The logarithmic Sobolev inequality on the Heisenberg group and applications to the uncertainty inequality and heat equation

Abstract

We consider the logarithmic Sobolev inequality on the Heisenberg group. One can derive the logarithmic Sobolev inequality from the Sobolev inequality, and we consider an application to the uncertainty inequality on the Heisenberg group. Moreover, one can also obtain a dissipative estimate of a solution of the heat equation on the Heisenberg group from the logarithmic Sobolev inequality.

1 Introduction

We consider Sobolev’s and a related inequality on the Heisenberg group $\mathbb{H}^{n}$. Sobolev’s inequality is an inequality that holds in the Sobolev space $W^{1,p}({\mathbb{R}}^{n})$ on n-dimensional Euclidean space ${\mathbb{R}}^{n}$, and is an essential tool in the study of nonlinear partial differential equations. The logarithmic Sobolev inequality is one of a variety of Sobolev’s inequality, an inequality that implies that the functional appearance in the Boltzmann–Gibbs–Shannon entropy is bounded by a Sobolev norm. In this paper, we derive the logarithmic Sobolev inequality on the Heisenberg group.

We first consider the n-dimensional Euclidean space case. The sharp form of Sobolev’s inequality is given in the following form: for any $f \in H^{1}({\mathbb{R}}^{n})$,

$$ \|f\|_{L^{2^{*}}({\mathbb{R}}^{n})}^{2} \leq S_{n} \|\nabla f\|_{L^{2}({ \mathbb{R}}^{n})}^{2}, $$

(1.1)

where $n \geq 3$, $2^{*}=2n/(n-2)$,

$$ \nabla = {^{t}\! \left (\frac{\partial }{\partial x_{1}}, \frac{\partial }{\partial x_{2}}, \ldots , \frac{\partial }{\partial x_{n}}\right )}, $$

and $H^{1}({\mathbb{R}}^{n}) \equiv \{f \in L^{2}({\mathbb{R}}^{n});\ | \nabla f| \in L^{2}({\mathbb{R}}^{n})\}$. The constant

$$ S_{n} = \frac{1}{n(n-2)} 2^{2(1-\frac{1}{n})} \pi ^{-(\frac{1}{n}+1)} \Gamma \left (\frac{n+1}{2}\right )^{\frac{2}{n}} $$

is the best possible, and this constant is attained by the Aubin–Talenti function:

$$ f(x) = (1+|x|^{2})^{-\frac{n-2}{2}} $$

(1.2)

up to conformal automorphism (see Aubin [1] and Talenti [18]). To be precise, Talenti [18] identified the sharp constant of Sobolev’s inequality in general cases $W^{1,p}({\mathbb{R}}^{n}) \subset L^{np/(n-p)}({\mathbb{R}}^{n})$ by finding that (1.2) attains the best possible constant. Lieb [13] proved the sharp version of the Hardy–Littlewood–Sobolev inequality and showed the sharp constant with the extremal function of Sobolev’s inequality (1.1) (see also [14]).

On the other hand, the following logarithmic Sobolev inequality was obtained by Stam [17] and Gross [10]: For any $f \in H^{1}({\mathbb{R}}^{n}) \setminus \{0\}$,

$$ \int _{{\mathbb{R}}^{n}} |f(x)|^{2} \log \frac{|f(x)|^{2}}{\|f\|_{L^{2}({\mathbb{R}}^{n})}^{2}}\, dx \leq \frac{n}{2} \|f\|_{L^{2}({\mathbb{R}}^{n})}^{2} \log \left ( \frac{2}{n\pi e \|f\|_{L^{2}({\mathbb{R}}^{n})}^{2}} \int _{{\mathbb{R}}^{n}} |\nabla f(x)|^{2}\, dx\right ), $$

(1.3)

where the constant appearing on the right-hand side is the best possible. Moreover, this constant is attained by the Gaussian function

$$ f(x) = e^{-|x|^{2}} $$

up to conformal automorphism. The optimal constant and extremal function of the inequality (1.3) were given by Weissler [22] and Carlen [3], respectively. In Lieb and Loss [14], we realize that the inequality (1.3) is equivalent to the $L^{p}$-$L^{q}$ dissipative estimate of a solution of the heat equation

$$ \left \{ \begin{aligned} &\partial _{t} u = \Delta u, &&t > 0,\ x \in{\mathbb{R}}^{n}, \\ &u(0, x) = u_{0}(x), &&x \in{\mathbb{R}}^{n}, \end{aligned} \right . $$

where $\partial _{t} = \partial /\partial t$ and $\Delta u = \nabla \cdot \nabla u$ (see also Davies [5], Carlen and Loss [4]). In fact, the extremal function of the inequality (1.3), that is, the Gaussian function coincides with the heat kernel by a conformal automorphism.

Concerning Sobolev’s inequality on the other manifold, we introduce the Heisenberg group:

Definition 1

(Heisenberg group)

The Heisenberg group $\mathbb{H}^{n} = {\mathbb{C}}^{n} \times {\mathbb{R}}$ is defined by

$$ (z, s) \cdot (z', s') = (z+z', t+t'+2\mathrm{Im}(z \cdot \bar{z'})) $$

for $(z, s), (z', s') \in {\mathbb{C}}^{n} \times {\mathbb{R}}$, where

$$ z \cdot \bar{z'} = \sum _{j=1}^{n} z_{j} \bar{z'_{j}} $$

when $z = (z_{1}, z_{2}, \dots , z_{n})$, $z' = (z'_{1}, z'_{2}, \dots , z'_{n})$.

For $(z, s) \in \mathbb{H}^{n}$ and $\lambda > 0$, the dilation is defined by $\lambda (z, s) = (\lambda z, \lambda ^{2} s)$, and we denote $|(z, s)|$ by

$$ |(z, s)| \equiv (|z|^{4} + s^{2})^{\frac{1}{4}}. $$

We denote the homogeneous dimension of $\mathbb{H}^{n}$ by $Q = 2n+2$. For $1 \leq p < \infty$, we define $L^{p}$ space on $\mathbb{H}^{n}$ by

$$ L^{p}(\mathbb{H}^{n}) \equiv \left \{f: \mathbb{H}^{n} \to{\mathbb{R}}; \ \|f\|_{p} = \left (\int _{\mathbb{H}^{n}} |f(z, s)|^{p}\, dz\, ds \right )^{\frac{1}{p}} < \infty \right \}. $$

For $j = 1, 2, \dots , n$, the vector field

$$ S = \frac{\partial }{\partial s}, \quad X_{j} = \frac{\partial }{\partial x_{j}} + 2y_{j} \frac{\partial }{\partial s}, \quad \mathrm{and}\quad Y_{j} = \frac{\partial }{\partial y_{j}} - 2x_{j} \frac{\partial }{\partial s} $$

satisfies the Hörmander condition. Then, the operator

$$ \Delta _{\mathbb{H}} = \sum _{j=1}^{n} (X_{j}^{2}+Y_{j}^{2}) $$

is a hypoelliptic operator. When we set $\nabla _{\mathbb{H}} = {^{t}\!(X_{1}, X_{2}, \dots , X_{n}, Y_{1}, Y_{2}, \dots , Y_{n})}$, we define Sobolev’s space $H^{1}(\mathbb{H}^{n})$ by

$$ H^{1}(\mathbb{H}^{n}) \equiv \left \{f \in L^{2}(\mathbb{H}^{n});\ \int _{\mathbb{H}^{n}} |\nabla _{\mathbb{H}} f(z, s)|^{2}\, dz\, ds < \infty \right \}, $$

where

$$ |\nabla _{\mathbb{H}} f(z, s)| = \left [\sum _{j=1}^{n} \left (|X_{j} f(z, s)|^{2} + |Y_{j} f(z, s)|^{2}\right )\right ]^{\frac{1}{2}}. $$

Sobolev’s inequality on Lie groups was studied by Folland [7] and Fischer and Ruzhansky [6]. For the Heisenberg group, Jerison and Lee [12] identified the sharp constant in Sobolev’s inequality, and Frank and Lieb [8] gave different proofs and derived the sharp Hardy–Littlewood–Sobolev inequality on the Heisenberg group.

Proposition 1.1

(Sobolev’s inequality on the Heisenberg group [8, 12])

For any $f \in H^{1}(\mathbb{H}^{n})$,

$$ \|f\|_{\frac{2Q}{Q-2}}^{2} \leq S_{\mathbb{H}} \|\nabla _{\mathbb{H}} f \|_{2}^{2}, $$

(1.4)

where the constant

$$ S_{\mathbb{H}} = 2^{-\frac{2}{n+1}} \pi ^{-1} n^{-2} (n!)^{ \frac{1}{n+1}} = 2^{2(1-\frac{2}{Q})} \pi ^{-1} (Q-2)^{-2} \Gamma \left (\frac{Q}{2}\right )^{\frac{2}{Q}} $$

is the best possible. Moreover, this constant is attained by

$$ f(z, s) = c H(\delta (a \cdot (z, s))), $$

where

$$ H(z, s) = \left ((1+|z|^{2})^{2}+s^{2}\right )^{-\frac{Q-2}{4}} $$

for $c \in {\mathbb{C}}$, $\delta > 0$, and $a \in \mathbb{H}^{n}$.

We derive the logarithmic Sobolev inequality on the Heisenberg group by using Sobolev’s inequality (1.4).

Theorem 1.2

(The logarithmic Sobolev inequality on the Heisenberg group)

For any $f \in H^{1}(\mathbb{H}^{n}) \setminus \{0\}$,

$$ \int _{\mathbb{H}^{n}} |f(z, s)|^{2} \log \frac{|f(z, s)|^{2}}{\|f\|_{2}^{2}}\, dz\, ds \leq \frac{Q}{2} \|f\|_{2}^{2} \log \left (\frac{S_{\mathbb{H}}}{\|f\|_{2}^{2}} \int _{\mathbb{H}^{n}} |\nabla _{\mathbb{H}} f(z,s)|^{2}\, dz\, ds\right ), $$

(1.5)

where the constant $S_{\mathbb{H}}$ is the same as the above.

For the Heisenberg group, Inglis and Papageorgiou [11] and Papageorgiou [15] showed the $L^{q}$-type logarithmic Sobolev inequality ($1 < q \leq 2$). They considered the logarithmic Sobolev inequality for Gibbs measures on the infinite product of Heisenberg groups. Bonnefont, Chafaï, and Herry [2] showed a variant of the logarithmic Sobolev inequality on the Heisenberg group, and Gordina and Luo [9] derived the logarithmic Sobolev inequality on nonisotropic Heisenberg groups.

One of the applications of the logarithmic Sobolev inequality (1.5) is to derive the Heisenberg uncertainty inequality on the Heisenberg group. The Heisenberg uncertainty implies that

$$ \frac{n}{2} \|f\|_{L^{2}({\mathbb{R}}^{n})}^{2} \leq \left (\int _{{ \mathbb{R}}^{n}} |x|^{2} |f(x)|^{2}\, dx\right )^{\frac{1}{2}} \left ( \int _{{\mathbb{R}}^{n}} |\nabla f(x)|^{2}\, dx\right )^{\frac{1}{2}}. $$

(1.6)

For the Heisenberg group $\mathbb{H}^{n}$, Thangavelu [19], Sitaram, Sundari, and Thangavelu [16], and Xiao and He [24] considered the Heisenberg uncertainty inequality (see also [20, 21]). In order to prove the Heisenberg uncertainty inequality on the Heisenberg group, we derive Shannon’s inequality. Define the weighted Lebesgue space $L^{2}_{1}(\mathbb{H}^{n})$ by

$$ L^{2}_{1}(\mathbb{H}^{n}) \equiv \{f \in L^{2}(\mathbb{H}^{n});\ |( \cdot , \cdot )| f \in L^{2}(\mathbb{H}^{n})\}. $$

Then, the following inequality holds:

Theorem 1.3

(The Shannon inequality)

For any $f \in L^{2}_{1}(\mathbb{H}^{n}) \setminus \{0\}$,

$$ - \int _{\mathbb{H}^{n}} |f(z,s)|^{2} \log \frac{|f(z,s)|^{2}}{\|f\|_{2}^{2}}\, dz\, ds \leq \frac{Q}{2} \|f\|_{2}^{2} \log \left (\frac{C_{n}}{\|f\|_{2}^{2}} \int _{\mathbb{H}^{n}} |(z, s)|^{2} |f(z,s)|^{2}\, dz\, ds\right ), $$

(1.7)

where the constant

$$ C_{n} = \frac{\pi e}{n + 1} n^{\frac{1}{n + 1}} \pi ^{- \frac{1}{2n+2}} \left ( \frac{\Gamma \left (\frac{n}{2}\right )}{\Gamma \left (\frac{n+1}{2}\right )} \right )^{\frac{1}{n + 1}} $$

is the best possible. Moreover, this constant is attained by

$$ f(x) = e^{-|(z, s)|^{2}} $$

up to conformal automorphism.

Combining the inequalities (1.3) and (1.7), we obtain the Heisenberg uncertainty inequality on the Heisenberg group and the explicit constant:

Corollary 1.4

For any $f \in L^{2}_{1}(\mathbb{H}^{n}) \cap H^{1}(\mathbb{H}^{n})$,

$$ D_{n} \|f\|_{2}^{2} \leq \left (\int _{\mathbb{H}^{n}} |(z,s)|^{2} |f(z,s)|^{2} \, dz\, ds\right )^{\frac{1}{2}} \left (\int _{\mathbb{H}^{n}} | \nabla _{\mathbb{H}}f(z,s)|^{2}\, dz\, ds\right )^{\frac{1}{2}}, $$

(1.8)

where the constant $D_{n}$ is given by

$$ D_{n} = S_{\mathbb{H}}^{- \frac{1}{2}} C_{n}^{- \frac{1}{2}}. $$

This paper is constructed in the following sections. In Sect. 2, we give the proofs of Theorems 1.2 and 1.3, and Corollary 1.4. We also consider the behavior of the constant appearing in the inequalities as $n \to \infty$. For other applications, we show the estimate of solutions to the heat equation on the Heisenberg group in Sect. 3. In the Appendix, we add some calculations.

In what follows, $\|\cdot \|_{p}$ denotes the $L^{p}({\mathbb{H}}^{n})$-norm for $1 \leq p < \infty$. We denote the gamma function by $\Gamma \left (\cdot \right )$.

2 Proof of the results and remarks on the constant

Proof of Theorem 1.2

If we set

$$ d\mu = \frac{|f(z, s)|^{2}}{\|f\|_{2}^{2}}\, dz\, ds, $$

then we obtain $\int _{\mathbb{H}^{n}} d\mu = 1$. Furthermore, since the logarithmic function logx is concave, by Jensen’s inequality, we have

$$\begin{aligned} \int _{\mathbb{H}^{n}} |f(z, s)|^{2} \log \frac{|f(z, s)|^{2}}{\|f\|_{2}^{2}}\, dz\, ds &= \frac{Q-2}{2} \|f\|_{2}^{2} \int _{\mathbb{H}^{n}} \log \frac{|f(z, s)|^{\frac{4}{Q-2}}}{\|f\|_{2}^{\frac{4}{Q-2}}}\, d\mu \\ &\leq \frac{Q-2}{2} \|f\|_{2}^{2} \log \int _{\mathbb{H}^{n}} \frac{|f(z, s)|^{\frac{4}{Q-2}}}{\|f\|_{2}^{\frac{4}{Q-2}}}\, d\mu \\ &= \frac{Q-2}{2} \|f\|_{2}^{2} \log \int _{\mathbb{H}^{n}} \frac{|f(z, s)|^{\frac{2Q}{Q-2}}}{\|f\|_{2}^{\frac{2Q}{Q-2}}} \, dz\, ds. \end{aligned}$$

By Sobolev’s inequality on the Heisenberg group (1.4)

$$ \int _{\mathbb{H}^{n}} |f(z, s)|^{\frac{2Q}{Q-2}}\, dz\, ds \leq S_{ \mathbb{H}}^{\frac{Q}{Q-2}} \|\nabla _{\mathbb{H}} f\|_{2}^{ \frac{2Q}{Q-2}}, $$

we obtain

$$\begin{aligned} \int _{\mathbb{H}^{n}} |f(z, s)|^{2} \log \frac{|f(z, s)|^{2}}{\|f\|_{2}^{2}} dz ds &\leq \frac{Q-2}{2} \|f\|_{2}^{2} \log \frac{S_{\mathbb{H}}^{\frac{Q}{Q-2}} \|\nabla _{\mathbb{H}} f\|_{2}^{\frac{2Q}{Q-2}}}{\|f\|_{2}^{\frac{2Q}{Q-2}}} \\ &\leq \frac{Q}{2} \|f\|_{2}^{2} \log \frac{S_{\mathbb{H}} \|\nabla _{\mathbb{H}} f\|_{2}^{2}}{\|f\|_{2}^{2}}, \end{aligned}$$

which implies the desired inequality (1.5). □

Remark 1

In Euclidean space ${\mathbb{R}}^{n}$, by the Stirling approximation for the Gamma function

$$ \Gamma \left (z\right ) \approx (2 \pi )^{\frac{1}{2}} e^{- z} z^{z - \frac{1}{2}} \quad \mathrm{for} \ z \gg 1, $$

the sharp constant $S_{n}$ in (1.1) is, for n large enough:

$$\begin{aligned} \frac{n \pi e}{2} S_{n} &= \frac{1}{n-2} 2^{1-\frac{2}{n}} \pi ^{- \frac{1}{n}} e \Gamma \left (\frac{n+1}{2}\right )^{\frac{2}{n}} \\ &\approx \frac{1}{n-2} 2^{1-\frac{2}{n}} \pi ^{-\frac{1}{n}} e \left ( \frac{4\pi}{n+1}\right )^{\frac{1}{n}} \left (\frac{n+1}{2e}\right )^{1+ \frac{1}{n}} \quad \mathrm{for} \ n \gg 1 \\ &= \frac{n+1}{n-2} (2e)^{-\frac{1}{n}} \approx 1 \quad \mathrm{for} \ n \gg 1, \end{aligned}$$

which implies that the sharp constant $S_{n}$ in (1.1) is approximated by the one in the logarithmic Sobolev inequality $2/(n\pi e)$ for n large enough.

On the other hand, in the Heisenberg group $\mathbb{H}^{n}$, the constant $S_{\mathbb{H}}$ in (1.4) is approximated by $2/(Q\pi e)$ as the following:

$$\begin{aligned} \frac{Q \pi e}{2} S_{\mathbb{H}} &= \frac{Q}{(Q-2)^{2}} 2^{1- \frac{4}{Q}} e \Gamma \left (\frac{Q}{2}\right )^{\frac{2}{Q}} \\ &\approx \frac{Q}{(Q-2)^{2}} 2^{1-\frac{4}{Q}} e \left ( \frac{4\pi}{Q}\right )^{\frac{1}{Q}} \frac{Q}{2e} \quad \mathrm{for} \ Q \gg 1 \\ &= \frac{Q^{2}}{(Q-2)^{2}} \left (\frac{\pi}{4Q}\right )^{\frac{1}{Q}} \approx 1 \quad \mathrm{for} \ Q \gg 1 \end{aligned}$$

when n is large enough. By the above observation, we conjugate that the sharp constant in (1.5) coincides with $2/(Q\pi e)$ or approximates it for n large enough.

As a corollary, one can obtain the parametric logarithmic Sobolev inequality on the Heisenberg group. The following inequality is useful for estimating solutions to the heat equation on the Heisenberg group:

Corollary 2.1

(The parametric logarithmic Sobolev inequality on the Heisenberg group)

For any $f \in H^{1}(\mathbb{H}^{n}) \setminus \{0\}$ and $a > 0$,

$$ \int _{\mathbb{H}^{n}} |f(z, s)|^{2} \log \frac{|f(z, s)|^{2}}{\|f\|_{2}^{2}}\, dz\, ds \leq a \|\nabla _{ \mathbb{H}} f\|_{2}^{2} + \frac{n}{2} \|f\|_{2}^{2} \log \frac{n S_{\mathbb{H}}}{2ae}. $$

(2.1)

Proof of Corollary 2.1

For any $a > 0$,

$$\begin{aligned} \int _{\mathbb{H}^{n}} |f(z, s)|^{2} \log \frac{|f(z, s)|^{2}}{\|f\|_{2}^{2}}\, dz\, ds &\leq \frac{Q}{2} \|f\|_{2}^{2} \log \frac{S_{\mathbb{H}} \|\nabla _{\mathbb{H}} f\|_{2}^{2}}{\|f\|_{2}^{2}} \\ &= \frac{Q}{2} \|f\|_{2}^{2} \log \frac{S_{\mathbb{H}}}{a} + \frac{n}{2} \|f\|_{2}^{2} \log \frac{a \|\nabla _{\mathbb{H}} f\|_{2}^{2}}{\|f\|_{2}^{2}}. \end{aligned}$$

Here, we use the essential inequality:

$$ \alpha \beta \leq e^{\alpha }+ \beta (\log{\beta }-1) $$

for $\alpha , \beta > 0$. By this inequality, we have

$$ \log \frac{a\|\nabla _{\mathbb{H}} f\|_{2}^{2}}{\|f\|_{2}^{2}} \leq a \frac{\|\nabla _{\mathbb{H}} f\|_{2}^{2}}{\|f\|_{2}^{2}} - 1. $$

Thus, we obtain

$$ \int _{\mathbb{H}^{n}} |f(z, s)|^{2} \log \frac{|f(z, s)|^{2}}{\|f\|_{2}^{2}}\, dz\, ds \leq \frac{Q}{2} \|f\|_{2}^{2} \log \frac{S_{\mathbb{H}}}{a} + \frac{n}{2} a \|\nabla _{\mathbb{H}} f \|_{2}^{2} - 1. $$

Replacing $n a/2 \to a$, we conclude that the inequality (2.1) holds. □

Proof of Theorem 1.3

Let ϕ be given by

$$ \phi (z, s) = k_{n, b} e^{-|(z, s)|^{2}}, $$

where the constant $k_{n, b}$ is defined below. By the polar transformation:

$$ \int _{\mathbb{H}^{n}} e^{-2|(z, s)|^{2}}\, dz\, ds = \left |\Sigma{}^{n-1} \right | \int _{0}^{\infty }e^{-2r^{2}} r^{2n+1}\, dr, $$

where

$$ \left |\Sigma{}^{n-1}\right | = \frac{2\pi ^{n+\frac{1}{2}} \Gamma \left (\frac{n}{2}\right )}{\Gamma \left (n\right ) \Gamma \left (\frac{n+1}{2}\right )}. $$

Changing variables with $2r^{2} = s$, since $r = \sqrt{s/2}$, $dr = 2^{-3/2} s^{-1/2}\, ds$:

$$ \int _{0}^{\infty }e^{-2r^{2}} r^{2n+1}\, dr = 2^{- n - 2} \int _{0}^{ \infty }e^{-s} s^{n}\, ds = 2^{-n-2} \Gamma \left (n+1\right ). $$

Thus, we have

$$ \int _{\mathbb{H}^{n}} e^{-2|(z, s)|^{2}}\, dz\, ds = 2^{-n-1} n \pi ^{n+ \frac{1}{2}} \frac{\Gamma \left (\frac{n}{2}\right )}{\Gamma \left (\frac{n+1}{2}\right )}, $$

that is,

$$ k_{n} = 2^{\frac{n+1}{2}} n^{-\frac{1}{2}} \pi ^{-\frac{2n+1}{4}} \left ( \frac{\Gamma \left (\frac{n+1}{2}\right )}{\Gamma \left (\frac{n}{2}\right )} \right )^{\frac{1}{2}}. $$

For any $f \in L^{2}_{1}(\mathbb{H}^{n})$ with $\|f\|_{2} = 1$, we consider the relative entropy of f and ϕ. By the Jensen inequality, we have

$$ \int _{\mathbb{H}^{n}} |f(z, s)|^{2} \log \frac{|\phi (z, s)|^{2}}{|f(z, s)|^{2}}\, dz\, ds \leq \log \int _{ \mathbb{H}^{n}} |\phi (z, s)|^{2}\, dz\, ds = 0. $$

Thus, we obtain

$$ - \int _{\mathbb{H}^{n}} |f(z, s)|^{2} \log |f(z, s)|^{2}\, dz\, ds \leq 2 \int _{\mathbb{H}^{n}} |(z, s)|^{2} |f(z, s)|^{2}\, dz\, ds - 2 \log{k_{n}}. $$

(2.2)

For $\lambda > 0$, set $f_{\lambda }$ to be the scaling of f by

$$ f_{\lambda }(z, s) \equiv \lambda ^{n+1} f(\lambda z, \lambda ^{2} s), $$

which preserves $L^{2}$-norm. Substituting $f_{\lambda }$ into inequality (2.2), the left-hand side of (2.2) is written by

$$ - \int _{\mathbb{H}^{n}} |f_{\lambda }(z, s)|^{2} \log |f_{\lambda }(z, s)|^{2}\, dz\, ds = - \int _{\mathbb{H}^{n}} |f(z, s)|^{2} \log |f(z, s)|^{2} \, dz\, ds - (2n+2) \log{\lambda }, $$

and the right-hand side is expressed by

$$ 2 \int _{\mathbb{H}^{n}} |(z, s)|^{2} |f_{\lambda }(z, s)|^{2}\, dz\, ds - 2 \log{k_{n}} = 2 \lambda ^{-2} \int _{\mathbb{H}^{n}} |(z, s)|^{2} |f(z, s)|^{2}\, dz\, ds - 2 \log{k_{n}}. $$

Combining both sides, we have

$$ - \int _{\mathbb{H}^{n}} |f(z, s)|^{2} \log |f(z, s)|^{2}\, dz\, ds \leq 2 \lambda ^{-2} \int _{\mathbb{H}^{n}} |(z, s)|^{2} |f(z, s)|^{2} \, dz\, ds + (2n+2) \log \frac{\lambda }{k_{n}^{\frac{1}{n + 1}}} $$

for any $\lambda > 0$. Optimizing the right-hand side in this inequality with

$$ \lambda _{0} = \left (\frac{2}{n + 1}\right )^{\frac{1}{2}} \left ( \int _{\mathbb{H}^{n}} |(z, s)|^{2} |f(z, s)|^{2}\, dz\, ds\right )^{ \frac{1}{2}}, $$

we obtain the desired inequality (1.7). □

Remark 2

We remark that the constant $C_{n}$ in the inequality (1.7) is approximated by $(2 \pi e)/Q$ as the following:

$$\begin{aligned} \frac{Q}{2 \pi e} C_{n} &= n^{\frac{1}{n + 1}} \pi ^{- \frac{1}{2n+2}} \left ( \frac{\Gamma \left (\frac{n}{2}\right )}{\Gamma \left (\frac{n+1}{2}\right )} \right )^{\frac{1}{n + 1}} \\ &\approx \left ( \frac{e^{- \frac{n}{2}} \left (\frac{n}{2}\right )^{\frac{n}{2} - \frac{1}{2}}}{e^{- \frac{n + 1}{2}} \left (\frac{n + 1}{2}\right )^{\frac{n}{2}}} \right )^{\frac{1}{n + 1}} \quad \mathrm{for} \ Q \gg 1 \\ &= \left (e^{\frac{1}{2}} \left (\frac{n}{2}\right )^{- \frac{1}{2}} \left (\frac{n}{n + 1}\right )^{\frac{n}{2}}\right )^{\frac{1}{n + 1}} \approx 1 \quad \mathrm{for} \ Q \gg 1 \end{aligned}$$

when n is large enough. The constant $(2 \pi e)/Q$ corresponds to the optimal constant of the Shannon inequality on the n-dimensional Euclidean space.

Proof of Corollary 1.4

Combining inequalities (1.3) and (1.7), we have

$$\begin{aligned} &- \frac{Q}{2} \|f\|_{2}^{2} \log \left (\frac{C_{n}}{\|f\|_{2}^{2}} \int _{\mathbb{H}^{n}} |(z, s)|^{2} |f(z, s)|^{2}\, dz\, ds\right )\\ &\quad \leq \int _{\mathbb{H}^{n}} |f(z, s)|^{2} \log \frac{|f(z, s)|^{2}}{\|f\|_{2}^{2}}\, dz\, ds \\ &\quad \leq \frac{Q}{2} \|f\|_{2}^{2} \log \left ( \frac{S_{\mathbb{H}}}{\|f\|_{2}^{2}} \int _{\mathbb{H}^{2}} |\nabla _{ \mathbb{H}}f(z, s)|^{2}\, dz\, ds\right ), \end{aligned}$$

and thus, we obtain

$$ \frac{Q}{2} \|f\|_{2}^{2} \log \left ( \frac{S_{\mathbb{H}}C_{n}}{\|f\|_{2}^{4}} \int _{\mathbb{H}^{2}} | \nabla _{\mathbb{H}}f(z, s)|^{2}\, dz\, ds \int _{\mathbb{H}^{n}} |(z, s)|^{2} |f(z, s)|^{2}\, dz\, ds\right ) \geq 0. $$

Thus, we obtain the desired inequality (1.8). □

Remark 3

By the above computations of the constants appearing in inequalities (1.5) and (1.7), we see that

$$ \mathop{\mathrm{lim}}\limits _{n \to \infty } \frac{2}{Q} D_{n} = 1. $$

We note that the constant $Q/2$ corresponds to the optimal constant of the Heisenberg uncertainty inequality (1.6) on the n-dimensional Euclidean space.

3 Application to the heat equation on the Heisenberg group

For other applications, one can derive the dissipative estimate of solutions to the heat equation on the Heisenberg group. We consider the Cauchy problem of the heat equation on the Heisenberg group:

$$ \left \{ \begin{aligned} &\partial _{t} u = \Delta _{\mathbb{H}} u, &&t > 0, \ (z, s) \in \mathbb{H}^{n}, \\ &\quad u(0, z, s) = u_{0}(z, s), && (z, s) \in \mathbb{H}^{n}, \end{aligned} \right . $$

(3.1)

where $u = u(t, z, s)$ is the unknown function, and $u_{0} \in L^{q}(\mathbb{H}^{n})$ ($q\geq 1$) is the given initial data. One can derive a $L^{p}$-$L^{q}$ dissipative estimate for a solution of (3.1) via the parametric logarithmic Sobolev inequality (2.1) on the Heisenberg group.

Theorem 3.1

Let u be a solution of (3.1) with $u_{0} \in L^{q}(\mathbb{H}^{n})$, $1 \leq q < \infty$. Then, for $t > 0$ and $q < p \leq \infty$,

$$ \|u(t)\|_{p} \leq D t^{-\frac{Q}{2}\left (\frac{1}{q}-\frac{1}{p} \right )} \|u_{0}\|_{q}, $$

(3.2)

where

$$ D = D(n, p, q) = \left (\frac{C_{q}}{C_{p}}\right )^{\frac{n}{2}} \left (\frac{2n S_{\mathbb{H}}}{e^{3} (\frac{1}{q}-\frac{1}{p})} \right )^{- \frac{n}{2} \left (\frac{1}{q}-\frac{1}{p}\right )}, \quad C_{p} = p^{\frac{1}{p}} \left (1-\frac{1}{p}\right )^{1- \frac{1}{p}} $$

and the constant $S_{\mathbb{H}}$ is the same as the above.

Proof of Theorem 3.1

The proof of Theorem 3.1 is based on the argument in Lieb and Loss [14]. We take a function $r: [0, t) \to{\mathbb{R}}$ such that

$$ r(0) = q, \quad r(t) = p, \quad r'(\tau ) > 0 \quad \mathrm{for} \ 0 \leq \tau < t. $$

(3.3)

This function has been given later. For a moment, suppose that $u(\tau ) \in L^{r(\tau )}(\mathbb{H}^{n})$, then we have formally,

$$\begin{aligned} \frac{d}{d\tau } \|u(\tau )\|_{r(\tau )} &= \frac{r'(\tau )}{r(\tau )^{2}} \int _{\mathbb{H}^{n}} |u(\tau )|^{r( \tau )} \log \frac{|u(\tau )|^{r(\tau )}}{\|u(\tau )\|_{r(\tau )}^{r(\tau )}}\, dz \,ds \\ &\quad + \int _{\mathbb{H}^{n}} |u(\tau )|^{r(\tau )-2} u(\tau ) \partial _{\tau }u(\tau )\, dz\, ds. \end{aligned}$$

From the second term on the right-hand side, using (3.1) and integrating by parts, we obtain

$$\begin{aligned} \int _{\mathbb{H}^{n}} |u(\tau )|^{r(\tau )-2} u(\tau ) \partial _{ \tau }u(\tau )\, dz\, ds &= \int _{\mathbb{H}^{n}} |u(\tau )|^{r( \tau )-2} u(\tau ) \Delta _{\mathbb{H}} u(\tau )\, dz\, ds \\ &= - \frac{4(r(\tau )-1)}{r(\tau )^{2}} \int _{\mathbb{H}^{n}} \left | \nabla _{\mathbb{H}}u(\tau )^{\frac{r(\tau )}{2}}\right |^{2}\, dz\, ds. \end{aligned}$$

Thus, we have

$$\begin{aligned} &\frac{d}{d\tau } \|u(\tau )\|_{r(\tau )} \\ &\quad = \frac{r'(\tau )}{r(\tau )^{2}} \left (\int _{\mathbb{H}^{n}} |u( \tau )|^{r(\tau )} \log \frac{|u(\tau )|^{r(\tau )}}{\|u(\tau )\|_{r(\tau )}^{r(\tau )}}\, dz \, ds - \frac{4(r(\tau )-1)}{r'(\tau )} \int _{\mathbb{H}^{n}} \left | \nabla _{\mathbb{H}}u(\tau )^{\frac{r(\tau )}{2}}\right |^{2}\, dz\, ds \right ). \end{aligned}$$

By (2.1) with

$$ a = \frac{4(r(\tau )-1)}{r'(\tau )}, $$

we have

$$ \frac{d}{d\tau } \|u(\tau )\|_{r(\tau )} \leq - \frac{n r'(\tau )}{2 r(\tau )^{2}} \|u(\tau )\|_{r(\tau )} \log \left (\frac{2n S_{\mathbb{H}}}{e} \frac{r(\tau )-1}{r'(\tau )} \right ). $$

(3.4)

Here, we set

$$ r(\tau ) = \frac{pqt}{pt-(p-q)\tau }, $$

then r satisfies the condition (3.3). Putting r to (3.4), we obtain

$$ \frac{\frac{d}{d\tau } \|u(\tau )\|_{r(\tau )}}{\|u(\tau )\|_{r(\tau )}} \leq - \frac{n}{2t} \left (\frac{1}{q}-\frac{1}{p}\right ) \log \left [\frac{2n S_{\mathbb{H}}}{e} \frac{\{pt-(p-q)\tau \} \{p q t - p t + (p - q)\tau \}}{pq(p-q)t} \right ]. $$

Integrating both sides from 0 to t, we obtain

$$ \log \frac{\|u(t)\|_{p}}{\|u_{0}\|_{q}} \leq - \frac{n}{2} \left ( \frac{1}{q}-\frac{1}{p}\right ) \log \frac{2n S_{\mathbb{H}} t}{e^{3} (\frac{1}{q}-\frac{1}{p})} - \frac{n}{2} \log \frac{C_{p}}{C_{q}}, $$

which implies the desired inequality (3.2):

$$ \|u(t)\|_{p} \leq D t^{- \frac{n}{2} \left (\frac{1}{q}-\frac{1}{p} \right )} \|u_{0}\|_{q}. $$

 □

Remark 4

The explicit form of the heat kernel of the heat equation (3.1) is known (see [21]), and Xiao and He [23] proved the Hardy–Littlewood–Sobolev inequality on the Heisenberg group by using the estimate of the heat kernel. One can also derive a $L^{p}$-$L^{q}$ dissipative estimate for a solution of the heat equation (3.1) by using this estimate and Young’s inequality. Furthermore, we conjugate that the heat kernel of the heat equation (3.1) is the extremal function of (1.5).

Appendix

For a measurable function $f(z, s)$, by the polar transformation, we have

$$\begin{aligned} \int _{\mathbb{H}^{n}} f(z, s)\, dz\, ds &= \int _{{\mathbb{R}}} \int _{{ \mathbb{S}}^{2n-1}} \int _{0}^{\infty }f(\rho \sigma , s) \rho ^{2n-1} \, d\rho \, d\sigma \, ds \\ &= \int _{{\mathbb{S}}^{2n-1}} \int _{-\frac{\pi}{2}}^{\frac{\pi}{2}} \int _{0}^{\infty }f(r \sigma \cos ^{\frac{1}{2}} \theta , r^{2} \sin \theta ) r^{2n+1} \cos ^{n-1} \theta \, dr\, d\theta \, d\sigma . \end{aligned}$$

In particular, we see that

$$\begin{aligned} \left |\Sigma{}^{n-1}\right | \equiv \int _{{\mathbb{S}}^{2n-1}} \int _{- \frac{\pi}{2}}^{\frac{\pi}{2}} \cos ^{n-1}{\theta }\, d\theta \, d \sigma = |{\mathbb{S}}^{2n-1}| \int _{-\frac{\pi}{2}}^{\frac{\pi}{2}} \cos ^{n-1}{\theta }\, d\theta . \end{aligned}$$

Here, since cosθ is even, we have

$$\begin{aligned} \int _{-\frac{\pi}{2}}^{\frac{\pi}{2}} \cos ^{n-1}{\theta }\, d \theta &= 2 \int _{0}^{\frac{\pi}{2}} \cos \theta (1-\sin ^{2}\theta )^{ \frac{n-2}{2}} d\theta = 2 \int _{0}^{1} (1-t^{2})^{\frac{n}{2}-1}\, dt \\ &= B\left (\frac{1}{2}, \frac{n}{2}\right ) = \frac{\pi ^{\frac{1}{2}} \Gamma \left (\frac{n}{2}\right )}{\Gamma \left (\frac{n+1}{2}\right )}. \end{aligned}$$

Thus, it holds that

$$ \left |\Sigma{}^{n-1}\right | = \frac{2\pi ^{n+\frac{1}{2}} \Gamma \left (\frac{n}{2}\right )}{\Gamma \left (n\right ) \Gamma \left (\frac{n+1}{2}\right )}. $$

Moreover, for a radial function $f(x) = f(r)$ ($r = |(z, s)|$),

$$ \int _{\mathbb{H}^{n}} f(z, s)\, dz\, ds = \left |\Sigma{}^{n-1}\right | \int _{0}^{\infty }f(r) r^{2n+1}\, dr. $$