0},\cdots,\lambda^{r}x_{r-1})$

and so $\Delta_{\mathbb{G}}$ is a special (but rather trivial) case of the
following corollary. Nonetheless this is a mild generalisation in that it
allows us to consider laplacian-like operators which are not a priori the
sublaplacian on a stratified Lie group but possess the same fundamental
structure. One may ask for a similar analogue of the horizontal Hardy
inequality but the point is that this result makes no assumption on whether
the underlying space is stratified and so the notion of the horizontal norm
outside of the realm of stratified Lie groups is somewhat ill-defined.

######  Corollary 4.13.

Let $L=\sum_{i=1}^{\ell}X_{i}^{2}$ be a hypoelliptic Hörmander operator acting
on $\mathbb{R}^{m}$ which is $2$-homogeneous with respect to a family
$(\delta_{\lambda})_{\lambda>0}$ of nonisotropic dilations

(4.5)
$\delta_{\lambda}(x_{1},\cdots,x_{m})=(\lambda^{\sigma_{1}}x_{1},\cdots,\lambda^{\sigma_{m}}x_{m}),\quad
1=\sigma_{1}\leq\sigma_{2}\leq\cdots\leq\sigma_{m},\quad\sigma_{i}\in\mathbb{Z}_{>0}$

with $q=\sum_{i=1}^{m}\sigma_{i}>2$. Denote by $u$ its fundamental solution
and set $N=u^{1/(2-q)}$. Let $f\in
C_{c}^{\infty}(\mathbb{R}^{m}\setminus\\{0\\})$. Then

$\int_{\mathbb{R}^{m}}N^{\alpha}\frac{\left\lvert\nabla_{L}N\right\rvert^{2}}{N^{2}}f^{2}dx\leq\left(\frac{2}{q+\alpha-2}\right)^{2}\int_{\mathbb{R}^{m}}N^{\alpha}\left\lvert\nabla_{L}f\right\rvert^{2}dx$

holds for any $\alpha$ such that $q+\alpha\neq 2$, and

$\int_{\mathbb{R}^{m}}N^{2-q}\frac{\left\lvert\nabla_{L}N\right\rvert^{2}}{N^{2}\log^{2}N}f^{2}dx\leq
4\int_{\mathbb{R}^{m}}N^{2-q}\left\lvert\nabla_{L}f\right\rvert^{2}dx,$

where $\nabla_{L}=(X_{1},\cdots,X_{\ell})$.

###### Proof.

$N$ verifies (3.8) with constant $Q=q$ for $L$ by following the proof of
Corollary 4.2. The carré du champ and reversible measure as before are
respectively $\Gamma(f)=\left\lvert\nabla_{L}f\right\rvert^{2}$ and Lebesgue
measure $dx$ on $\mathbb{R}^{m}$. ∎

There are in fact degenerate operators appearing in the literature which are
$2$-homogeneous with respect to a family of dilations with $\sigma_{i}$
possibly nonintegral and that which may not be hypoelliptic. One important
operator belonging to this class is the Grushin operator

$\Delta_{\gamma}=\Delta_{x}+\left\lvert
x\right\rvert^{2\gamma}\Delta_{\gamma}=\nabla_{\gamma}\cdot\nabla_{\gamma},\quad\gamma\geq
0$

on $\mathbb{R}_{x}^{m}\times\mathbb{R}_{y}^{k}$ for
$\nabla_{\gamma}=(\nabla_{x},\left\lvert x\right\rvert^{\gamma}\nabla_{y})$,
and another is the Heisenberg-Greiner operator

$\Delta_{\delta}=\sum_{i=1}^{m}X_{i}^{2}+Y_{i}^{2}=\nabla_{\delta}\cdot\nabla_{\delta},\quad\delta\geq
1$

on
$\mathbb{H}^{m}\cong\mathbb{R}_{x}^{m}\times\mathbb{R}_{y}^{m}\times\mathbb{R}_{z}^{1}$
where $X_{i}=\partial_{x_{i}}+2\delta y_{i}r^{2\delta-2}\partial_{z}$ and
$Y_{i}=\partial_{y_{i}}-2\delta x_{i}r^{2\delta-2}\partial_{z}$ for $r$ the
horizontal norm and $\nabla_{\delta}=(X_{1},Y_{1},\cdots,X_{m},Y_{m})$.

Although these two operators for general $\gamma$ and $\delta$ do not fall
under the scope of the previous corollary, they nonetheless exhibit many of
the same properties. This may be somewhat unsurprising given the Grushin and
Heisenberg-Greiner operators respectively generalise the euclidean laplacian
when $\gamma=0$ and the Heisenberg sublaplacian when $\delta=1$. For instance,
both the Grushin and Heisenberg-Greiner operators are respectively
$2$-homogeneous with respect to the dilations $\delta_{\lambda}(x,y)=(\lambda
x,\lambda^{1+\gamma}y)$ and $\delta_{\lambda}(x,y,z)=(\lambda x,\lambda
y,\lambda^{2\delta}z)$, and in this way the underlying space
$\mathbb{R}_{x}^{m}\times\mathbb{R}_{y}^{k}$ on which the Grushin operator
acts may also be regarded as being stratified. Moreover, they are hypoelliptic
when $\gamma\in 2\mathbb{Z}_{\geq 0}$ and $\delta\in\mathbb{Z}_{\geq 1}$.

The main point is that $\Delta_{\gamma}$ and $\Delta_{\delta}$ are very
similar to sublaplacians on a stratified Lie group and indeed their associated
Hardy inequalities first proven in [D’A05, §3] are essentially identical to
those presented earlier in the chapter; the above “stratification” induces a
“horizontal norm” on $\mathbb{R}_{x}^{m}\times\mathbb{R}_{y}^{k}$ for the
Grushin operator (whereas the Heisenberg-Greiner operator enjoys the
stratification of $\mathbb{H}^{m}$) and there exists an analogue of the
Korányi-Folland gauge $N$ in both cases, that is a function that is a suitable
power of the fundamental solution. Moreover, there is a corresponding analogue
of $Q$. One can think of the Grushin and Heisenberg-Greiner operators as
respectively having possibly nonintegral homogeneous dimension
$Q(m,k,\gamma)=m+(1+\gamma)k$ and $Q(m,\delta)=2m+2\delta$, that is the second
stratum $\mathbb{R}_{y}^{k}$ and $\mathbb{R}_{z}^{1}$ as respectively having
“dimension” $(1+\gamma)k$ and $2\delta$.

###  4.2. Hyperbolic space

In this brief section, we shall present an interesting example of a Hardy
inequality on a manifold. In fact, although the setting of a stratified Lie
group can be regarded as a subriemannian manifold, the analysis in the
previous section requires no explicit description of a metric or the geometry
of the underlying space, which is instead expressed through the laplacian
comparison (3.8).

The reader may notice the constant $Q$ representing homogeneous dimension has
so far always been nonnegative, at least in the “usual” setting. For instance,
if $\alpha$ is sufficiently large and negative in the context of the weighted
Hardy inequalities (3.11), then certainly $Q+\alpha$ can be made negative. Or,
if $\gamma$ and $\delta$ is sufficiently large and negative, the homogeneous
dimensions $Q(m,k,\gamma)$ and $Q(m,\delta)$ associated respectively to the
Grushin and Heisenberg-Greiner operators can be made negative also. One may
however think this is rather contrived and whether the case $Q\leq 0$ may
arise more “naturally”.

Consider the realisation of the $m$-dimensional hyperbolic space $H^{m}$ as
the Poincaré (upper) half-plane model
$H^{m}\cong\mathbb{R}^{m-1}\times(0,\infty)\subset\mathbb{R}^{m}$ for $m\geq
2$. Here the hyperbolic laplacian $\Delta_{H^{m}}$ has carré du champ
$\Gamma_{H^{m}}(f)=x_{m}^{2}\left\lvert\nabla f\right\rvert^{2}$ where
$\nabla$ is the euclidean gradient, and reversible measure
$d\mu(x)=x_{m}^{-m}dx$ where $dx$ is Lebesgue measure on $\mathbb{R}^{m}$. We
refer the reader to [BGL+14, S4.10] for more details on the hyperbolic
laplacian and where the following Hardy inequality is referred to as a
Poincaré inequality [BGL+14, Corollary 4.10.2].

######  Corollary 4.14.

Let $m\geq 2$ and $f\in C_{c}^{\infty}(H^{m})$. Then

$\int_{H^{m}}f^{2}d\mu\leq\frac{4}{(m-1)^{2}}\int_{H^{m}}\Gamma_{H^{m}}(f)d\mu.$

###### Proof.

The coordinate function $x_{m}$ satisfies $\Delta_{H^{m}}x_{m}=-(m-2)x_{m}$
and $\Gamma_{H^{m}}(x_{m})=x_{m}^{2}$, so that

$\frac{x_{m}\Delta_{H^{m}}x_{m}}{\Gamma_{H^{m}}(x_{m})}=2-m=(3-m)-1,$

i.e. $x_{m}$ verifies (3.8) with constant $Q=3-m\in\mathbb{Z}_{\leq 1}$ for
$\Delta_{H^{m}}$. Then

$\displaystyle\int_{H^{m}}\frac{\Gamma_{H^{m}}(x_{m})}{x_{m}^{2}}f^{2}d\mu=\int_{H^{m}}f^{2}d\mu$
$\displaystyle\leq\frac{4}{(m-1)^{2}}\int_{H^{m}}\Gamma_{H^{m}}(f)d\mu$
$\displaystyle=\left(\frac{2}{Q-2}\right)^{2}\int_{H^{m}}\Gamma_{H^{m}}(f)d\mu.\qed$

###  4.3. Rellich inequalities

There is a proof of the euclidean Rellich inequality starting from the
euclidean Hardy inequality which relies only on integration by parts and
Young’s inequality. We can generalise to the Bakry-Émery setting and obtain a
Rellich inequality with $\left\lvert\nabla\psi\right\rvert^{2}$ replaced by
$\Gamma(\psi)$ and weights $\psi^{\alpha}$ appearing on both sides. Note
although the following proof of the Rellich inequality will not require any
semigroup theory, it may be possible, since
$\int\Gamma_{2}(f)d\mu=\int(Lf)^{2}d\mu$, to prove a Rellich inequality
starting from a $\Gamma_{2}$-calculus argument. However, since this is likely
to fail in the subelliptic setting where $\Gamma_{2}\geq 0$ fails to hold, we
choose to instead recover $(Lf)^{2}$ through integration by parts and Young’s
inequality.

###### Proof.

Recall the weighted Hardy inequality (3.11)

$\int\psi^{\alpha}\frac{\Gamma(\psi)}{\psi^{2}}f^{2}d\mu\leq
c(Q,\alpha)\int\psi^{\alpha}\Gamma(f)d\mu$

where $c(Q,\alpha)=(\frac{2}{Q+\alpha-2})^{2}$ for brevity. Replacing $f$ with
$f/\psi$, we find

$\psi^{\alpha}\Gamma(f\psi^{-1})=\psi^{\alpha}(\Gamma(f)\psi^{-2}+2f\psi^{-1}\Gamma(f,\psi^{-1})+f^{2}\Gamma(\psi^{-1})).$

The third addend is $\psi^{\alpha}\Gamma(\psi)\psi^{-4}f^{2}$, which can be
carried over to the left hand side with constant $c(Q,\alpha)$. The first two
addends are

$\displaystyle\psi^{\alpha}(\Gamma(f)\psi^{-2}+2f\psi^{-1}\Gamma(f,\psi^{-1})$
$\displaystyle=\psi^{\alpha}\Gamma(f,f\psi^{-2})$
$\displaystyle=\Gamma(f,f\psi^{\alpha-2})-\alpha
f\psi^{\alpha-3}\Gamma(f,\psi)$
$\displaystyle=\Gamma(f,f\psi^{\alpha-2})-D(\alpha)\Gamma(f^{2},\psi^{\alpha-2})$

where $D(\alpha)=\frac{\alpha}{2(\alpha-2)}$. The first addend, after
integration by parts (2.8), is

$\displaystyle\int\Gamma(f,f\psi^{\alpha-2})d\mu$
$\displaystyle=-\int(Lf)f\psi^{\alpha-2}d\mu$
$\displaystyle=-\int\frac{(Lf)}{\Gamma(\psi)^{1/2}}\psi^{\alpha-2}\Gamma(\psi)^{1/2}d\mu$
$\displaystyle\leq\frac{1}{2\eta}\int\psi^{\alpha}\frac{(Lf)^{2}}{\Gamma(\psi)}+\frac{\eta}{2}\int\psi^{\alpha}\frac{\Gamma(\psi)}{\psi^{4}}f^{2}d\mu$

by Young’s inequality and writing
$\psi^{\alpha-2}=\psi^{\alpha/2}\psi^{\alpha/2-2}$.

The second addend, again after integration by parts, is

(4.6) $\displaystyle-\int D(\alpha)\Gamma(f^{2},\psi^{\alpha-2})d\mu$
$\displaystyle=D(\alpha)\int L(\psi^{\alpha-2})f^{2}d\mu$
$\displaystyle=D(\alpha)(\alpha-2)(Q+\alpha-4)\int\psi^{\alpha}\frac{\Gamma(\psi)}{\psi^{4}}f^{2}d\mu$

by the chain rule (2.4). After optimising the constant $\eta$ in terms of $Q$
and $\alpha$ we find the Rellich inequality

$\int\psi^{\alpha}\frac{\Gamma(\psi)}{\psi^{2}}f^{2}d\mu\leq\left(\frac{4}{(Q-\alpha)(Q+\alpha-4)}\right)^{2}\int\psi^{\alpha}\frac{(Lf)^{2}}{\Gamma(\psi)}d\mu.$

Note the case $\alpha=2$ is special since $D(\alpha)=\infty$, although it
would appear from the Rellich constant that it should pose no immediate
problem, at least not independently of $Q$. The problem lies with the term
$-\alpha
f\psi^{\alpha-3}\Gamma(f,\psi)\rvert_{\alpha=2}=-2f\psi^{-1}\Gamma(f,\psi)=-\Gamma(f^{2},\log\psi)$,
since the integration by parts here should actually read

(4.7) $-\int\Gamma(f^{2},\log\psi)d\mu=\int
L(\log\psi)f^{2}d\mu=(Q-2)\int\psi^{2}\frac{\Gamma(\psi)}{\psi^{4}}f^{2}d\mu.$

Nonetheless the argument works in exactly the same way since the apparent
singularity in $D(\alpha)(\alpha-2)(Q+\alpha-4)$ is removable, and at
$\alpha=2$ it equals $Q-2$. Lastly, we just need to verify $\eta>0$. The
solution of $\eta$ we obtain is

$\eta=\frac{(Q-\alpha)(Q+\alpha-4)}{4}$

so the argument works for any $\alpha$ for which $(Q-\alpha)(Q+\alpha-4)>0$. ∎

######  Theorem 4.5.

Let $\psi\geq 0$ verify (3.8) with constant $Q\in\mathbb{R}$, and let $f\in
C_{c}^{\infty}(X\setminus\\{\psi=0\\})$. Then

(4.8)
$\int\psi^{\alpha}\frac{\Gamma(\psi)}{\psi^{4}}f^{2}d\mu\leq\left(\frac{4}{(Q-\alpha)(Q+\alpha-4)}\right)^{2}\int\psi^{\alpha}\frac{(Lf)^{2}}{\Gamma(\psi)}d\mu$

holds for any $\alpha$ such that $Q+\alpha\neq 2$ and
$(Q-\alpha)(Q+\alpha-4)>0$.

In the euclidean setting, this inequality first appeared in [Rel54, p. 247-9]
for $\alpha=0$, was later generalised by [DH98, Theorem 12] to allow for
weights $\psi^{\alpha}$, and then generalised again by [Kom06, Theorem 5.1] to
the subelliptic setting. Moreover, as one might expect (or hope), it is
possible to obtain a generalisation in the case where only a one-sided
laplacian comparison is available.

######  Corollary 4.15.

Let $\psi\geq 0$ verify (3.8) from below with constant $Q\in\mathbb{R}$, and
let $f\in C_{c}^{\infty}(X\setminus\\{\psi=0\\})$. Then (4.8) holds if
additionally $Q+\alpha>2$, and $\alpha\leq 0$.

######  Corollary 4.16.

Let $\psi\geq 0$ verify (3.8) from above with constant $Q\in\mathbb{R}$, and
let $f\in C_{c}^{\infty}(X\setminus\\{\psi=0\\})$. Then (4.8) holds if
additionally $Q+\alpha<2$ and $\alpha\geq 0$.

###### Proof.

Let us handle the case of the lower bound. The proof is identical, but
requirements on $\alpha$ enter in the following way. The first is that
$Q+\alpha>2$ since our starting point is the weighted Hardy inequality of
Corollary 3.2. The second relates to the behaviour of (4.6); no longer do we
have equality but we seek an upper bound on $D(\alpha)L(\psi^{\alpha-2})$. If
$\alpha=2$ then the integration by parts follows (4.7) and a lower bound on
$L\psi$ furnishes a lower bound on $L\log\psi$, not an upper bound. On the
other hand, if $\alpha\neq 2$ then $D(\alpha)L(\psi^{\alpha-2})\leq
D(\alpha)(\alpha-2)(Q+\alpha-4)\psi^{\alpha-4}\Gamma(\psi)$ if
$D(\alpha)(\alpha-2)=\frac{\alpha}{2}\leq 0$, that is if $\alpha\leq 0$. The
case of the upper bound is exactly opposite. ∎

Note unlike in the setting of Hardy inequalities that introducing weights into
the Rellich inequality does not follow from the replacement of $L$ with
$L_{\omega}$ or $L_{\sigma}$ for $\omega=e^{\sigma}$ as was done in the proof
of Corollary 3.1, since $L$ itself appears on the right hand side. That is,
unlike for Hardy inequalities, the dependence of the Rellich inequality on the
operator and not just the carré du champ and reversible measure alone implies
a weighted Rellich inequality for $L$ is _not_ the same as an unweighted
Rellich inequality for $L_{\omega}$ or $L_{\sigma}$. This also explains why a
Rellich inequality for $L_{\omega}$ is not equivalent to a Rellich inequality
for $L_{\sigma}$, namely the appearance of $\Gamma(\psi)$ in the denominator
on the right hand side implies a weight on the carré du champ is not
equivalent to a weight on the reversible measure.

###  4.4. Weak Hardy inequalities

The diffusions $L$ studied in the previous chapter were all laplacian-like,
sums of squares type, operators whose reversible measure $\mu$ was always
either the Lebesgue measure on $\mathbb{R}^{m}$ or the riemannian volume form
on $(M,g)$. One generic way in which a reversible _probability_ measure
manifests follows an earlier remark on the proof of Corollary 3.1, namely the
operator $L_{\sigma}=L+\Gamma(\sigma,-)$, for $\sigma$ a smooth real-valued
function, has the same carré du champ $\Gamma_{\sigma}=\Gamma$ but has a
weighted reversible measure $d\mu_{\sigma}=e^{\sigma}d\mu$. In particular, if
$e^{\sigma}\in L^{1}(\mu)$ then $\mu_{\sigma}$ is a probability.

Operators of the form $L_{\sigma}=L+\Gamma(\sigma,-)$ are typically called
Kolmogorov type operators in the literature, and $\Gamma(\sigma,-)$ is called
a “gradient drift”. As a standard example, the euclidean Ornstein-Uhlenbeck
semigroup on $\mathbb{R}^{m}$ corresponds to the case $L=\Delta$ and
$\sigma=-\frac{1}{2}\left\lvert x\right\rvert^{2}$, in which case
$L_{\sigma}=\Delta-x\cdot\nabla$ and $\mu_{\sigma}$ is the standard gaussian
measure on $\mathbb{R}^{m}$. In the euclidean probability setting, the
“correct” Hardy inequality reads

(4.9) $\int_{\mathbb{R}^{m}}\frac{f^{2}}{\left\lvert
x\right\rvert^{2}}d\mu\leq C_{0}\int_{\mathbb{R}^{m}}\left\lvert\nabla
f\right\rvert^{2}d\mu+C_{1}\int_{\mathbb{R}^{m}}f^{2}d\mu$

where $C_{0},C_{1}>0$. Such an inequality is called a weak Hardy inequality in
[Dav99] but is also called a weighted Hardy inequality in, for instance,
[Can+19]. Nonetheless we shall adopt the convention of the former to
distinguish from the weighted Hardy inequality (3.11).

We will be interested in generalising (4.9) to the Bakry-Émery setting by
replacing the singularity $\left\lvert x\right\rvert^{-2}$ with
$\Gamma(\psi)\psi^{-2}$ where $\psi$ verifies (3.8) with constant $Q\neq 2$,
and replacing the euclidean gradient $\left\lvert\nabla f\right\rvert^{2}$
with the carré du champ $\Gamma(f)$. That is, the desired analogue of (4.9)
reads

(4.10) $\int\frac{\Gamma(\psi)}{\psi^{2}}f^{2}d\mu\leq
C_{0}\int\Gamma(f)d\mu+C_{1}\int f^{2}d\mu.$

[Can+19, Theorem 1.2] recently proved the inequality (4.9) with
$C_{0}=(\frac{2}{m-2})^{2}$ the optimal constant appearing in the usual
euclidean Hardy inequality on $\mathbb{R}^{m}$ for $m\geq 3$ and a large class
of probabilities $\mu$ which, in particular, included the family
$d\mu(x)=e^{-\alpha\left\lvert x\right\rvert^{\beta}}dx$ for any
$\alpha,\beta>0$. Expressed in the language of Kolmogorov type operators, this
is a weak Hardy inequality for the family $\Delta-\alpha\nabla\left\lvert
x\right\rvert^{\beta}\cdot\nabla$, so a standard example we will be interested
in studying is the analogous family
$L_{\alpha,\beta}=L-\alpha\Gamma(\psi^{\beta},-)$ where $L$ is one of the
laplacian-like operators studied in this chapter and $\Gamma$ is its carré du
champ.

From the view of the functional inequality (3.5) it is tempting to consider
again the multiplier $W=\psi^{p}$ for $p=\frac{Q-2}{2}$, but the introduction
of the gradient drift term $-\alpha\Gamma(\psi^{\beta},-)$ means the
sufficient condition (3.3) reads

$\frac{L_{\alpha,\beta}W}{W}-3\frac{\Gamma(W)}{W^{2}}=\frac{LW}{W}-3\frac{\Gamma(W)}{W^{2}}-\frac{\alpha\Gamma(\psi^{\beta},W)}{W}=-\alpha\beta
p\Gamma(\psi)\psi^{\beta-2}$

since the original terms $\frac{LW}{W}-3\frac{\Gamma(W)}{W^{2}}$ vanish for
the choice $W=\psi^{(Q-2)/2}$ by Lemma 3.2 and $\psi$ verifying (3.8) for $L$.
But obviously if $\alpha,\beta>0$ and, in the generic high dimensional case,
$p>0$, then $-\alpha\beta p\Gamma(\psi)\psi^{\beta-2}$ will typically blow up
(along level sets of $\psi$) either at the origin if $\beta<2$, or at infinity
if $\beta>2$. In such a situation one cannot verify (3.3) for any
$\gamma\in\mathbb{R}$. Thankfully, we can sidestep the need to verify (3.3),
since throughout this paper we have never been interested in the pointwise
subcommutation (3.1), but always the integrated subcommutation (3.4).

######  Lemma 4.1.

Let $\psi\geq 0$ verify (3.8) with constant $Q\neq 2$. Let
$L_{\sigma}=L+\Gamma(\sigma,-)$ where $\sigma$ is smooth, and let $f\in
C_{c}^{\infty}(X\setminus\\{\psi=0\\})$. Then

(4.11)
$\int\frac{\Gamma(\psi)}{\psi^{2}}f^{2}d\mu_{\sigma}\leq\left(\frac{2}{Q-2}\right)^{2}\int\Gamma(f)d\mu_{\sigma}-(Q-2)\int\frac{\Gamma(\sigma,\psi)}{\psi}f^{2}d\mu_{\sigma}.$

###### Proof.

If $\psi\geq 0$ verifies (3.8) with constant $Q\neq 2$, then $W=\psi^{p}$ for
$p=\frac{Q-2}{2}$ verifies (3.3) with constant $\gamma=0$ so
$\Gamma^{W}_{L}(f)\geq 0$ where $\Gamma^{W}_{L}$ is the $\Gamma^{W}$-calculus
for $L$, not $L_{\sigma}$. But

$\displaystyle 2\Gamma^{W}_{L_{\sigma}}(f)$
$\displaystyle=L_{\sigma}(W^{2})f^{2}+2\Gamma_{\sigma}(W^{2},f^{2})+2W^{2}\Gamma_{\sigma}(f)$
$\displaystyle=(L(W^{2})f^{2}+2\Gamma(W^{2},f^{2})+2W^{2}\Gamma(f))+\Gamma(\sigma,W^{2})f^{2}$
$\displaystyle\geq\Gamma(\sigma,W^{2})f^{2}$

since $\Gamma_{\sigma}=\Gamma_{0}$. Integrating both sides with respect to
$\mu_{\sigma}$, we find

$-\int L_{\sigma}(W^{2})f^{2}d\mu_{\sigma}+2\int
W^{2}\Gamma_{\sigma}(f)d\mu_{\sigma}\geq\int\Gamma(\sigma,W^{2})f^{2}d\mu_{\sigma}$

which implies

$\int L(W^{2})f^{2}d\mu_{\sigma}\leq 2\int
W^{2}\Gamma(f)d\mu_{\sigma}-2\int\Gamma(\sigma,W^{2})f^{2}d\mu_{\sigma}.$

This is just the starting point of Lemma 3.1 and repeating the same
calculations we recover the conclusion of Theorem 3.3 with the additional term

$-\int\frac{\Gamma(\sigma,W^{2})}{W^{2}}f^{2}d\mu=-(Q-2)\int\frac{\Gamma(\sigma,\psi)}{\psi}f^{2}d\mu_{\sigma}.\qed$

######  Remark 16.

In fact, this is also a second proof of Theorem 3.3; originally it was proved
by differentiating (3.4), whereas here it has been proved by integrating
(3.2).

Now although the lemma relaxes the requirement that
$\frac{\Gamma(\sigma,W^{2})}{W^{2}}$ be _pointwise_ bounded below everywhere,
even in the simple setting where $\sigma=-\alpha\psi^{\beta}$ one has

$-(Q-2)\frac{\Gamma(\sigma,\psi)}{\psi}=\alpha\beta(Q-2)\Gamma(\psi)\psi^{\beta-2},$

that is there is a singularity both at the origin and at infinity depending on
$\beta$. But since this weight is tested against the measure
$d\mu_{\sigma}=e^{-\alpha\psi^{\beta}}d\mu$, the singularity at infinity can
be controlled provided $\Gamma(\psi)\psi^{-2}$ grows slowly enough at
infinity. The standard examples, for instance, have bounded carré du champs
and therefore $\Gamma(\psi)\psi^{-2}$ decays at infinity.

######  Lemma 4.2.

Let $\psi\geq 0$ verify (3.8) with constant $Q>2$. Let
$L_{\sigma}=L+\Gamma(\sigma,-)$ where $\sigma$ is smooth. Suppose also there
exists $R>0$ for which the density $e^{\sigma}\Gamma(\psi)\psi^{-2}$ is
bounded in $\\{\psi\geq R\\}$, and let $f\in
C_{c}^{\infty}(X\setminus\\{\psi=0\\})$. Then

$\int\frac{\Gamma(\psi)}{\psi^{2}}f^{2}d\mu_{\sigma}\leq\left(\frac{2}{Q-2}\right)^{2}\int\Gamma(f,f)d\mu_{\sigma}-(Q-2)\int\frac{\Gamma(\sigma,\psi)}{\psi}f^{2}\chi
d\mu_{\sigma}+C\int f^{2}d\mu_{\sigma}$

for $C>0$ a constant and $\chi=\mathbbm{1}\\{\psi\leq R\\}$.

###### Proof.

First, note the idea of replacing $f$ with $f(\chi_{0}+\chi_{\infty})$ where
$\\{\chi_{0},\chi_{\infty}\\}$ is a smooth partition of unity localised at
zero and infinity fails in the sense it produces a worse constant. Instead we
proceed as follows. Form the decomposition

$\int\frac{\Gamma(\psi)}{\psi^{2}}f^{2}d\mu_{\sigma}=\int_{B(R)}\frac{\Gamma(\psi)}{\psi^{2}}f^{2}d\mu_{\sigma}+\int_{B(R)^{c}}\frac{\Gamma(\psi)}{\psi^{2}}f^{2}d\mu_{\sigma}$

where $B(R)=\\{\psi\leq R\\}$. The second term is controlled by $\int
f^{2}d\mu_{\sigma}$ by the assumed boundedness of
$e^{\sigma}\Gamma(\psi)\psi^{-2}$ in $B(R)^{c}$. To control the first term, we
replace $W$ with

$V=\begin{cases}\psi^{(Q-2)/2},&\psi\leq R,\\\ 1,&\psi\geq 2R\end{cases}$

and with $V$ smooth and $\psi$-radial in $\\{R\leq\psi\leq 2R\\}$. Then $V$
verifies (3.3) with some $\gamma<0$ since the quantity
$\eta\vcentcolon=\frac{LV}{V}-3\frac{\Gamma(V)}{V}$ vanishes away from
$\\{R\leq\psi\leq 2R\\}$ and is smooth in between. By Lemma 3.1,

$\int\eta f^{2}d\mu_{\sigma}\leq
2\int\Gamma(f)d\mu_{\sigma}-(Q-2)\int\frac{\Gamma(\sigma,V^{2})}{V^{2}}f^{2}d\mu_{\sigma}-\gamma\int
f^{2}d\mu_{\sigma}.$

Since $V\equiv W$ for $\psi\leq R$ and everything is otherwise smooth and
bounded in $\\{R\leq\psi\leq 2R\\}$, it follows there is a constant $C$ for
which

$\int_{B(R)}\frac{\Gamma(\psi)}{\psi^{2}}f^{2}d\mu_{\sigma}\leq\left(\frac{2}{Q-2}\right)^{2}\int\Gamma(f)d\mu_{\sigma}-(Q-2)\int\frac{\Gamma(\sigma,\psi)}{\psi}f^{2}\chi
d\mu_{\sigma}+C\int f^{2}d\mu_{\sigma}.\qed$

Presently, this appears to be as far as we can go; the rest depends on the
behaviour of $\Gamma(\sigma,\psi)\psi^{-1}$ near the origin. If
$\sigma=-\alpha\psi^{\beta}$ then
$\Gamma(\sigma,\psi)\psi^{-1}=-\alpha\beta\Gamma(\psi)\psi^{\beta-2}$ is
bounded near the origin for $\beta\geq 2$ and blows up there for $\beta<2$.
That said, in the subquadratic case we can always estimate
$\psi^{\beta-2}\leq\varepsilon\psi^{-2}+K(\varepsilon)$ near the origin for
any $\varepsilon>0$. It follows we can recover a “near optimal” weak Hardy
inequality, although the price we pay is the constant $C_{1}$ on the defective
term $\int f^{2}d\mu$ must be large.

######  Corollary 4.17.

Let $\psi\geq 0$ verify (3.8) with constant $Q\neq 2$. Let
$L_{\alpha,\beta}=L-\alpha\Gamma(\psi^{\beta},-)$ for any $\alpha,\beta>0$, in
which case $L_{\alpha,\beta}$ has reversible measure
$d\mu_{\alpha,\beta}=e^{-\alpha\psi^{\beta}}d\mu$. Suppose also
$e^{-\alpha\psi^{\beta}}\Gamma(\psi)\psi^{-2}$ is bounded for $\psi$
sufficiently large, and let $f\in C_{c}^{\infty}(X\setminus\\{\psi=0\\})$.
Then there are two cases:

  1. (1)

if $\beta\geq 2$ then there exists a constant $C_{\beta}>0$ for which

$\int\frac{\Gamma(\psi)}{\psi^{2}}f^{2}d\mu_{\alpha,\beta}\leq\left(\frac{2}{Q-2}\right)^{2}\Gamma(f)d\mu_{\alpha,\beta}+C_{q}\int
f^{2}d\mu_{\alpha,\beta}$

and

  2. (2)

if $0<\beta<2$ and $\varepsilon>0$ then there exists a constant
$C_{\beta,\varepsilon}>0$ for which

$\int\frac{\Gamma(\psi)}{\psi^{2}}f^{2}d\mu_{\alpha,\beta}\leq(1+\varepsilon)\left(\frac{2}{Q-2}\right)^{2}\Gamma(f)d\mu_{\alpha,\beta}+C_{q,\varepsilon}\int
f^{2}d\mu_{\alpha,\beta}.$

The conclusion here coincides with [Can+19, Theorem 3.3] in the euclidean
setting. To obtain the weak Hardy inequality with the optimal constant and
without $C_{1}=C_{q,\varepsilon}$ blowing up in the subquadratic case, it
appears one needs a more precise Hardy inequality (see [Can+19, Theorem 3.5])
with remainder which, to the best of our knowledge, is only known to hold in
the euclidean setting and in the subelliptic setting for the Heisenberg group
and the horizontal norm [Xia11, Theorem 1.1]. This would obtain the optimal
weak Hardy inequality (4.10) for $\psi$ the horizontal norm, but such a
choice, although the argument still works, does not give a probability $\mu$.

######  Acknowledgements.

We would like to thank the warm advice and encouragement of Professor Boguslaw
Zegarliński. The author is supported by the President’s Ph.D. Scholarship of
Imperial College London.

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