Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 14-Day Trial for You or Your Team.

Learn More →

Local and non‐local Poincaré inequalities on Lie groups

Local and non‐local Poincaré inequalities on Lie groups INTRODUCTIONThe aim of this paper is to establish two forms of Poincaré inequality on non‐compact connected Lie groups. On the one hand, we shall obtain the Lie group analogue of the classical inequality on Rd$\mathbb {R}^d$1.1∥f−fB∥Lp(B)⩽Cr∥∇f∥Lp(B),$$\begin{equation} \Vert f -f_B\Vert _{L^p(B)} \leqslant C r \Vert \nabla f\Vert _{L^p(B)}, \end{equation}$$where p∈[1,∞)$p\in [1,\infty )$, f∈C∞(Rd)$f\in C^\infty (\mathbb {R}^d)$, B$B$ is a ball of radius r$r$ and fB$f_B$ is the average of f$f$ on B$B$. On the other hand, we shall consider a non‐local L2$L^2$‐version of such an inequality, which takes the form1.2∥f∥L2(Rd,μ)⩽C∥∇f∥L2(Rd,μ)$$\begin{equation} \Vert f \Vert _{L^2(\mathbb {R}^d\!,\,\mu )} \leqslant C \Vert \nabla f\Vert _{L^2(\mathbb {R}^d\!,\,\mu )} \end{equation}$$for certain finite measures μ$\mu$ which are absolutely continuous with respect to the Lebesgue measure and whose densities satisfy a certain decay condition at infinity. One should think, for example, to the case when μ$\mu$ is a Gaussian measure.Extensions of the classical Poincaré inequality (1.1) to non‐Euclidean settings have widely been studied in the last decades. A thorough overview of the literature would go out of the scope of the present paper, so we refer the reader to the milestone [11] and the references therein. For what concerns Lie groups, a Poincaré inequality on unimodular groups can be obtained by combining [16, §8.3] and [11, Theorem 9.7]. In this paper we prove that a Poincaré inequality also holds on non‐unimodular Lie groups endowed with a relatively invariant measure, and we also describe the behaviour of the Poincaré constant in a quantitative way. We show that this grows at most exponentially with respect to the radius of the ball, and that if the group is non‐doubling, then such growth is, in general, exponential. More precisely, in a class of Lie groups including the real hyperbolic spaces as a subclass, we estimate from below the constant involved in the Poincaré inequality by a quantity which grows exponentially with respect to the radius of the ball.Non‐local inequalities such as (1.2) have been introduced more recently [2] on Rd$\mathbb {R}^d$, for densities satisfying suitable differential inequalities expressed in terms of the Laplacian Δ$\Delta$ (cf. [2, Corollary 1.6]) whose prototype is a Gaussian function. After its establishment on Rd$\mathbb {R}^d$, such non‐local inequalities were extended to unimodular Lie groups of polynomial growth in [15], where a sum‐of‐squares subelliptic sub‐Laplacian plays the role of Δ$\Delta$. In this paper, we extend their method to the non‐doubling regime where the sub‐Laplacian has an additional drift term, in a setting where we previously studied various function spaces [4–6] and the Sobolev and Moser–Trudinger inequalities [7].As a classical application of the local Poincaré inequality, we show the so‐called local parabolic Harnack principle for the sub‐Laplacian with drift. Another application of our inequality is given in [3] to the study of spectral properties of Schrödinger operators on Lie groups.SETTING AND PRELIMINARIESLet G$G$ be a non‐compact connected Lie group with identity e$e$. We denote by ρ$\rho$ a right Haar measure, by χ$\chi$ a continuous positive character of G$G$ and by μχ$\mu _\chi$ the measure with density χ$\chi$ with respect to ρ$\rho$. As the modular function on G$G$, which we denote by δ$\delta$, is such a character, μδ$\mu _\delta$ is a left Haar measure on G$G$. We denote it by λ$\lambda$. Observe also that μ1=ρ$\mu _1=\rho$.Let X={X1,⋯,Xℓ}$\mathbf {X}= \lbrace X_1,\dots , X_\ell \rbrace$ be a family of left‐invariant linearly independent vector fields which satisfy Hörmander's condition. Let dC(·,·)$d_C(\, \cdot \, ,\, \cdot \, )$ be its associated left‐invariant Carnot–Carathéodory distance. We let |x|=dC(x,e)$|x|=d_C(x,e)$, and denote by Br$B_r$ the ball centred at e$e$ of radius r$r$. The measure of Br$B_r$ with respect to ρ$\rho$ will be denoted by V(r)=ρ(Br)$V(r)=\rho (B_r)$; we recall that V(r)=λ(Br)$V(r)= \lambda (B_r)$. It is well known, cf. [10, 19], that there exist d∈N∗$d\in \mathbb {N}^*$ depending on G$G$ and X${\bf X}$, and C>0$C&gt;0$, such that2.1C−1rd⩽V(r)⩽Crd∀r∈(0,1],$$\begin{equation} C^{-1} r^d \leqslant V(r) \leqslant C r^d\qquad \forall r\in (0,1], \end{equation}$$and D0,D>0$D_0,D&gt;0$ depending only on G$G$, such that, either C−1rD⩽V(r)⩽CrD$C^{-1} r^D \leqslant V(r)\leqslant C r^D$ for all r⩾1$r\geqslant 1$, or2.2C−1eD0r⩽V(r)⩽CeDr$$\begin{equation} C^{-1}\mathrm{e}^{D_0 r} \leqslant V(r)\leqslant C \mathrm{e}^{Dr} \end{equation}$$for all r⩾1$r\geqslant 1$. In the former case, the group G$G$ is said to be of polynomial growth, while in the latter case of exponential growth.For any character χ$\chi$ and r>0$r&gt;0$, one has (see [12, Proposition 5.7])2.3supBrχ=ec(χ)r,wherec(χ)=∑j=1ℓ|Xjχ(e)|21/2.$$\begin{equation} \sup _{B_r} \chi = \mathrm{e}^{c(\chi ) r}, \qquad \mbox{where} \quad c(\chi ) = {\left(\sum _{j=1}^\ell |X_j\chi (e)|^2\right)}^{1/2}. \end{equation}$$Since χ$\chi$ is a character, by (2.3) one also has2.4infBrχ=e−c(χ)r.$$\begin{equation} \inf _{B_r} \chi = \mathrm{e}^{-c(\chi ) r}. \end{equation}$$Given a ball B$B$ with respect to dC$d_C$, we denote by cB$c_B$ its center and by rB$r_B$ its radius, and we write B=B(cB,rB)$B=B(c_B,r_B)$; we also set 2B=B(cB,2rB)$2B=B(c_B,2r_B)$. Moreover, for R>0$R&gt;0$ let BR$\mathcal {B}_R$ be the family of all balls of radius ⩽R$\leqslant R$ and2.5D(R,χ)=supB∈BRμχ(2B)μχ(B)=sup0<r⩽Rμχ(B2r)μχ(Br),$$\begin{equation} D(R,\chi )= \sup _{B\in \mathcal {B}_R} \frac{\mu _\chi (2B)}{\mu _\chi (B)} = \sup _{0&lt;r\leqslant R}\frac{\mu _\chi (B_{2r})}{\mu _\chi (B_r)}, \end{equation}$$where the latter equality holds since μχ(B(cB,r))=(χδ−1)(cB)μχ(Br)$\mu _\chi (B(c_B,r)) = (\chi \delta ^{-1})(c_B) \mu _\chi (B_r)$ for all r>0$r&gt;0$ and cB∈G$c_B\in G$.In the following lemma we estimate the local doubling constant D(R,χ)$D(R,\chi )$.2.1LemmaThe metric measure space (G,dC,μχ)$(G, d_C, \mu _\chi )$ is doubling if and only if χ=1$\chi =1$ and (G,dC,ρ)$(G, d_C, \rho )$ is doubling, in which case there exists C>0$C&gt;0$ such that D(R,χ)⩽C$D(R,\chi )\leqslant C$. If χ≠1$\chi \ne 1$ and (G,dC,ρ)$(G, d_C, \rho )$ is doubling, then there exists C>0$C&gt;0$ such thatD(R,χ)⩽Ce3c(χ)R∀R>0,$$\begin{equation*} D(R,\chi ) \leqslant C \mathrm{e}^{3c(\chi )R} \qquad \forall R&gt;0, \end{equation*}$$while if χ≠1$\chi \ne 1$ and (G,dC,ρ)$(G, d_C, \rho )$ is non‐doubling, then there exists C>0$C&gt;0$ such thatD(R,χ)⩽Ce(2D−D0+3c(χ))R∀R>0.$$\begin{equation*} D(R,\chi ) \leqslant C \mathrm{e}^{(2D - D_0 + 3c(\chi ))R} \qquad \forall R&gt;0. \end{equation*}$$ProofIf χ=1$\chi =1$, the first statement is obvious since μχ=ρ$\mu _\chi =\rho$.Assume then that χ≠1$\chi \ne 1$, so that there is x∈G$x\in G$ with χ(x)>1$\chi (x)&gt;1$. If N$N$ denotes the lowest integer such that N⩾|x|$N\geqslant |x|$, then BNxn⊆B(n+1)N$B_N x^n\subseteq B_{(n+1)N}$. If r>N$r&gt;N$ and n$n$ is the largest integer such that (n+1)N⩽[r]$(n+1)N\leqslant [r]$, thenμχ(Br)⩾μχ(BNxn)=χ(x)nμχ(BN)⩾χ(x)[r]/N−2μχ(BN),$$\begin{equation*} \mu _\chi (B_r) \geqslant \mu _\chi (B_N x^{n}) = \chi (x)^{n} \mu _\chi (B_N)\geqslant \chi (x)^{[r]/N-2} \mu _\chi (B_N), \end{equation*}$$whence μχ(Br)$\mu _\chi (B_r)$ grows exponentially with r$r$ and the space (G,dC,μχ)$(G, d_C, \mu _\chi )$ is non‐doubling.We now show the two bounds on D(R,χ)$D(R,\chi )$. First, observe that by (2.1), (2.3) and (2.4)μχ(B2r)μχ(Br)⩽C∀r∈(0,1],$$\begin{equation*} \frac{\mu _\chi (B_{2r})}{\mu _\chi (B_r)}\leqslant C \qquad \forall r\in (0,1], \end{equation*}$$for some C>0$C&gt;0$. Moreover, if (G,dC,ρ)$(G, d_C, \rho )$ is doubling, then by (2.3) and (2.4)μχ(B2r)μχ(Br)⩽e3c(χ)rV(2r)V(r)⩽Ce3c(χ)r∀r⩾1,$$\begin{equation*} \frac{\mu _\chi (B_{2r})}{\mu _\chi (B_{r})} \leqslant \mathrm{e}^{3c(\chi )r}\frac{V(2r)}{V(r)}\leqslant C\mathrm{e}^{3c(\chi )r}\qquad \forall r\geqslant 1, \end{equation*}$$while if (G,dC,ρ)$(G, d_C, \rho )$ is non‐doubling, then the stated estimate follows similarly by (2.2),  (2.3) and (2.4).□$\Box$THE LOCAL POINCARÉ INEQUALITY ON LIE GROUPSIn this section we prove the Lp$L^p$‐Poincaré inequality for smooth functions on (G,dC,μχ)$(G, d_C, \mu _\chi )$. Given a ball B$B$ and f∈C∞(G)$f\in C^\infty (G)$, we denote by fBχ$f_B^\chi$ its average over B$B$ with respect to μχ$\mu _\chi$,fBχ=1μχ(B)∫Bfdμχ,$$\begin{equation*} f_B^\chi = \frac{1}{\mu _\chi (B)}\int _Bf \, {d}\mu _\chi , \end{equation*}$$and we let |∇f|2=∑j=1ℓ|Xjf|2${|\nabla f |}^{2}={\sum}_{j=1}^{\ell}{|{X}_{j}f |}^{2}$. If S$S$ is a set of variables, we denote by C(S)$C(S)$ a constant depending only on the elements of S$S$.3.1TheoremThere exist a constant C=C(G,X)>0$C=C(G,\mathbf {X})&gt;0$ and a universal constant α>0$\alpha &gt;0$ such that, for all p∈[1,∞)$p\in [1,\infty )$, R>0$R&gt;0$, all balls B$B$ of radius r∈(0,R]$r\in (0,R]$ and f∈C∞(G)$f\in C^\infty (G)$,3.1∥f−fBχ∥Lp(B,μχ)⩽Ce1p[2c(χ)+c(χδ−1)]RD(R,χ)αr∥|∇f|∥Lp(B,μχ).$$\begin{equation} \Vert f - f_B^\chi \Vert _{L^p(B,\mu _\chi )} \leqslant C\,e^{ \frac{1}{p} [2c(\chi ) + c(\chi \delta ^{-1})] R}\, D(R, \chi )^{\alpha } \, r \,\Vert |\nabla f|\Vert _{L^p(B,\mu _\chi )}. \end{equation}$$Notice that the Poincaré constant grows at most exponentially with respect to the radius of the ball. The exponential term cannot, in general, be removed. After establishing the theorem, indeed, we show that when G$G$ is the so‐called “ax+b$ax+b$” group and μχ=λ$\mu _\chi =\lambda$ is a left Haar measure, the growth of the constant is indeed exponential.ProofLet p∈[1,∞)$p\in [1,\infty )$ be given. We shall prove that for every ball B$B$ of radius r>0$r &gt;0$ and f∈C∞(G)$f\in C^\infty (G)$3.2∫B|f−fBχ|pdμχ⩽2pec(χδ−1)re2c(χ)rμχ(B2r)μχ(Br)rp∫2B|∇f|pdμχ.$$\begin{equation} \int _B |f - f_B^\chi |^p\, {d}\mu _\chi \leqslant 2^p\, \mathrm{e}^{c(\chi \delta ^{-1})r} \mathrm{e}^{2c(\chi ) r} \, \frac{\mu _\chi (B_{2r})}{\mu _\chi (B_r)} \,r^p \, \int _{2B} |\nabla f|^p \, {d}\mu _\chi . \end{equation}$$Once (3.2) is at disposal, the Poincaré inequality can be obtained by classical arguments, see, for example, [11, Theorem 9.7]. A careful inspection of [13, Section 5], in particular, shows how a Whitney decomposition of B$B$ brings to the constant given in the statement. We omit the details, which would be tedious and an almost verbatim repetition of the arguments that the reader can find in [13].We then show (3.2). For z∈G$z\in G$, let γz:[0,|z|]→G$\gamma _z \colon [0,|z|] \rightarrow G$ be a C1$C^1$‐geodesic such that γz(0)=e$\gamma _z(0)=e$, γz(|z|)=z$\gamma _z({|z|})=z$, γz(s)∈B|z|$\gamma _z(s)\in B_{|z|}$ and |γz′(s)|⩽1$|\gamma _z^{\prime }(s)|\leqslant 1$ for every s∈[0,|z|]$s\in [0,|z|]$.Let B$B$ be a ball of radius r>0$r&gt;0$. Observe that if x,y∈B$x,y \in B$, and z=x−1y$z= x^{-1}y$, then |z|<2r$|z|&lt;2r$. For every x,z∈G$x,z\in G$, by Hölder's inequality3.3|f(x)−f(xz)|p⩽∫0|z||∇f(xγz(s))|dsp⩽|z|p−1∫0|z||∇f(xγz(s))|pds.$$\begin{equation} |f(x) - f(xz)|^p\leqslant {\left(\int _0^{|z|} |\nabla f(x\gamma _z(s))|\, {d}s\right)}^p \leqslant |z|^{p-1}\int _0^{|z|} |\nabla f(x\gamma _z(s))|^p\, {d}s. \end{equation}$$We then have∫B|f−fBχ|pdμχ=∫B1μχ(B)∫Bf(x)−f(y)dμχ(y)pdμχ(x)⩽1μχ(B)∫B∫Bf(x)−f(y)pdμχ(y)dμχ(x),$$\begin{align*} \int _B |f - f_B^\chi |^p\, {d}\mu _\chi & = \int _B {\left| \frac{1}{\mu _\chi (B)} \int _B {\left(f(x)-f(y)\right)}\, {d}\mu _\chi (y)\right|}^p \, {d}\mu _\chi (x)\\ & \leqslant \frac{1}{\mu _\chi (B)} \int _B \int _B {\left| f(x)-f(y)\right|}^p\, {d}\mu _\chi (y) \, {d}\mu _\chi (x), \end{align*}$$and after the change of variables y=xz$y=xz$, we get∫B|f−fBχ|pdμχ⩽1μχ(B)∫G∫G1B(x)1B(xz)f(x)−f(xz)p(χδ−1)(x)dμχ(x)dμχ(z).$$\begin{align*} \int _B |f - f_B^\chi |^p\, {d}\mu _\chi \leqslant \frac{1}{\mu _\chi (B)} \int _G \int _G \mathbf {1}_B(x)\mathbf {1}_B(xz){\left| f(x)-f(xz)\right|}^p (\chi \delta ^{-1})(x)\, {d}\mu _\chi (x)\, {d}\mu _\chi (z). \end{align*}$$Observe now that by (3.3) and Fubini's theorem, we get∫G1B(x)1B(xz)f(x)−f(xz)p(χδ−1)(x)dμχ(x)⩽1μχ(B)|z|p−1∫0|z|∫G1B(x)1B(xz)|∇f(xγz(s))|p(χδ−1)(x)dμχ(x)ds.$$\begin{align*} &\int _G \mathbf {1}_B(x)\mathbf {1}_B(xz){\left| f(x)-f(xz)\right|}^p (\chi \delta ^{-1})(x)\, {d}\mu _\chi (x)\\ & \quad \leqslant \frac{1}{\mu _\chi (B)} |z|^{p-1} \int _0^{|z|} \int _G \mathbf {1}_B(x)\mathbf {1}_B(xz)|\nabla f(x\gamma _z(s))|^p\, (\chi \delta ^{-1})(x)\,{d}\mu _\chi (x)\, {d}s . \end{align*}$$We make the change of variables ζ=xγz(s)$\zeta = x\gamma _z(s)$ and observe that by (2.3), if x∈B$x\in B$, then(χδ−1)(x)⩽(χδ−1)(cB)supBr(χδ−1)⩽(χδ−1)(cB)ec(χδ−1)r,$$\begin{equation*} (\chi \delta ^{-1})(x) \leqslant (\chi \delta ^{-1})(c_B) \, \sup _{B_r }(\chi \delta ^{-1}) \leqslant (\chi \delta ^{-1})(c_B)\, \mathrm{e}^{c(\chi \delta ^{-1})r}, \end{equation*}$$and χ(γz(s))⩽e2c(χ)r$\chi (\gamma _z(s)) \leqslant \mathrm{e}^{2 c(\chi )r}$. We obtain∫G1B(x)1B(xz)(χδ−1)(x)|∇f(xγz(s))|pdμχ(x)⩽(χδ−1)(cB)ec(χδ−1)re2c(χ)r∫G1Bγz(s)(ζ)1Bz−1γz(s)(ζ)|∇f(ζ)|pdμχ(ζ).$$\begin{align*} &\int _G \mathbf {1}_B(x)\mathbf {1}_B(xz) (\chi \delta ^{-1})(x) |\nabla f(x\gamma _z(s))|^p\, {d}\mu _\chi (x)\\ & \quad \leqslant (\chi \delta ^{-1})(c_B) \, \mathrm{e}^{c(\chi \delta ^{-1})r}\, \mathrm{e}^{2 c(\chi )r} \int _G \mathbf {1}_{B\gamma _z(s)}(\zeta )\mathbf {1}_{Bz^{-1}\gamma _z(s)}(\zeta ) |\nabla f(\zeta )|^p\, {d}\mu _\chi (\zeta ). \end{align*}$$Notice that Bγz(s)∩Bz−1γz(s)⊆2B$B\gamma _z(s) \cap Bz^{-1}\gamma _z(s) \subseteq {2B}$ for all s∈[0,|z|]$s\in [0,|z|]$. This is straightforward by the triangle inequality when |z|<r$|z|&lt; r$. Otherwise, let s0∈[0,|z|]$s_0\in [0,|z|]$ be such that |γz(s0)|=r$|\gamma _z(s_0)|=r$. Then Bγz(s)⊆2B$B\gamma _z(s)\subseteq 2B$ for s∈[0,s0]$s\in [0,s_0]$ and Bz−1γz(s)⊆2B$Bz^{-1}\gamma _z(s)\subseteq 2B$ for s∈(s0,|z|]$s\in (s_0,|z|]$. Therefore∫G1B(x)1B(xz)(χδ−1)(x)|∇f(xγz(s))|pdμχ(x)⩽(χδ−1)(cB)ec(χδ−1)re2c(χ)r1B2r(z)∫2B|∇f(ζ)|pdμχ(ζ).$$\begin{align*} &\int _G \mathbf {1}_B(x)\mathbf {1}_B(xz) (\chi \delta ^{-1})(x) |\nabla f(x\gamma _z(s))|^p\, {d}\mu _\chi (x)\\ & \quad \leqslant (\chi \delta ^{-1})(c_B) \, \mathrm{e}^{c(\chi \delta ^{-1})r}\, \mathrm{e}^{2 c(\chi )r} \mathbf {1}_{B_{2r}}(z) \int _{2B} |\nabla f(\zeta )|^p\, {d}\mu _\chi (\zeta ). \end{align*}$$Since (χδ−1)(cB)μχ(B2r)=μχ(2B)$(\chi \delta ^{-1})(c_B) \mu _\chi (B_{2r}) = \mu _\chi (2B)$, by integrating with respect to z$z$ we get∫B|f−fBχ|pdμχ⩽e2c(χ)rec(χδ−1)r(2r)pμχ(2B)μχ(B)∫2B|∇f(ζ)|pdμχ(ζ),$$\begin{align*} \int _B |f - f_B^\chi |^p\, {d}\mu _\chi & \leqslant \mathrm{e}^{2 c(\chi )r} \mathrm{e}^{c(\chi \delta ^{-1})r} (2r)^p \, {\frac{\mu _\chi (2B)}{\mu _\chi (B)} } \int _{2B} |\nabla f(\zeta )|^p \, {d}\mu _\chi (\zeta ), \end{align*}$$which concludes the proof. □$\Box$As a corollary, we obtain the so‐called local parabolic Harnack principle. We introduce the operator3.4Δχ=−∑j=1ℓ(Xj2+(Xjχ)(e)Xj),$$\begin{equation} \Delta _{\chi } =-\sum _{j=1}^{\ell }(X_j^2 +(X_j\chi )(e)X_j ), \end{equation}$$which is essentially self‐adjoint on L2(μχ)$L^2(\mu _\chi )$ and non‐negative; see, for example, [4, 12]. We say that Δχ$\Delta _\chi$ satisfies the local parabolic Harnack principle up to distance R>0$R&gt;0$ if there is C(R)>0$C(R)&gt;0$ such that, for all x∈G$x\in G$, r∈(0,R]$r\in (0,R]$, s∈R$s\in {\mathbb {R}}$, and any positive solutions u$u$ of (∂t+Δχ)u=0$(\partial _t +\Delta _\chi )u=0$ on (s,s+r2)×B(x,r)$(s,s+r^2)\times B(x,r)$, we have thatsupQ−u⩽C(R)infQ+u,$$\begin{equation*} \sup _{Q_-} u \leqslant C(R) \inf _{Q_+} u, \end{equation*}$$whereQ−=s+r2/6,s+r2/3×B(x,r/2),Q+=s+2r2/3,s+r2×B(x,r/2).$$\begin{equation*} Q_- = {\left(s+r^2/6, s + r^2/3\right)} \times B(x,r/2),\qquad Q_+ = {\left(s+2r^2/3, s+r^2\right)} \times B(x,r/2). \end{equation*}$$The following result follows at once from Theorem 3.1 and [16, Theorem 2.1].3.2CorollaryFor every R>0$R&gt;0$, Δχ$\Delta _\chi$ satisfies the local parabolic Harnack principle up to distance R$R$. In particular, the positive Δχ$\Delta _\chi$‐harmonic functions satisfy the local elliptic Harnack inequality.Exponential growth of the constantFor r>0$r&gt;0$ and p∈[1,∞)$p\in [1,\infty )$, define3.5C(r,p)=inf∫Br|f−fBχ|pdμχ∫Br|∇f|pdμχ,$$\begin{equation} C(r,p)=\inf \, \frac{\int _{B_r} |f - f_B^\chi |^p\, {d}\mu _\chi }{\int _{B_r} |\nabla f|^p \, {d}\mu _\chi }, \end{equation}$$where the infimum runs over all functions f∈C∞(G)$f \in C^{\infty }(G)$. In this section we show that the exponential bound of C(r,p)$C(r,p)$ appearing in inequality (3.1) is in general optimal, in the sense that such constant cannot grow less than exponentially with respect to r$r$. Indeed, in the particular case of ax+b$ax+b$ groups of arbitrary dimension, we provide a lower bound of exponential type for C(r,p)$C(r,p)$. For notational convenience, we shall write A≲B$A \lesssim B$ to indicate that there is a constant C$C$ such that A⩽CB$A \leqslant CB$. If A≲B$A\lesssim B$ and B≲A$B\lesssim A$, then we write A≈B$A\approx B$.Let G=Rn−1⋊R+$G=\mathbb {R}^{n-1} \rtimes \mathbb {R}^+$ and let (x,a)$(x,a)$ be its generic element. Recall thatdλ(x,a)=dxdaananddρ(x,a)=dxdaa,$$\begin{equation*} {d}\lambda (x,a) = \frac{{d}x\, {d}a}{a^n} \qquad {\rm {and}}\qquad {d}\rho (x,a) = \frac{{d}x\, {d}a}{a}, \end{equation*}$$since δ(x,a)=a−n+1$\delta (x,a) = a^{-n+1}$; all positive characters of G$G$ are of the form χγ(x,a)=aγ$ \chi _\gamma (x,a) = a^\gamma$ for some γ∈R$\gamma \in \mathbb {R}$. We shall write μγ$\mu _\gamma$ for the measure μχγ$\mu _{\chi _\gamma }$. In particular, λ=μ1−n$\lambda = \mu _{1-n}$ is the hyperbolic measure. We consider the left‐invariant vector fields Xi=a∂i$X_i=a\partial _i$, i=1,⋯,n−1$i=1,\dots ,n-1$, and X0=a∂a$X_0=a\partial _a$ which form a basis of the Lie algebra of G$G$. The distance induced by such vector fields is the hyperbolic metric which is given bycosh|(x,a)|=12(a+a−1+a−1|x|2),$$\begin{equation*} \cosh |(x,a)| = \tfrac{1}{2}(a+a^{-1} + a^{-1}|x|^2), \end{equation*}$$where |x|$|x|$ is the Euclidean norm of x∈Rn−1$x\in \mathbb {R}^{n-1}$ (see [1, (2.18)], [17, (1.1)]). ThenBr=(x,a):e−r<a<er,|x|2<2a(coshr−coshloga).$$\begin{equation*} B_r = {\left\lbrace (x,a)\colon \mathrm{e}^{-r}&lt;a&lt;\mathrm{e}^r, \; |x|^2 &lt; 2a(\cosh r - \cosh \log a)\right\rbrace} . \end{equation*}$$In the case of the real hyperbolic space, that is, the ax+b$ax+b$ group endowed with the measure λ$\lambda$ and the metric defined above, the constant C(r,p)$C(r,p)$ in (3.5) was estimated from above in [11, Section 10.1]. We now estimate such constant from below.Consider the function ϕ:G→R$\phi \colon G\rightarrow \mathbb {R}$ defined byϕ(x,a)=x1,(x,a)∈G.$$\begin{equation*} \phi (x,a) = x_1,\quad (x,a)\in G. \end{equation*}$$Observe that ∫Brϕdμγ=0$\int _{B_r} \phi \, {d}\mu _\gamma =0$ for all γ∈R$\gamma \in \mathbb {R}$ and |∇ϕ(x,a)|p=ap$|\nabla \phi (x,a) |^p = a^p$. Moreover,∫Br|ϕ|pdμγ≈∫e−reraγ−1+p+n−12(coshr−coshloga)p+n−12da,$$\begin{equation*} \int _{B_r} |\phi |^p \, {d}\mu _\gamma \approx \int _{\mathrm{e}^{-r}}^{\mathrm{e}^r} a^{\gamma -1 + \frac{p+n-1}{2}} (\cosh r - \cosh \log a)^{\frac{p+n-1}{2}} {{d}a}, \end{equation*}$$while∫Br|∇ϕ|pdμγ≈∫e−reraγ−1+p+n−12(coshr−coshloga)n−12da.$$\begin{equation*} \int _{B_r} |\nabla \phi |^p \, {d}\mu _\gamma \approx \int _{\mathrm{e}^{-r}}^{\mathrm{e}^r} a^{\gamma -1 +p+ \frac{n-1}{2}}{(\cosh r - \cosh \log a)^{\frac{n-1}{2}}} \, {d}a . \end{equation*}$$3.3LemmaLet δ∈R$\delta \in \mathbb {R}$ and ε>0$\epsilon &gt;0$. Then∫e−reraδ(coshr−coshloga)εda≈er(|δ+1|+ε).$$\begin{equation*} \int _{\mathrm{e}^{-r}}^{\mathrm{e}^r} a^{\delta }{(\cosh r - \cosh \log a)^{\epsilon }} \, {d}a \approx \mathrm{e}^{r(|\delta +1| + \epsilon ) }. \end{equation*}$$ProofWe first make a change of variables∫e−reraδ(coshr−coshloga)εda=∫−rret(δ+1)(coshr−cosht)εdt.$$\begin{equation*} \int _{\mathrm{e}^{-r}}^{\mathrm{e}^r} a^{\delta }{(\cosh r - \cosh \log a)^{\epsilon }} \, {d}a = \int _{-r}^r \mathrm{e}^{t(\delta +1)} (\cosh r - \cosh t)^{\epsilon } \, {d}t. \end{equation*}$$Since coshr−cosht≈er$\cosh r - \cosh t \approx \mathrm{e}^r$ if |t|<r−1$ |t|&lt;r-1$, while coshr−cosht≈(r−|t|)er$\cosh r - \cosh t \approx (r-|t|)\mathrm{e}^r$ if r−1<|t|<r$ r-1&lt;|t|&lt;r$, we get∫−r+1r−1et(δ+1)(coshr−cosht)εdt≈eεr∫−r+1r−1et(δ+1)dt≈e(ε+|δ+1|)r$$\begin{equation*} \int _{-r+1}^{r-1} \mathrm{e}^{t(\delta +1)} (\cosh r - \cosh t)^{\epsilon } \, {d}t\approx \mathrm{e}^{\epsilon r} \int _{-r+1}^{r-1} \mathrm{e}^{t(\delta +1)}\, {d}t \approx \mathrm{e}^{(\epsilon + |\delta +1|) r} \end{equation*}$$while∫r−1<|t|<ret(δ+1)(coshr−cosht)εdt≈e(ε+|δ+1|)r∫r−1<|t|<r(r−|t|)εdt≈e(ε+|δ+1|)r,$$\begin{equation*} \int _{r-1&lt;|t|&lt;r} \mathrm{e}^{t(\delta +1)} (\cosh r - \cosh t)^{\epsilon } \, {d}t\approx \mathrm{e}^{(\epsilon +|\delta +1|) r} \int _{r-1&lt;|t|&lt;r} (r-|t|)^\epsilon \, {d}t \approx \mathrm{e}^{(\epsilon + |\delta +1|) r} , \end{equation*}$$as required.□$\Box$From the lemma above, we get that∫Br|ϕ|pdμγ≈e(|γ+p+n−12|+p+n−12)r,$$\begin{equation*} \int _{B_r} |\phi |^p \, {d}\mu _\gamma \approx \mathrm{e}^{({|\gamma + \frac{p+n-1}{2}|} + \frac{p+n-1}{2})r}, \end{equation*}$$while∫Br|∇ϕ|pdμγ≈e(|γ+p+n−12|+n−12)r.$$\begin{equation*} \int _{B_r} |\nabla \phi |^p \, {d}\mu _\gamma \approx \mathrm{e}^{({|\gamma +p+ \frac{n-1}{2}| }+ \frac{n-1}{2})r}. \end{equation*}$$We observe that, if γ<−p+n−12$\gamma &lt; {-\frac{p+n-1}{2}}$, then|γ+p+n−12|+p+n−12>|γ+p+n−12|+n−12.$$\begin{equation*} {\bigg |\gamma + \frac{p+n-1}{2}\bigg |} + \frac{p+n-1}{2} &gt; { \bigg |\gamma +p+ \frac{n-1}{2} \bigg |} + \frac{n-1}{2}. \end{equation*}$$Thus for such γ$\gamma$C(r,p)⩾Cerγ+p+n−12+p2−γ+p+n−12.$$\begin{equation*} C(r,p) \geqslant C \mathrm{e}^{ r\left({{\left|\gamma + \frac{p+n-1}{2}\right|} }+ \frac{p}{2} - { {\left|\gamma +p+ \frac{n-1}{2} \right|}} \right)}. \end{equation*}$$If in particular γ=−n+1$\gamma = -n+1$, hence μγ$\mu _\gamma$ is the left measure, and n>p+1$n&gt;p+1$, thenC(r,p)⩾e−n+p+12−−n+12+p+p2r=eprn⩾2p+1e(n−p−1)rp+1<n⩽2p+1.$$\begin{equation*} C(r,p) { \geqslant \mathrm{e}^{ \left({\left| \frac{-n+p+1}{2} \right|}- {\left| \frac{-n+1}{2}+p \right|} +\frac{p}{2}\right)r } = {\begin{cases} \mathrm{e}^{pr} & n \geqslant 2p+1\\ \mathrm{e}^{(n-p-1)r} & p+1 &lt;n\leqslant 2p+1. \end{cases}} } \end{equation*}$$NON‐LOCAL POINCARÉ INEQUALITYIn this second part of the paper we prove a non‐local L2$L^2$‐Poincaré inequality for suitable finite measures on G$G$ in the spirit of [14, 15]. More precisely, let M$M$ be a positive C2$C^2$‐function in L1(μχ)$L^1(\mu _\chi )$ and μχ,M$\mu _{\chi ,M}$ be the finite measure whose density is M$M$ with respect to μχ$\mu _\chi$. We shall prove L2$L^2$‐global Poincaré inequalities for the measure μχ,M$\mu _{\chi ,M}$ for a large family of functions M$M$. In order to do this, we letL12(μχ,M)={f∈L2(μχ,M):|∇f|∈L2(μχ,M)}$$\begin{equation*} L^2_1(\mu _{\chi ,M}) = \lbrace f\in L^2(\mu _{\chi ,M}) \colon |\nabla f| \in L^2(\mu _{\chi ,M}) \rbrace \end{equation*}$$and introduce the operator4.1Δχ,M=Δχ−∇(logM)·∇,$$\begin{equation} \Delta _{\chi ,M} = \Delta _\chi - \nabla (\log M) \cdot \nabla , \end{equation}$$where Δχ$\Delta _\chi$ is that of (3.4), Dom(Δχ,M)={f∈L12(μχ,M):Δχ,Mf∈L2(μχ,M)}$\mathrm{Dom}(\Delta _{\chi ,M}) = \lbrace f \in L^2_1(\mu _{\chi ,M})\colon \Delta _{\chi ,M} f \in L^2(\mu _{\chi ,M}) \rbrace$ and the derivatives are meant in the distributional sense. Observe that Δχ,M$\Delta _{\chi ,M}$ is symmetric on L2(μχ,M)$L^2(\mu _{\chi ,M})$; in particular, for all f∈Dom(Δχ,M)$f\in \mathrm{Dom}(\Delta _{\chi ,M})$ and g∈L12(μχ,M)$g\in L^2_1(\mu _{\chi ,M})$,∫G∇f·∇gdμχ,M=∫GΔχ,Mf·gdμχ,M,$$\begin{equation*} \int _{G} \nabla f \cdot \nabla g\, {d}\mu _{\chi ,M}=\int _{G} \Delta _{\chi ,M} f \cdot g\, {d}\mu _{\chi ,M}, \end{equation*}$$where ∇f·∇g=∑j=1ℓ(Xjf)(Xjg)$\nabla f \cdot \nabla g = \sum _{j=1}^\ell (X_j f)(X_j g)$.We say that the couple (Δχ,M)$(\Delta _\chi ,M)$ admits a Lyapunov function if there exist a C2$C^2$ function W:G→[1,∞)$W\colon G\rightarrow [1,\infty )$ and constants θ>0$\theta &gt;0$, b⩾0$b\geqslant 0$, R>0$R&gt;0$ such that4.2−Δχ,MW(x)⩽−θW(x)+b1BR(x)∀x∈G.$$\begin{equation} - \Delta _{\chi ,M} W(x)\leqslant -\theta W(x)+b{\bf 1}_{B_R}(x) \qquad \forall x\in G. \end{equation}$$Observe that the existence of a Lyapunov function depends on G$G$, X$\mathbf {X}$, χ$\chi$ and M$M$. For f∈L1(μχ,M)$f\in L^1(\mu _{\chi ,M})$ we letfχ,M=1μχ,M(G)∫Gfdμχ,M.$$\begin{equation*} f_{\chi ,M} = \frac{1}{\mu _{\chi ,M}(G)}\int _G f\, {d}\mu _{\chi ,M}. \end{equation*}$$Our second main result is the following global L2$L^2$‐Poincaré inequality for μχ,M$\mu _{\chi ,M}$.4.1TheoremIf (Δχ,M)$(\Delta _\chi ,M)$ admits a Lyapunov function, then there exists a constant C=C(G,X,χ,M)$C=C(G, \mathbf {X}, \chi , M)$ such that for all f∈L12(μχ,M)$f\in L^2_1(\mu _{\chi ,M})$4.3∥f−fχ,M∥L2(μχ,M)⩽C∥|∇f|∥L2(μχ,M).$$\begin{equation} \Vert f- f_{\chi ,M}\Vert _{L^2(\mu _{\chi ,M})} \leqslant C \Vert |\nabla f|\Vert _{L^2(\mu _{\chi ,M})}. \end{equation}$$Theorem 4.1 is a generalization to any connected non‐compact possibly non‐unimodular Lie group of the non‐local L2$L^2$‐Poincaré inequalities proved in [2, Theorem 1.4] in the Euclidean setting and in [15, Theorem 1.1] in unimodular Lie groups of polynomial growth, by which our proof is inspired. The validity of other versions of non‐local Poincaré inequalities in the current setting, such as those in [15, Theorem 1.4], is still an open problem. More general versions of non‐local Poincaré inequalities of this kind were proved in [9] in the setting of a topological measure space endowed with a family of sets which play the role of unit balls and satisfy suitable assumptions. Recently in [8] non‐local Lp$L^p$‐Poincaré inequalities were obtained on Carnot groups of Engel type in the case when the density of the measure depends on a homogeneous norm of the group; we note, however, that the case p=2$p=2$ is always excluded.ProofLet f∈L12(μχ,M)$f\in L^2_1(\mu _{\chi ,M})$, and observe first that4.4∫Gf−fχ,M2dμχ,M=minc∈R∫Gf−c2dμχ,M.$$\begin{equation} \int _G {\left| f- f_{\chi ,M} \right|}^2 {d}\mu _{\chi ,M} = \min _{c\in \mathbb {R}} \int _G {\left| f- c\right|}^2 {d}\mu _{\chi ,M}. \end{equation}$$Let now g=f−c$g=f-c$ for a positive c$c$ to be determined, and W$W$ be a Lyapunov function for (Δχ,M)$(\Delta _\chi ,M)$. By (4.2)4.5∫G|g|2dμχ,M⩽∫G|g|2Δχ,MWθWdμχ,M+∫BR|g|2bθWdμχ,M.$$\begin{equation} \int _G |g|^2 \, {d}\mu _{\chi ,M} \leqslant \int _G |g|^2 \frac{\Delta _{\chi ,M} W}{\theta W}\, {d}\mu _{\chi ,M} +\int _{B_R} |g|^2 \frac{b}{\theta W}\, {d}\mu _{\chi ,M}. \end{equation}$$We treat the two terms separately.Let us consider the first term and prove that4.6∫GΔχ,MWWg2dμχ,M⩽∫G|∇g|2dμχ,M.$$\begin{equation} \int _G \frac{\Delta _{\chi ,M} W}{W}g^2 {d}\mu _{\chi ,M} \leqslant \int _G |\nabla g |^2{d}\mu _{\chi ,M}. \end{equation}$$We prove it by density, and firstly assume that g$g$ is compactly supported. By definition of Δχ,M$\Delta _{\chi ,M}$,∫GΔχ,MWWg2dμχ,M=∫G∇g2W·∇Wdμχ,M=2∫GgW∇g·∇Wdμχ,M−∫Gg2W2|∇W|2dμχ,M=∫G|∇g|2dμχ,M−∫G|∇g−gW∇W|2dμχ,M⩽∫G|∇g|2dμχ,M.$$\begin{align*} \int _G \frac{\Delta _{\chi ,M} W}{W} g^2 \,{d}\mu _{\chi ,M} &= \int _G \nabla {\left(\frac{g^2}{W}\right)}\cdot \nabla W\, {d}\mu _{\chi ,M} \\ & = 2 \int _G \frac{g}{W} \nabla g\cdot \nabla W\, {d}\mu _{\chi ,M} -\int _G \frac{g^2}{W^2} |\nabla W|^2 \, {d}\mu _{\chi ,M}\\ & = \int _G |\nabla g|^2\, {d}\mu _{\chi ,M} - \int _G |\nabla g-\frac{g}{W} \nabla W|^2 \, {d}\mu _{\chi ,M} \\ & \leqslant \int _G |\nabla g|^2{d}\mu _{\chi ,M}. \end{align*}$$Let now g∈L12(μχ,M)$g\in L^2_1(\mu _{\chi ,M})$, and consider a non‐decreasing sequence of functions ψn∈Cc∞(G)$\psi _n\in C_c^\infty (G)$ such that1BnR⩽ψn⩽1,|∇ψn|⩽1.$$\begin{equation*} {\bf 1}_{B_{nR}} \leqslant \psi _n\leqslant 1, \quad |\nabla \psi _n| \leqslant 1. \end{equation*}$$By applying  (4.6) to gψn$g\psi _n$, the monotone convergence theorem in the left‐hand side and the dominated convergence theorem in the right‐hand side, one gets (4.6).To deal with the second term, we choose c$c$ such that ∫BRgdμχ=0$\int _{B_R}g\, {d}\mu _\chi =0$. By (3.2) applied to g$g$ on BR$B_R$, and the fact that M$M$ is bounded from above and below on B2R$B_{2R}$, one has∫BR|g|2dμχ,M⩽C∫BR|g|2dμχ⩽C∫B2R|∇g|2dμχ⩽C∫B2R|∇g|2dμχ,M,$$\begin{align*} \int _{B_R} |g|^2 \, {d}\mu _{\chi ,M} \leqslant C \int _{B_R} |g|^2 \, {d}\mu _\chi \leqslant C \int _{B_{2R}} |\nabla g|^2\, {d}\mu _\chi \leqslant C \int _{B_{2R}} |\nabla g|^2\, {d}\mu _{\chi ,M}, \end{align*}$$where the constant C$C$ depends on R$R$ and M$M$. Therefore, since W⩾1$W\geqslant 1$,∫BR|g|2bθWdμχ,M⩽C∫B2R|∇g|2dμχ,M⩽C∫G|∇g|2dμχ,M,$$\begin{equation*} \int _{B_R} |g|^2 \frac{b}{\theta W}\, {d}\mu _{\chi ,M} \leqslant C \int _{B_{2R}} |\nabla g|^2\, {d}\mu _{\chi ,M} \leqslant C \int _{G} |\nabla g|^2\, {d}\mu _{\chi ,M} , \end{equation*}$$which completes the proof.□$\Box$4.2CorollaryLet v=−logM$v= -\log M$. If there exist a∈(0,1)$a\in (0,1)$, c>0$c&gt;0$ and R>0$R&gt;0$ such that4.7a|∇v|2(x)+Δχv(x)⩾c∀x∈BRc,$$\begin{equation} a|\nabla v|^2(x) + \Delta _\chi v(x)\geqslant c \qquad \forall \, x \in B_R^c, \end{equation}$$then (Δχ,M)$(\Delta _\chi ,M)$ admits a Lyapunov function, and (4.3) holds.ProofLet W(x)=e(1−a)(v(x)−infGv)$W(x)= \mathrm{e}^{(1-a)(v(x)-\inf _Gv)}$, so that−Δχ,MW=(1−a)W−Δχv−a|∇v|2.$$\begin{equation*} -\Delta _{\chi ,M} W =(1-a)W {\left(- \Delta _\chi v-a|\nabla v|^2\right)}. \end{equation*}$$Then W$W$ is a Lyapunov function with θ=c(1−a)$\theta =c (1-a)$ and b=maxBR(−Δχ,MW+θW)$b= \max _{B_R} (-\Delta _{\chi ,M} W+\theta W)$.□$\Box$One can actually show that if (4.7) holds with a<1/2$a&lt;1/2$, then (4.3) self‐improves as follows.4.3PropositionLet v=−logM$v= -\log M$. If there exist c>0$c&gt;0$, R>0$R&gt;0$ and ε∈(0,1)$\epsilon \in (0,1)$ such that4.81−ε2|∇v|2(x)+Δχv(x)⩾c∀x∈BRc,$$\begin{equation} \frac{1-\epsilon }{2}|\nabla v|^2(x) + \Delta _\chi v(x)\geqslant c \qquad \forall \, x\in B_R^c, \end{equation}$$then there exists C>0$C&gt;0$ such that for all f∈L12(μχ,M)$f\in L^2_1(\mu _{\chi ,M})$4.9∥|f−fχ,M|1+|∇v|∥L2(μχ,M)⩽C∥|∇f|∥L2(μχ,M).$$\begin{equation} \Vert |f - f_{\chi ,M}|{\left(1+| \nabla v|\right)}\Vert _{L^2(\mu _{\chi ,M})} \leqslant C \Vert |\nabla f|\Vert _{L^2(\mu _{\chi ,M})}. \end{equation}$$ProofObserve firstly that, since v$v$ is C2$C^2$ and (4.8) holds,4.101−ε2|∇v|2−Δχv⩾α$$\begin{equation} \frac{1-\epsilon }{2}|\nabla v|^2 - \Delta _\chi v \geqslant \alpha \end{equation}$$for some α∈R$\alpha \in \mathbb {R}$. Let f∈L12(μχ,M)$f\in L^2_1(\mu _{\chi ,M})$ and let g=fM$g=f\sqrt {M}$. Since∇f=1M∇g+12g1M∇v,$$\begin{equation*} \nabla f=\frac{1}{\sqrt {M}} \nabla g+\frac{1}{2}g \frac{1}{\sqrt {M}} \nabla v, \end{equation*}$$by (4.10)∫G|∇f|2dμχ,M=∫G|∇g|2+14|g|2|∇v|2+g∇g·∇vdμχ=∫G|∇g|2+14|g|2|∇v|2+12∇(|g|2)·∇vdμχ⩾∫G|g|214|∇v|2+12Δχvdμχ⩾12∫G|f|2ε2|∇v|2+αdμχ,M.$$\begin{align*} \int _{G}|\nabla f|^2 \, {d}\mu _{\chi ,M} &= \int _{G} {\left(|\nabla g|^2 +\frac{1}{4}|g|^2|\nabla v|^2+ g \nabla g\cdot \nabla v\right)} \, {d}\mu _\chi \\ &= \int _{G} {\left(|\nabla g|^2 +\frac{1}{4} |g|^2|\nabla v|^2+\frac{1}{2}\nabla (|g|^2)\cdot \nabla v\right)} \, {d}\mu _\chi \\ &\geqslant \int _{G} |g|^2 {\left(\frac{1}{4}| \nabla v|^2+ \frac{1}{2} \Delta _\chi v \right)} \, {d}\mu _\chi \\ &\geqslant {\frac{1}{2} \int _{G} |f|^2 {\left(\frac{\epsilon }{2}|\nabla v|^2 + \alpha \right)}\, {d}\mu _{\chi ,M}.} \end{align*}$$Since (4.3) holds by (4.8) and Corollary 4.2, the conclusion follows.□$\Box$ACKNOWLEDGEMENTSThe authors wish to thank the anonymous referee for their careful reading of the paper and interesting comments. All authors are partially supported by the GNAMPA 2020 project “Fractional Laplacians and subLaplacians on Lie groups and trees” and are members of the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). T. Bruno acknowledges support by the Research Foundation–Flanders (FWO) through the postdoctoral grant 12ZW120N.Open Access Funding provided by Universita degli Studi di Genova within the CRUI‐CARE Agreement.JOURNAL INFORMATIONThe Bulletin of the London Mathematical Society is wholly owned and managed by the London Mathematical Society, a not‐for‐profit Charity registered with the UK Charity Commission. All surplus income from its publishing programme is used to support mathematicians and mathematics research in the form of research grants, conference grants, prizes, initiatives for early career researchers and the promotion of mathematics.REFERENCESJ.‐Ph. Anker, E. Damek, and C. Yacoub, Spherical analysis on harmonic AN$AN$ groups, Ann. Sc. Norm. Super. Pisa 33 (1996), 643–679.D. Bakry, F. Barthe, P. Cattiaux, and A. Guillin, A simple proof of the Poincaré inequality for a large class of probability measures including the log‐concave case, Electron. Comm. Probab. 13 (2008), 60–66.T. Bruno and M. Calzi, Schrödinger operators on Lie groups with purely discrete spectrum, Adv. Math. 404 (2022) n. 108444.T. Bruno, M. M. Peloso, A. Tabacco, and M. Vallarino, Sobolev spaces on Lie groups: embedding theorems and algebra properties, J. Funct. Anal. 276 (2019), no. 10, 3014–3050.T. Bruno, M. M. Peloso, and M. Vallarino, Besov and Triebel–Lizorkin spaces on Lie groups, Math. Ann. 377 (2020), no. 1–2, 335–377.T. Bruno, M. M. Peloso, and M. Vallarino, Potential spaces on Lie groups. Geometric Aspects of Harmonic Analysis, Springer INdAM Series, vol. 45, 2021, pp. 149–192.T. Bruno, M. M. Peloso, and M. Vallarino, The Sobolev embedding constant on Lie groups, Nonlinear Anal. 216 (2022), 112707.M. Chatzakou, S. Federico, and B. Zegarlinski, q‐Poincarè inequalities on Carnot Groups with a filiform Lie algebra, arXiv:2007.04689.P. T. Gressman, Fractional Poincaré and logarithmic Sobolev inequalities for measure spaces, J. Funct. Anal. 265 (2013), no. 6, 867–889.Y. Guivarc'h, Croissance polynomiale et périodes des fonctions harmoniques, Bull. Soc. Math. France 101 (1973), 333–379.P. Hajlasz and P. Koskela, Sobolev met Poincaré, Mem. Amer. Math. Soc. 145 (2000), no. 688, x+101.W. Hebisch, G. Mauceri, and S. Meda, Spectral multipliers for Sub‐Laplacians with drift on Lie groups, Math. Z. 251 (2005), no. 4, 899–927.D. Jerison, The Poincaré inequality for vector fields satisfying Hörmander's condition, Duke Math. J. 53 (1986), no. 2, 503–523.C. Mouhot, E. Russ, and Y. Sire, Fractional Poincaré inequalities for general measures, J. Math. Pure Appl. (9) 95 (2011), no. 1, 72–84.E. Russ and Y. Sire, Nonlocal Poincaré inequalities on Lie groups with polynomial volume growth and Riemannian manifolds, Studia Math. 203 (2011), no. 2, 105–127.L. Saloff‐Coste, Parabolic Harnack inequality for divergence‐form second‐order differential operators. Potential theory and degenerate partial differential operators, Potential Anal. 4 (1995), no. 4, 429–467.P. Sjögren and M. Vallarino, Boundedness from H1$H^1$ to L1$L^1$ of Riesz transforms on a Lie group of exponential growth, Ann. Inst. Fourier, Grenoble 58 (2008), no. 4, 1117–1151.N. Th. Varopoulos, Fonctions harmoniques sur les groupes de Lie, C. R. Acad. Sci. Paris, Série I, Math. 309 (1987), 519–521.N. Th. Varopoulos, Analysis on Lie groups, J. Funct. Anal. 76 (1988), no. 2, 346–410. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

Local and non‐local Poincaré inequalities on Lie groups

Loading next page...
 
/lp/wiley/local-and-non-local-poincar-inequalities-on-lie-groups-OAvWSnXa8A

References (23)

Publisher
Wiley
Copyright
© 2022 London Mathematical Society.
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms.12684
Publisher site
See Article on Publisher Site

Abstract

INTRODUCTIONThe aim of this paper is to establish two forms of Poincaré inequality on non‐compact connected Lie groups. On the one hand, we shall obtain the Lie group analogue of the classical inequality on Rd$\mathbb {R}^d$1.1∥f−fB∥Lp(B)⩽Cr∥∇f∥Lp(B),$$\begin{equation} \Vert f -f_B\Vert _{L^p(B)} \leqslant C r \Vert \nabla f\Vert _{L^p(B)}, \end{equation}$$where p∈[1,∞)$p\in [1,\infty )$, f∈C∞(Rd)$f\in C^\infty (\mathbb {R}^d)$, B$B$ is a ball of radius r$r$ and fB$f_B$ is the average of f$f$ on B$B$. On the other hand, we shall consider a non‐local L2$L^2$‐version of such an inequality, which takes the form1.2∥f∥L2(Rd,μ)⩽C∥∇f∥L2(Rd,μ)$$\begin{equation} \Vert f \Vert _{L^2(\mathbb {R}^d\!,\,\mu )} \leqslant C \Vert \nabla f\Vert _{L^2(\mathbb {R}^d\!,\,\mu )} \end{equation}$$for certain finite measures μ$\mu$ which are absolutely continuous with respect to the Lebesgue measure and whose densities satisfy a certain decay condition at infinity. One should think, for example, to the case when μ$\mu$ is a Gaussian measure.Extensions of the classical Poincaré inequality (1.1) to non‐Euclidean settings have widely been studied in the last decades. A thorough overview of the literature would go out of the scope of the present paper, so we refer the reader to the milestone [11] and the references therein. For what concerns Lie groups, a Poincaré inequality on unimodular groups can be obtained by combining [16, §8.3] and [11, Theorem 9.7]. In this paper we prove that a Poincaré inequality also holds on non‐unimodular Lie groups endowed with a relatively invariant measure, and we also describe the behaviour of the Poincaré constant in a quantitative way. We show that this grows at most exponentially with respect to the radius of the ball, and that if the group is non‐doubling, then such growth is, in general, exponential. More precisely, in a class of Lie groups including the real hyperbolic spaces as a subclass, we estimate from below the constant involved in the Poincaré inequality by a quantity which grows exponentially with respect to the radius of the ball.Non‐local inequalities such as (1.2) have been introduced more recently [2] on Rd$\mathbb {R}^d$, for densities satisfying suitable differential inequalities expressed in terms of the Laplacian Δ$\Delta$ (cf. [2, Corollary 1.6]) whose prototype is a Gaussian function. After its establishment on Rd$\mathbb {R}^d$, such non‐local inequalities were extended to unimodular Lie groups of polynomial growth in [15], where a sum‐of‐squares subelliptic sub‐Laplacian plays the role of Δ$\Delta$. In this paper, we extend their method to the non‐doubling regime where the sub‐Laplacian has an additional drift term, in a setting where we previously studied various function spaces [4–6] and the Sobolev and Moser–Trudinger inequalities [7].As a classical application of the local Poincaré inequality, we show the so‐called local parabolic Harnack principle for the sub‐Laplacian with drift. Another application of our inequality is given in [3] to the study of spectral properties of Schrödinger operators on Lie groups.SETTING AND PRELIMINARIESLet G$G$ be a non‐compact connected Lie group with identity e$e$. We denote by ρ$\rho$ a right Haar measure, by χ$\chi$ a continuous positive character of G$G$ and by μχ$\mu _\chi$ the measure with density χ$\chi$ with respect to ρ$\rho$. As the modular function on G$G$, which we denote by δ$\delta$, is such a character, μδ$\mu _\delta$ is a left Haar measure on G$G$. We denote it by λ$\lambda$. Observe also that μ1=ρ$\mu _1=\rho$.Let X={X1,⋯,Xℓ}$\mathbf {X}= \lbrace X_1,\dots , X_\ell \rbrace$ be a family of left‐invariant linearly independent vector fields which satisfy Hörmander's condition. Let dC(·,·)$d_C(\, \cdot \, ,\, \cdot \, )$ be its associated left‐invariant Carnot–Carathéodory distance. We let |x|=dC(x,e)$|x|=d_C(x,e)$, and denote by Br$B_r$ the ball centred at e$e$ of radius r$r$. The measure of Br$B_r$ with respect to ρ$\rho$ will be denoted by V(r)=ρ(Br)$V(r)=\rho (B_r)$; we recall that V(r)=λ(Br)$V(r)= \lambda (B_r)$. It is well known, cf. [10, 19], that there exist d∈N∗$d\in \mathbb {N}^*$ depending on G$G$ and X${\bf X}$, and C>0$C&gt;0$, such that2.1C−1rd⩽V(r)⩽Crd∀r∈(0,1],$$\begin{equation} C^{-1} r^d \leqslant V(r) \leqslant C r^d\qquad \forall r\in (0,1], \end{equation}$$and D0,D>0$D_0,D&gt;0$ depending only on G$G$, such that, either C−1rD⩽V(r)⩽CrD$C^{-1} r^D \leqslant V(r)\leqslant C r^D$ for all r⩾1$r\geqslant 1$, or2.2C−1eD0r⩽V(r)⩽CeDr$$\begin{equation} C^{-1}\mathrm{e}^{D_0 r} \leqslant V(r)\leqslant C \mathrm{e}^{Dr} \end{equation}$$for all r⩾1$r\geqslant 1$. In the former case, the group G$G$ is said to be of polynomial growth, while in the latter case of exponential growth.For any character χ$\chi$ and r>0$r&gt;0$, one has (see [12, Proposition 5.7])2.3supBrχ=ec(χ)r,wherec(χ)=∑j=1ℓ|Xjχ(e)|21/2.$$\begin{equation} \sup _{B_r} \chi = \mathrm{e}^{c(\chi ) r}, \qquad \mbox{where} \quad c(\chi ) = {\left(\sum _{j=1}^\ell |X_j\chi (e)|^2\right)}^{1/2}. \end{equation}$$Since χ$\chi$ is a character, by (2.3) one also has2.4infBrχ=e−c(χ)r.$$\begin{equation} \inf _{B_r} \chi = \mathrm{e}^{-c(\chi ) r}. \end{equation}$$Given a ball B$B$ with respect to dC$d_C$, we denote by cB$c_B$ its center and by rB$r_B$ its radius, and we write B=B(cB,rB)$B=B(c_B,r_B)$; we also set 2B=B(cB,2rB)$2B=B(c_B,2r_B)$. Moreover, for R>0$R&gt;0$ let BR$\mathcal {B}_R$ be the family of all balls of radius ⩽R$\leqslant R$ and2.5D(R,χ)=supB∈BRμχ(2B)μχ(B)=sup0<r⩽Rμχ(B2r)μχ(Br),$$\begin{equation} D(R,\chi )= \sup _{B\in \mathcal {B}_R} \frac{\mu _\chi (2B)}{\mu _\chi (B)} = \sup _{0&lt;r\leqslant R}\frac{\mu _\chi (B_{2r})}{\mu _\chi (B_r)}, \end{equation}$$where the latter equality holds since μχ(B(cB,r))=(χδ−1)(cB)μχ(Br)$\mu _\chi (B(c_B,r)) = (\chi \delta ^{-1})(c_B) \mu _\chi (B_r)$ for all r>0$r&gt;0$ and cB∈G$c_B\in G$.In the following lemma we estimate the local doubling constant D(R,χ)$D(R,\chi )$.2.1LemmaThe metric measure space (G,dC,μχ)$(G, d_C, \mu _\chi )$ is doubling if and only if χ=1$\chi =1$ and (G,dC,ρ)$(G, d_C, \rho )$ is doubling, in which case there exists C>0$C&gt;0$ such that D(R,χ)⩽C$D(R,\chi )\leqslant C$. If χ≠1$\chi \ne 1$ and (G,dC,ρ)$(G, d_C, \rho )$ is doubling, then there exists C>0$C&gt;0$ such thatD(R,χ)⩽Ce3c(χ)R∀R>0,$$\begin{equation*} D(R,\chi ) \leqslant C \mathrm{e}^{3c(\chi )R} \qquad \forall R&gt;0, \end{equation*}$$while if χ≠1$\chi \ne 1$ and (G,dC,ρ)$(G, d_C, \rho )$ is non‐doubling, then there exists C>0$C&gt;0$ such thatD(R,χ)⩽Ce(2D−D0+3c(χ))R∀R>0.$$\begin{equation*} D(R,\chi ) \leqslant C \mathrm{e}^{(2D - D_0 + 3c(\chi ))R} \qquad \forall R&gt;0. \end{equation*}$$ProofIf χ=1$\chi =1$, the first statement is obvious since μχ=ρ$\mu _\chi =\rho$.Assume then that χ≠1$\chi \ne 1$, so that there is x∈G$x\in G$ with χ(x)>1$\chi (x)&gt;1$. If N$N$ denotes the lowest integer such that N⩾|x|$N\geqslant |x|$, then BNxn⊆B(n+1)N$B_N x^n\subseteq B_{(n+1)N}$. If r>N$r&gt;N$ and n$n$ is the largest integer such that (n+1)N⩽[r]$(n+1)N\leqslant [r]$, thenμχ(Br)⩾μχ(BNxn)=χ(x)nμχ(BN)⩾χ(x)[r]/N−2μχ(BN),$$\begin{equation*} \mu _\chi (B_r) \geqslant \mu _\chi (B_N x^{n}) = \chi (x)^{n} \mu _\chi (B_N)\geqslant \chi (x)^{[r]/N-2} \mu _\chi (B_N), \end{equation*}$$whence μχ(Br)$\mu _\chi (B_r)$ grows exponentially with r$r$ and the space (G,dC,μχ)$(G, d_C, \mu _\chi )$ is non‐doubling.We now show the two bounds on D(R,χ)$D(R,\chi )$. First, observe that by (2.1), (2.3) and (2.4)μχ(B2r)μχ(Br)⩽C∀r∈(0,1],$$\begin{equation*} \frac{\mu _\chi (B_{2r})}{\mu _\chi (B_r)}\leqslant C \qquad \forall r\in (0,1], \end{equation*}$$for some C>0$C&gt;0$. Moreover, if (G,dC,ρ)$(G, d_C, \rho )$ is doubling, then by (2.3) and (2.4)μχ(B2r)μχ(Br)⩽e3c(χ)rV(2r)V(r)⩽Ce3c(χ)r∀r⩾1,$$\begin{equation*} \frac{\mu _\chi (B_{2r})}{\mu _\chi (B_{r})} \leqslant \mathrm{e}^{3c(\chi )r}\frac{V(2r)}{V(r)}\leqslant C\mathrm{e}^{3c(\chi )r}\qquad \forall r\geqslant 1, \end{equation*}$$while if (G,dC,ρ)$(G, d_C, \rho )$ is non‐doubling, then the stated estimate follows similarly by (2.2),  (2.3) and (2.4).□$\Box$THE LOCAL POINCARÉ INEQUALITY ON LIE GROUPSIn this section we prove the Lp$L^p$‐Poincaré inequality for smooth functions on (G,dC,μχ)$(G, d_C, \mu _\chi )$. Given a ball B$B$ and f∈C∞(G)$f\in C^\infty (G)$, we denote by fBχ$f_B^\chi$ its average over B$B$ with respect to μχ$\mu _\chi$,fBχ=1μχ(B)∫Bfdμχ,$$\begin{equation*} f_B^\chi = \frac{1}{\mu _\chi (B)}\int _Bf \, {d}\mu _\chi , \end{equation*}$$and we let |∇f|2=∑j=1ℓ|Xjf|2${|\nabla f |}^{2}={\sum}_{j=1}^{\ell}{|{X}_{j}f |}^{2}$. If S$S$ is a set of variables, we denote by C(S)$C(S)$ a constant depending only on the elements of S$S$.3.1TheoremThere exist a constant C=C(G,X)>0$C=C(G,\mathbf {X})&gt;0$ and a universal constant α>0$\alpha &gt;0$ such that, for all p∈[1,∞)$p\in [1,\infty )$, R>0$R&gt;0$, all balls B$B$ of radius r∈(0,R]$r\in (0,R]$ and f∈C∞(G)$f\in C^\infty (G)$,3.1∥f−fBχ∥Lp(B,μχ)⩽Ce1p[2c(χ)+c(χδ−1)]RD(R,χ)αr∥|∇f|∥Lp(B,μχ).$$\begin{equation} \Vert f - f_B^\chi \Vert _{L^p(B,\mu _\chi )} \leqslant C\,e^{ \frac{1}{p} [2c(\chi ) + c(\chi \delta ^{-1})] R}\, D(R, \chi )^{\alpha } \, r \,\Vert |\nabla f|\Vert _{L^p(B,\mu _\chi )}. \end{equation}$$Notice that the Poincaré constant grows at most exponentially with respect to the radius of the ball. The exponential term cannot, in general, be removed. After establishing the theorem, indeed, we show that when G$G$ is the so‐called “ax+b$ax+b$” group and μχ=λ$\mu _\chi =\lambda$ is a left Haar measure, the growth of the constant is indeed exponential.ProofLet p∈[1,∞)$p\in [1,\infty )$ be given. We shall prove that for every ball B$B$ of radius r>0$r &gt;0$ and f∈C∞(G)$f\in C^\infty (G)$3.2∫B|f−fBχ|pdμχ⩽2pec(χδ−1)re2c(χ)rμχ(B2r)μχ(Br)rp∫2B|∇f|pdμχ.$$\begin{equation} \int _B |f - f_B^\chi |^p\, {d}\mu _\chi \leqslant 2^p\, \mathrm{e}^{c(\chi \delta ^{-1})r} \mathrm{e}^{2c(\chi ) r} \, \frac{\mu _\chi (B_{2r})}{\mu _\chi (B_r)} \,r^p \, \int _{2B} |\nabla f|^p \, {d}\mu _\chi . \end{equation}$$Once (3.2) is at disposal, the Poincaré inequality can be obtained by classical arguments, see, for example, [11, Theorem 9.7]. A careful inspection of [13, Section 5], in particular, shows how a Whitney decomposition of B$B$ brings to the constant given in the statement. We omit the details, which would be tedious and an almost verbatim repetition of the arguments that the reader can find in [13].We then show (3.2). For z∈G$z\in G$, let γz:[0,|z|]→G$\gamma _z \colon [0,|z|] \rightarrow G$ be a C1$C^1$‐geodesic such that γz(0)=e$\gamma _z(0)=e$, γz(|z|)=z$\gamma _z({|z|})=z$, γz(s)∈B|z|$\gamma _z(s)\in B_{|z|}$ and |γz′(s)|⩽1$|\gamma _z^{\prime }(s)|\leqslant 1$ for every s∈[0,|z|]$s\in [0,|z|]$.Let B$B$ be a ball of radius r>0$r&gt;0$. Observe that if x,y∈B$x,y \in B$, and z=x−1y$z= x^{-1}y$, then |z|<2r$|z|&lt;2r$. For every x,z∈G$x,z\in G$, by Hölder's inequality3.3|f(x)−f(xz)|p⩽∫0|z||∇f(xγz(s))|dsp⩽|z|p−1∫0|z||∇f(xγz(s))|pds.$$\begin{equation} |f(x) - f(xz)|^p\leqslant {\left(\int _0^{|z|} |\nabla f(x\gamma _z(s))|\, {d}s\right)}^p \leqslant |z|^{p-1}\int _0^{|z|} |\nabla f(x\gamma _z(s))|^p\, {d}s. \end{equation}$$We then have∫B|f−fBχ|pdμχ=∫B1μχ(B)∫Bf(x)−f(y)dμχ(y)pdμχ(x)⩽1μχ(B)∫B∫Bf(x)−f(y)pdμχ(y)dμχ(x),$$\begin{align*} \int _B |f - f_B^\chi |^p\, {d}\mu _\chi & = \int _B {\left| \frac{1}{\mu _\chi (B)} \int _B {\left(f(x)-f(y)\right)}\, {d}\mu _\chi (y)\right|}^p \, {d}\mu _\chi (x)\\ & \leqslant \frac{1}{\mu _\chi (B)} \int _B \int _B {\left| f(x)-f(y)\right|}^p\, {d}\mu _\chi (y) \, {d}\mu _\chi (x), \end{align*}$$and after the change of variables y=xz$y=xz$, we get∫B|f−fBχ|pdμχ⩽1μχ(B)∫G∫G1B(x)1B(xz)f(x)−f(xz)p(χδ−1)(x)dμχ(x)dμχ(z).$$\begin{align*} \int _B |f - f_B^\chi |^p\, {d}\mu _\chi \leqslant \frac{1}{\mu _\chi (B)} \int _G \int _G \mathbf {1}_B(x)\mathbf {1}_B(xz){\left| f(x)-f(xz)\right|}^p (\chi \delta ^{-1})(x)\, {d}\mu _\chi (x)\, {d}\mu _\chi (z). \end{align*}$$Observe now that by (3.3) and Fubini's theorem, we get∫G1B(x)1B(xz)f(x)−f(xz)p(χδ−1)(x)dμχ(x)⩽1μχ(B)|z|p−1∫0|z|∫G1B(x)1B(xz)|∇f(xγz(s))|p(χδ−1)(x)dμχ(x)ds.$$\begin{align*} &\int _G \mathbf {1}_B(x)\mathbf {1}_B(xz){\left| f(x)-f(xz)\right|}^p (\chi \delta ^{-1})(x)\, {d}\mu _\chi (x)\\ & \quad \leqslant \frac{1}{\mu _\chi (B)} |z|^{p-1} \int _0^{|z|} \int _G \mathbf {1}_B(x)\mathbf {1}_B(xz)|\nabla f(x\gamma _z(s))|^p\, (\chi \delta ^{-1})(x)\,{d}\mu _\chi (x)\, {d}s . \end{align*}$$We make the change of variables ζ=xγz(s)$\zeta = x\gamma _z(s)$ and observe that by (2.3), if x∈B$x\in B$, then(χδ−1)(x)⩽(χδ−1)(cB)supBr(χδ−1)⩽(χδ−1)(cB)ec(χδ−1)r,$$\begin{equation*} (\chi \delta ^{-1})(x) \leqslant (\chi \delta ^{-1})(c_B) \, \sup _{B_r }(\chi \delta ^{-1}) \leqslant (\chi \delta ^{-1})(c_B)\, \mathrm{e}^{c(\chi \delta ^{-1})r}, \end{equation*}$$and χ(γz(s))⩽e2c(χ)r$\chi (\gamma _z(s)) \leqslant \mathrm{e}^{2 c(\chi )r}$. We obtain∫G1B(x)1B(xz)(χδ−1)(x)|∇f(xγz(s))|pdμχ(x)⩽(χδ−1)(cB)ec(χδ−1)re2c(χ)r∫G1Bγz(s)(ζ)1Bz−1γz(s)(ζ)|∇f(ζ)|pdμχ(ζ).$$\begin{align*} &\int _G \mathbf {1}_B(x)\mathbf {1}_B(xz) (\chi \delta ^{-1})(x) |\nabla f(x\gamma _z(s))|^p\, {d}\mu _\chi (x)\\ & \quad \leqslant (\chi \delta ^{-1})(c_B) \, \mathrm{e}^{c(\chi \delta ^{-1})r}\, \mathrm{e}^{2 c(\chi )r} \int _G \mathbf {1}_{B\gamma _z(s)}(\zeta )\mathbf {1}_{Bz^{-1}\gamma _z(s)}(\zeta ) |\nabla f(\zeta )|^p\, {d}\mu _\chi (\zeta ). \end{align*}$$Notice that Bγz(s)∩Bz−1γz(s)⊆2B$B\gamma _z(s) \cap Bz^{-1}\gamma _z(s) \subseteq {2B}$ for all s∈[0,|z|]$s\in [0,|z|]$. This is straightforward by the triangle inequality when |z|<r$|z|&lt; r$. Otherwise, let s0∈[0,|z|]$s_0\in [0,|z|]$ be such that |γz(s0)|=r$|\gamma _z(s_0)|=r$. Then Bγz(s)⊆2B$B\gamma _z(s)\subseteq 2B$ for s∈[0,s0]$s\in [0,s_0]$ and Bz−1γz(s)⊆2B$Bz^{-1}\gamma _z(s)\subseteq 2B$ for s∈(s0,|z|]$s\in (s_0,|z|]$. Therefore∫G1B(x)1B(xz)(χδ−1)(x)|∇f(xγz(s))|pdμχ(x)⩽(χδ−1)(cB)ec(χδ−1)re2c(χ)r1B2r(z)∫2B|∇f(ζ)|pdμχ(ζ).$$\begin{align*} &\int _G \mathbf {1}_B(x)\mathbf {1}_B(xz) (\chi \delta ^{-1})(x) |\nabla f(x\gamma _z(s))|^p\, {d}\mu _\chi (x)\\ & \quad \leqslant (\chi \delta ^{-1})(c_B) \, \mathrm{e}^{c(\chi \delta ^{-1})r}\, \mathrm{e}^{2 c(\chi )r} \mathbf {1}_{B_{2r}}(z) \int _{2B} |\nabla f(\zeta )|^p\, {d}\mu _\chi (\zeta ). \end{align*}$$Since (χδ−1)(cB)μχ(B2r)=μχ(2B)$(\chi \delta ^{-1})(c_B) \mu _\chi (B_{2r}) = \mu _\chi (2B)$, by integrating with respect to z$z$ we get∫B|f−fBχ|pdμχ⩽e2c(χ)rec(χδ−1)r(2r)pμχ(2B)μχ(B)∫2B|∇f(ζ)|pdμχ(ζ),$$\begin{align*} \int _B |f - f_B^\chi |^p\, {d}\mu _\chi & \leqslant \mathrm{e}^{2 c(\chi )r} \mathrm{e}^{c(\chi \delta ^{-1})r} (2r)^p \, {\frac{\mu _\chi (2B)}{\mu _\chi (B)} } \int _{2B} |\nabla f(\zeta )|^p \, {d}\mu _\chi (\zeta ), \end{align*}$$which concludes the proof. □$\Box$As a corollary, we obtain the so‐called local parabolic Harnack principle. We introduce the operator3.4Δχ=−∑j=1ℓ(Xj2+(Xjχ)(e)Xj),$$\begin{equation} \Delta _{\chi } =-\sum _{j=1}^{\ell }(X_j^2 +(X_j\chi )(e)X_j ), \end{equation}$$which is essentially self‐adjoint on L2(μχ)$L^2(\mu _\chi )$ and non‐negative; see, for example, [4, 12]. We say that Δχ$\Delta _\chi$ satisfies the local parabolic Harnack principle up to distance R>0$R&gt;0$ if there is C(R)>0$C(R)&gt;0$ such that, for all x∈G$x\in G$, r∈(0,R]$r\in (0,R]$, s∈R$s\in {\mathbb {R}}$, and any positive solutions u$u$ of (∂t+Δχ)u=0$(\partial _t +\Delta _\chi )u=0$ on (s,s+r2)×B(x,r)$(s,s+r^2)\times B(x,r)$, we have thatsupQ−u⩽C(R)infQ+u,$$\begin{equation*} \sup _{Q_-} u \leqslant C(R) \inf _{Q_+} u, \end{equation*}$$whereQ−=s+r2/6,s+r2/3×B(x,r/2),Q+=s+2r2/3,s+r2×B(x,r/2).$$\begin{equation*} Q_- = {\left(s+r^2/6, s + r^2/3\right)} \times B(x,r/2),\qquad Q_+ = {\left(s+2r^2/3, s+r^2\right)} \times B(x,r/2). \end{equation*}$$The following result follows at once from Theorem 3.1 and [16, Theorem 2.1].3.2CorollaryFor every R>0$R&gt;0$, Δχ$\Delta _\chi$ satisfies the local parabolic Harnack principle up to distance R$R$. In particular, the positive Δχ$\Delta _\chi$‐harmonic functions satisfy the local elliptic Harnack inequality.Exponential growth of the constantFor r>0$r&gt;0$ and p∈[1,∞)$p\in [1,\infty )$, define3.5C(r,p)=inf∫Br|f−fBχ|pdμχ∫Br|∇f|pdμχ,$$\begin{equation} C(r,p)=\inf \, \frac{\int _{B_r} |f - f_B^\chi |^p\, {d}\mu _\chi }{\int _{B_r} |\nabla f|^p \, {d}\mu _\chi }, \end{equation}$$where the infimum runs over all functions f∈C∞(G)$f \in C^{\infty }(G)$. In this section we show that the exponential bound of C(r,p)$C(r,p)$ appearing in inequality (3.1) is in general optimal, in the sense that such constant cannot grow less than exponentially with respect to r$r$. Indeed, in the particular case of ax+b$ax+b$ groups of arbitrary dimension, we provide a lower bound of exponential type for C(r,p)$C(r,p)$. For notational convenience, we shall write A≲B$A \lesssim B$ to indicate that there is a constant C$C$ such that A⩽CB$A \leqslant CB$. If A≲B$A\lesssim B$ and B≲A$B\lesssim A$, then we write A≈B$A\approx B$.Let G=Rn−1⋊R+$G=\mathbb {R}^{n-1} \rtimes \mathbb {R}^+$ and let (x,a)$(x,a)$ be its generic element. Recall thatdλ(x,a)=dxdaananddρ(x,a)=dxdaa,$$\begin{equation*} {d}\lambda (x,a) = \frac{{d}x\, {d}a}{a^n} \qquad {\rm {and}}\qquad {d}\rho (x,a) = \frac{{d}x\, {d}a}{a}, \end{equation*}$$since δ(x,a)=a−n+1$\delta (x,a) = a^{-n+1}$; all positive characters of G$G$ are of the form χγ(x,a)=aγ$ \chi _\gamma (x,a) = a^\gamma$ for some γ∈R$\gamma \in \mathbb {R}$. We shall write μγ$\mu _\gamma$ for the measure μχγ$\mu _{\chi _\gamma }$. In particular, λ=μ1−n$\lambda = \mu _{1-n}$ is the hyperbolic measure. We consider the left‐invariant vector fields Xi=a∂i$X_i=a\partial _i$, i=1,⋯,n−1$i=1,\dots ,n-1$, and X0=a∂a$X_0=a\partial _a$ which form a basis of the Lie algebra of G$G$. The distance induced by such vector fields is the hyperbolic metric which is given bycosh|(x,a)|=12(a+a−1+a−1|x|2),$$\begin{equation*} \cosh |(x,a)| = \tfrac{1}{2}(a+a^{-1} + a^{-1}|x|^2), \end{equation*}$$where |x|$|x|$ is the Euclidean norm of x∈Rn−1$x\in \mathbb {R}^{n-1}$ (see [1, (2.18)], [17, (1.1)]). ThenBr=(x,a):e−r<a<er,|x|2<2a(coshr−coshloga).$$\begin{equation*} B_r = {\left\lbrace (x,a)\colon \mathrm{e}^{-r}&lt;a&lt;\mathrm{e}^r, \; |x|^2 &lt; 2a(\cosh r - \cosh \log a)\right\rbrace} . \end{equation*}$$In the case of the real hyperbolic space, that is, the ax+b$ax+b$ group endowed with the measure λ$\lambda$ and the metric defined above, the constant C(r,p)$C(r,p)$ in (3.5) was estimated from above in [11, Section 10.1]. We now estimate such constant from below.Consider the function ϕ:G→R$\phi \colon G\rightarrow \mathbb {R}$ defined byϕ(x,a)=x1,(x,a)∈G.$$\begin{equation*} \phi (x,a) = x_1,\quad (x,a)\in G. \end{equation*}$$Observe that ∫Brϕdμγ=0$\int _{B_r} \phi \, {d}\mu _\gamma =0$ for all γ∈R$\gamma \in \mathbb {R}$ and |∇ϕ(x,a)|p=ap$|\nabla \phi (x,a) |^p = a^p$. Moreover,∫Br|ϕ|pdμγ≈∫e−reraγ−1+p+n−12(coshr−coshloga)p+n−12da,$$\begin{equation*} \int _{B_r} |\phi |^p \, {d}\mu _\gamma \approx \int _{\mathrm{e}^{-r}}^{\mathrm{e}^r} a^{\gamma -1 + \frac{p+n-1}{2}} (\cosh r - \cosh \log a)^{\frac{p+n-1}{2}} {{d}a}, \end{equation*}$$while∫Br|∇ϕ|pdμγ≈∫e−reraγ−1+p+n−12(coshr−coshloga)n−12da.$$\begin{equation*} \int _{B_r} |\nabla \phi |^p \, {d}\mu _\gamma \approx \int _{\mathrm{e}^{-r}}^{\mathrm{e}^r} a^{\gamma -1 +p+ \frac{n-1}{2}}{(\cosh r - \cosh \log a)^{\frac{n-1}{2}}} \, {d}a . \end{equation*}$$3.3LemmaLet δ∈R$\delta \in \mathbb {R}$ and ε>0$\epsilon &gt;0$. Then∫e−reraδ(coshr−coshloga)εda≈er(|δ+1|+ε).$$\begin{equation*} \int _{\mathrm{e}^{-r}}^{\mathrm{e}^r} a^{\delta }{(\cosh r - \cosh \log a)^{\epsilon }} \, {d}a \approx \mathrm{e}^{r(|\delta +1| + \epsilon ) }. \end{equation*}$$ProofWe first make a change of variables∫e−reraδ(coshr−coshloga)εda=∫−rret(δ+1)(coshr−cosht)εdt.$$\begin{equation*} \int _{\mathrm{e}^{-r}}^{\mathrm{e}^r} a^{\delta }{(\cosh r - \cosh \log a)^{\epsilon }} \, {d}a = \int _{-r}^r \mathrm{e}^{t(\delta +1)} (\cosh r - \cosh t)^{\epsilon } \, {d}t. \end{equation*}$$Since coshr−cosht≈er$\cosh r - \cosh t \approx \mathrm{e}^r$ if |t|<r−1$ |t|&lt;r-1$, while coshr−cosht≈(r−|t|)er$\cosh r - \cosh t \approx (r-|t|)\mathrm{e}^r$ if r−1<|t|<r$ r-1&lt;|t|&lt;r$, we get∫−r+1r−1et(δ+1)(coshr−cosht)εdt≈eεr∫−r+1r−1et(δ+1)dt≈e(ε+|δ+1|)r$$\begin{equation*} \int _{-r+1}^{r-1} \mathrm{e}^{t(\delta +1)} (\cosh r - \cosh t)^{\epsilon } \, {d}t\approx \mathrm{e}^{\epsilon r} \int _{-r+1}^{r-1} \mathrm{e}^{t(\delta +1)}\, {d}t \approx \mathrm{e}^{(\epsilon + |\delta +1|) r} \end{equation*}$$while∫r−1<|t|<ret(δ+1)(coshr−cosht)εdt≈e(ε+|δ+1|)r∫r−1<|t|<r(r−|t|)εdt≈e(ε+|δ+1|)r,$$\begin{equation*} \int _{r-1&lt;|t|&lt;r} \mathrm{e}^{t(\delta +1)} (\cosh r - \cosh t)^{\epsilon } \, {d}t\approx \mathrm{e}^{(\epsilon +|\delta +1|) r} \int _{r-1&lt;|t|&lt;r} (r-|t|)^\epsilon \, {d}t \approx \mathrm{e}^{(\epsilon + |\delta +1|) r} , \end{equation*}$$as required.□$\Box$From the lemma above, we get that∫Br|ϕ|pdμγ≈e(|γ+p+n−12|+p+n−12)r,$$\begin{equation*} \int _{B_r} |\phi |^p \, {d}\mu _\gamma \approx \mathrm{e}^{({|\gamma + \frac{p+n-1}{2}|} + \frac{p+n-1}{2})r}, \end{equation*}$$while∫Br|∇ϕ|pdμγ≈e(|γ+p+n−12|+n−12)r.$$\begin{equation*} \int _{B_r} |\nabla \phi |^p \, {d}\mu _\gamma \approx \mathrm{e}^{({|\gamma +p+ \frac{n-1}{2}| }+ \frac{n-1}{2})r}. \end{equation*}$$We observe that, if γ<−p+n−12$\gamma &lt; {-\frac{p+n-1}{2}}$, then|γ+p+n−12|+p+n−12>|γ+p+n−12|+n−12.$$\begin{equation*} {\bigg |\gamma + \frac{p+n-1}{2}\bigg |} + \frac{p+n-1}{2} &gt; { \bigg |\gamma +p+ \frac{n-1}{2} \bigg |} + \frac{n-1}{2}. \end{equation*}$$Thus for such γ$\gamma$C(r,p)⩾Cerγ+p+n−12+p2−γ+p+n−12.$$\begin{equation*} C(r,p) \geqslant C \mathrm{e}^{ r\left({{\left|\gamma + \frac{p+n-1}{2}\right|} }+ \frac{p}{2} - { {\left|\gamma +p+ \frac{n-1}{2} \right|}} \right)}. \end{equation*}$$If in particular γ=−n+1$\gamma = -n+1$, hence μγ$\mu _\gamma$ is the left measure, and n>p+1$n&gt;p+1$, thenC(r,p)⩾e−n+p+12−−n+12+p+p2r=eprn⩾2p+1e(n−p−1)rp+1<n⩽2p+1.$$\begin{equation*} C(r,p) { \geqslant \mathrm{e}^{ \left({\left| \frac{-n+p+1}{2} \right|}- {\left| \frac{-n+1}{2}+p \right|} +\frac{p}{2}\right)r } = {\begin{cases} \mathrm{e}^{pr} & n \geqslant 2p+1\\ \mathrm{e}^{(n-p-1)r} & p+1 &lt;n\leqslant 2p+1. \end{cases}} } \end{equation*}$$NON‐LOCAL POINCARÉ INEQUALITYIn this second part of the paper we prove a non‐local L2$L^2$‐Poincaré inequality for suitable finite measures on G$G$ in the spirit of [14, 15]. More precisely, let M$M$ be a positive C2$C^2$‐function in L1(μχ)$L^1(\mu _\chi )$ and μχ,M$\mu _{\chi ,M}$ be the finite measure whose density is M$M$ with respect to μχ$\mu _\chi$. We shall prove L2$L^2$‐global Poincaré inequalities for the measure μχ,M$\mu _{\chi ,M}$ for a large family of functions M$M$. In order to do this, we letL12(μχ,M)={f∈L2(μχ,M):|∇f|∈L2(μχ,M)}$$\begin{equation*} L^2_1(\mu _{\chi ,M}) = \lbrace f\in L^2(\mu _{\chi ,M}) \colon |\nabla f| \in L^2(\mu _{\chi ,M}) \rbrace \end{equation*}$$and introduce the operator4.1Δχ,M=Δχ−∇(logM)·∇,$$\begin{equation} \Delta _{\chi ,M} = \Delta _\chi - \nabla (\log M) \cdot \nabla , \end{equation}$$where Δχ$\Delta _\chi$ is that of (3.4), Dom(Δχ,M)={f∈L12(μχ,M):Δχ,Mf∈L2(μχ,M)}$\mathrm{Dom}(\Delta _{\chi ,M}) = \lbrace f \in L^2_1(\mu _{\chi ,M})\colon \Delta _{\chi ,M} f \in L^2(\mu _{\chi ,M}) \rbrace$ and the derivatives are meant in the distributional sense. Observe that Δχ,M$\Delta _{\chi ,M}$ is symmetric on L2(μχ,M)$L^2(\mu _{\chi ,M})$; in particular, for all f∈Dom(Δχ,M)$f\in \mathrm{Dom}(\Delta _{\chi ,M})$ and g∈L12(μχ,M)$g\in L^2_1(\mu _{\chi ,M})$,∫G∇f·∇gdμχ,M=∫GΔχ,Mf·gdμχ,M,$$\begin{equation*} \int _{G} \nabla f \cdot \nabla g\, {d}\mu _{\chi ,M}=\int _{G} \Delta _{\chi ,M} f \cdot g\, {d}\mu _{\chi ,M}, \end{equation*}$$where ∇f·∇g=∑j=1ℓ(Xjf)(Xjg)$\nabla f \cdot \nabla g = \sum _{j=1}^\ell (X_j f)(X_j g)$.We say that the couple (Δχ,M)$(\Delta _\chi ,M)$ admits a Lyapunov function if there exist a C2$C^2$ function W:G→[1,∞)$W\colon G\rightarrow [1,\infty )$ and constants θ>0$\theta &gt;0$, b⩾0$b\geqslant 0$, R>0$R&gt;0$ such that4.2−Δχ,MW(x)⩽−θW(x)+b1BR(x)∀x∈G.$$\begin{equation} - \Delta _{\chi ,M} W(x)\leqslant -\theta W(x)+b{\bf 1}_{B_R}(x) \qquad \forall x\in G. \end{equation}$$Observe that the existence of a Lyapunov function depends on G$G$, X$\mathbf {X}$, χ$\chi$ and M$M$. For f∈L1(μχ,M)$f\in L^1(\mu _{\chi ,M})$ we letfχ,M=1μχ,M(G)∫Gfdμχ,M.$$\begin{equation*} f_{\chi ,M} = \frac{1}{\mu _{\chi ,M}(G)}\int _G f\, {d}\mu _{\chi ,M}. \end{equation*}$$Our second main result is the following global L2$L^2$‐Poincaré inequality for μχ,M$\mu _{\chi ,M}$.4.1TheoremIf (Δχ,M)$(\Delta _\chi ,M)$ admits a Lyapunov function, then there exists a constant C=C(G,X,χ,M)$C=C(G, \mathbf {X}, \chi , M)$ such that for all f∈L12(μχ,M)$f\in L^2_1(\mu _{\chi ,M})$4.3∥f−fχ,M∥L2(μχ,M)⩽C∥|∇f|∥L2(μχ,M).$$\begin{equation} \Vert f- f_{\chi ,M}\Vert _{L^2(\mu _{\chi ,M})} \leqslant C \Vert |\nabla f|\Vert _{L^2(\mu _{\chi ,M})}. \end{equation}$$Theorem 4.1 is a generalization to any connected non‐compact possibly non‐unimodular Lie group of the non‐local L2$L^2$‐Poincaré inequalities proved in [2, Theorem 1.4] in the Euclidean setting and in [15, Theorem 1.1] in unimodular Lie groups of polynomial growth, by which our proof is inspired. The validity of other versions of non‐local Poincaré inequalities in the current setting, such as those in [15, Theorem 1.4], is still an open problem. More general versions of non‐local Poincaré inequalities of this kind were proved in [9] in the setting of a topological measure space endowed with a family of sets which play the role of unit balls and satisfy suitable assumptions. Recently in [8] non‐local Lp$L^p$‐Poincaré inequalities were obtained on Carnot groups of Engel type in the case when the density of the measure depends on a homogeneous norm of the group; we note, however, that the case p=2$p=2$ is always excluded.ProofLet f∈L12(μχ,M)$f\in L^2_1(\mu _{\chi ,M})$, and observe first that4.4∫Gf−fχ,M2dμχ,M=minc∈R∫Gf−c2dμχ,M.$$\begin{equation} \int _G {\left| f- f_{\chi ,M} \right|}^2 {d}\mu _{\chi ,M} = \min _{c\in \mathbb {R}} \int _G {\left| f- c\right|}^2 {d}\mu _{\chi ,M}. \end{equation}$$Let now g=f−c$g=f-c$ for a positive c$c$ to be determined, and W$W$ be a Lyapunov function for (Δχ,M)$(\Delta _\chi ,M)$. By (4.2)4.5∫G|g|2dμχ,M⩽∫G|g|2Δχ,MWθWdμχ,M+∫BR|g|2bθWdμχ,M.$$\begin{equation} \int _G |g|^2 \, {d}\mu _{\chi ,M} \leqslant \int _G |g|^2 \frac{\Delta _{\chi ,M} W}{\theta W}\, {d}\mu _{\chi ,M} +\int _{B_R} |g|^2 \frac{b}{\theta W}\, {d}\mu _{\chi ,M}. \end{equation}$$We treat the two terms separately.Let us consider the first term and prove that4.6∫GΔχ,MWWg2dμχ,M⩽∫G|∇g|2dμχ,M.$$\begin{equation} \int _G \frac{\Delta _{\chi ,M} W}{W}g^2 {d}\mu _{\chi ,M} \leqslant \int _G |\nabla g |^2{d}\mu _{\chi ,M}. \end{equation}$$We prove it by density, and firstly assume that g$g$ is compactly supported. By definition of Δχ,M$\Delta _{\chi ,M}$,∫GΔχ,MWWg2dμχ,M=∫G∇g2W·∇Wdμχ,M=2∫GgW∇g·∇Wdμχ,M−∫Gg2W2|∇W|2dμχ,M=∫G|∇g|2dμχ,M−∫G|∇g−gW∇W|2dμχ,M⩽∫G|∇g|2dμχ,M.$$\begin{align*} \int _G \frac{\Delta _{\chi ,M} W}{W} g^2 \,{d}\mu _{\chi ,M} &= \int _G \nabla {\left(\frac{g^2}{W}\right)}\cdot \nabla W\, {d}\mu _{\chi ,M} \\ & = 2 \int _G \frac{g}{W} \nabla g\cdot \nabla W\, {d}\mu _{\chi ,M} -\int _G \frac{g^2}{W^2} |\nabla W|^2 \, {d}\mu _{\chi ,M}\\ & = \int _G |\nabla g|^2\, {d}\mu _{\chi ,M} - \int _G |\nabla g-\frac{g}{W} \nabla W|^2 \, {d}\mu _{\chi ,M} \\ & \leqslant \int _G |\nabla g|^2{d}\mu _{\chi ,M}. \end{align*}$$Let now g∈L12(μχ,M)$g\in L^2_1(\mu _{\chi ,M})$, and consider a non‐decreasing sequence of functions ψn∈Cc∞(G)$\psi _n\in C_c^\infty (G)$ such that1BnR⩽ψn⩽1,|∇ψn|⩽1.$$\begin{equation*} {\bf 1}_{B_{nR}} \leqslant \psi _n\leqslant 1, \quad |\nabla \psi _n| \leqslant 1. \end{equation*}$$By applying  (4.6) to gψn$g\psi _n$, the monotone convergence theorem in the left‐hand side and the dominated convergence theorem in the right‐hand side, one gets (4.6).To deal with the second term, we choose c$c$ such that ∫BRgdμχ=0$\int _{B_R}g\, {d}\mu _\chi =0$. By (3.2) applied to g$g$ on BR$B_R$, and the fact that M$M$ is bounded from above and below on B2R$B_{2R}$, one has∫BR|g|2dμχ,M⩽C∫BR|g|2dμχ⩽C∫B2R|∇g|2dμχ⩽C∫B2R|∇g|2dμχ,M,$$\begin{align*} \int _{B_R} |g|^2 \, {d}\mu _{\chi ,M} \leqslant C \int _{B_R} |g|^2 \, {d}\mu _\chi \leqslant C \int _{B_{2R}} |\nabla g|^2\, {d}\mu _\chi \leqslant C \int _{B_{2R}} |\nabla g|^2\, {d}\mu _{\chi ,M}, \end{align*}$$where the constant C$C$ depends on R$R$ and M$M$. Therefore, since W⩾1$W\geqslant 1$,∫BR|g|2bθWdμχ,M⩽C∫B2R|∇g|2dμχ,M⩽C∫G|∇g|2dμχ,M,$$\begin{equation*} \int _{B_R} |g|^2 \frac{b}{\theta W}\, {d}\mu _{\chi ,M} \leqslant C \int _{B_{2R}} |\nabla g|^2\, {d}\mu _{\chi ,M} \leqslant C \int _{G} |\nabla g|^2\, {d}\mu _{\chi ,M} , \end{equation*}$$which completes the proof.□$\Box$4.2CorollaryLet v=−logM$v= -\log M$. If there exist a∈(0,1)$a\in (0,1)$, c>0$c&gt;0$ and R>0$R&gt;0$ such that4.7a|∇v|2(x)+Δχv(x)⩾c∀x∈BRc,$$\begin{equation} a|\nabla v|^2(x) + \Delta _\chi v(x)\geqslant c \qquad \forall \, x \in B_R^c, \end{equation}$$then (Δχ,M)$(\Delta _\chi ,M)$ admits a Lyapunov function, and (4.3) holds.ProofLet W(x)=e(1−a)(v(x)−infGv)$W(x)= \mathrm{e}^{(1-a)(v(x)-\inf _Gv)}$, so that−Δχ,MW=(1−a)W−Δχv−a|∇v|2.$$\begin{equation*} -\Delta _{\chi ,M} W =(1-a)W {\left(- \Delta _\chi v-a|\nabla v|^2\right)}. \end{equation*}$$Then W$W$ is a Lyapunov function with θ=c(1−a)$\theta =c (1-a)$ and b=maxBR(−Δχ,MW+θW)$b= \max _{B_R} (-\Delta _{\chi ,M} W+\theta W)$.□$\Box$One can actually show that if (4.7) holds with a<1/2$a&lt;1/2$, then (4.3) self‐improves as follows.4.3PropositionLet v=−logM$v= -\log M$. If there exist c>0$c&gt;0$, R>0$R&gt;0$ and ε∈(0,1)$\epsilon \in (0,1)$ such that4.81−ε2|∇v|2(x)+Δχv(x)⩾c∀x∈BRc,$$\begin{equation} \frac{1-\epsilon }{2}|\nabla v|^2(x) + \Delta _\chi v(x)\geqslant c \qquad \forall \, x\in B_R^c, \end{equation}$$then there exists C>0$C&gt;0$ such that for all f∈L12(μχ,M)$f\in L^2_1(\mu _{\chi ,M})$4.9∥|f−fχ,M|1+|∇v|∥L2(μχ,M)⩽C∥|∇f|∥L2(μχ,M).$$\begin{equation} \Vert |f - f_{\chi ,M}|{\left(1+| \nabla v|\right)}\Vert _{L^2(\mu _{\chi ,M})} \leqslant C \Vert |\nabla f|\Vert _{L^2(\mu _{\chi ,M})}. \end{equation}$$ProofObserve firstly that, since v$v$ is C2$C^2$ and (4.8) holds,4.101−ε2|∇v|2−Δχv⩾α$$\begin{equation} \frac{1-\epsilon }{2}|\nabla v|^2 - \Delta _\chi v \geqslant \alpha \end{equation}$$for some α∈R$\alpha \in \mathbb {R}$. Let f∈L12(μχ,M)$f\in L^2_1(\mu _{\chi ,M})$ and let g=fM$g=f\sqrt {M}$. Since∇f=1M∇g+12g1M∇v,$$\begin{equation*} \nabla f=\frac{1}{\sqrt {M}} \nabla g+\frac{1}{2}g \frac{1}{\sqrt {M}} \nabla v, \end{equation*}$$by (4.10)∫G|∇f|2dμχ,M=∫G|∇g|2+14|g|2|∇v|2+g∇g·∇vdμχ=∫G|∇g|2+14|g|2|∇v|2+12∇(|g|2)·∇vdμχ⩾∫G|g|214|∇v|2+12Δχvdμχ⩾12∫G|f|2ε2|∇v|2+αdμχ,M.$$\begin{align*} \int _{G}|\nabla f|^2 \, {d}\mu _{\chi ,M} &= \int _{G} {\left(|\nabla g|^2 +\frac{1}{4}|g|^2|\nabla v|^2+ g \nabla g\cdot \nabla v\right)} \, {d}\mu _\chi \\ &= \int _{G} {\left(|\nabla g|^2 +\frac{1}{4} |g|^2|\nabla v|^2+\frac{1}{2}\nabla (|g|^2)\cdot \nabla v\right)} \, {d}\mu _\chi \\ &\geqslant \int _{G} |g|^2 {\left(\frac{1}{4}| \nabla v|^2+ \frac{1}{2} \Delta _\chi v \right)} \, {d}\mu _\chi \\ &\geqslant {\frac{1}{2} \int _{G} |f|^2 {\left(\frac{\epsilon }{2}|\nabla v|^2 + \alpha \right)}\, {d}\mu _{\chi ,M}.} \end{align*}$$Since (4.3) holds by (4.8) and Corollary 4.2, the conclusion follows.□$\Box$ACKNOWLEDGEMENTSThe authors wish to thank the anonymous referee for their careful reading of the paper and interesting comments. All authors are partially supported by the GNAMPA 2020 project “Fractional Laplacians and subLaplacians on Lie groups and trees” and are members of the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). T. Bruno acknowledges support by the Research Foundation–Flanders (FWO) through the postdoctoral grant 12ZW120N.Open Access Funding provided by Universita degli Studi di Genova within the CRUI‐CARE Agreement.JOURNAL INFORMATIONThe Bulletin of the London Mathematical Society is wholly owned and managed by the London Mathematical Society, a not‐for‐profit Charity registered with the UK Charity Commission. All surplus income from its publishing programme is used to support mathematicians and mathematics research in the form of research grants, conference grants, prizes, initiatives for early career researchers and the promotion of mathematics.REFERENCESJ.‐Ph. Anker, E. Damek, and C. Yacoub, Spherical analysis on harmonic AN$AN$ groups, Ann. Sc. Norm. Super. Pisa 33 (1996), 643–679.D. Bakry, F. Barthe, P. Cattiaux, and A. Guillin, A simple proof of the Poincaré inequality for a large class of probability measures including the log‐concave case, Electron. Comm. Probab. 13 (2008), 60–66.T. Bruno and M. Calzi, Schrödinger operators on Lie groups with purely discrete spectrum, Adv. Math. 404 (2022) n. 108444.T. Bruno, M. M. Peloso, A. Tabacco, and M. Vallarino, Sobolev spaces on Lie groups: embedding theorems and algebra properties, J. Funct. Anal. 276 (2019), no. 10, 3014–3050.T. Bruno, M. M. Peloso, and M. Vallarino, Besov and Triebel–Lizorkin spaces on Lie groups, Math. Ann. 377 (2020), no. 1–2, 335–377.T. Bruno, M. M. Peloso, and M. Vallarino, Potential spaces on Lie groups. Geometric Aspects of Harmonic Analysis, Springer INdAM Series, vol. 45, 2021, pp. 149–192.T. Bruno, M. M. Peloso, and M. Vallarino, The Sobolev embedding constant on Lie groups, Nonlinear Anal. 216 (2022), 112707.M. Chatzakou, S. Federico, and B. Zegarlinski, q‐Poincarè inequalities on Carnot Groups with a filiform Lie algebra, arXiv:2007.04689.P. T. Gressman, Fractional Poincaré and logarithmic Sobolev inequalities for measure spaces, J. Funct. Anal. 265 (2013), no. 6, 867–889.Y. Guivarc'h, Croissance polynomiale et périodes des fonctions harmoniques, Bull. Soc. Math. France 101 (1973), 333–379.P. Hajlasz and P. Koskela, Sobolev met Poincaré, Mem. Amer. Math. Soc. 145 (2000), no. 688, x+101.W. Hebisch, G. Mauceri, and S. Meda, Spectral multipliers for Sub‐Laplacians with drift on Lie groups, Math. Z. 251 (2005), no. 4, 899–927.D. Jerison, The Poincaré inequality for vector fields satisfying Hörmander's condition, Duke Math. J. 53 (1986), no. 2, 503–523.C. Mouhot, E. Russ, and Y. Sire, Fractional Poincaré inequalities for general measures, J. Math. Pure Appl. (9) 95 (2011), no. 1, 72–84.E. Russ and Y. Sire, Nonlocal Poincaré inequalities on Lie groups with polynomial volume growth and Riemannian manifolds, Studia Math. 203 (2011), no. 2, 105–127.L. Saloff‐Coste, Parabolic Harnack inequality for divergence‐form second‐order differential operators. Potential theory and degenerate partial differential operators, Potential Anal. 4 (1995), no. 4, 429–467.P. Sjögren and M. Vallarino, Boundedness from H1$H^1$ to L1$L^1$ of Riesz transforms on a Lie group of exponential growth, Ann. Inst. Fourier, Grenoble 58 (2008), no. 4, 1117–1151.N. Th. Varopoulos, Fonctions harmoniques sur les groupes de Lie, C. R. Acad. Sci. Paris, Série I, Math. 309 (1987), 519–521.N. Th. Varopoulos, Analysis on Lie groups, J. Funct. Anal. 76 (1988), no. 2, 346–410.

Journal

Bulletin of the London Mathematical SocietyWiley

Published: Dec 1, 2022

There are no references for this article.