Abstract
In this paper we present \(L^2\) and \(L^p\) versions of the geometric Hardy inequalities in half-spaces and convex domains on stratified (Lie) groups. As a consequence, we obtain the geometric uncertainty principles. We give examples of the obtained results for the Heisenberg and the Engel groups.
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1 Introduction
In the Euclidean setting, a geometric Hardy inequality in a (Euclidean) convex domain \(\Omega\) has the following form
for \(u \in C_0^{\infty }(\Omega )\) with the sharp constant 1/4. There is a number of studies related to this subject, see e.g. [1,2,3, 5, 6, 13].
In the case of the Heisenberg group \(\mathbb {H}\), Luan and Yang [12] obtained the following Hardy inequality on the half space \(\mathbb {H}^+ := \{(x_1,x_2,x_3)\in \mathbb {H}, | \, x_3>0 \}\) for \(u \in C_0^{\infty }(\mathbb {H}^+)\)
Moreover, the geometric \(L^p\)-Hardy inequalities for the sub-Laplacian on the convex domain in the Heisenberg group was obtained by Larson [11] which also generalises the previous result in [12]. In this note by using the approach in [11] we obtain the geometric Hardy type inequalities on the half-spaces and the convex domains on general stratified groups, so our results extend known results of Abelian (Euclidean) and Heisenberg groups.
Thus, the main aim of this paper is to prove the geometric Hardy type inequalities on general stratified groups. As consequences, the geometric uncertainty principles are obtained. In Sect. 2 we present \(L^2\) and \(L^p\) versions of the subelliptic geometric Hardy type inequalities on the half-space. In Sect. 3, we show subelliptic \(L^2\) and \(L^p\) versions of the geometric Hardy type inequalities on the convex domains.
1.1 Preliminaries
Let \(\mathbb {G}=(\mathbb {R}^n,\circ ,\delta _{\lambda })\) be a stratified Lie group (or a homogeneous Carnot group), with dilation structure \(\delta _{\lambda }\) and Jacobian generators \(X_{1},\ldots ,X_{N}\), so that N is the dimension of the first stratum of \(\mathbb {G}\). We denote by Q the homogeneous dimension of \(\mathbb {G}\). We refer to [9], or to the recent books [4, 8] for extensive discussions of stratified Lie groups and their properties.
The sub-Laplacian on \(\mathbb {G}\) is given by
We also recall that the standard Lebesque measure dx on \(\mathbb R^{n}\) is the Haar measure for \(\mathbb {G}\) (see, e.g. [8, Proposition 1.6.6]). Each left invariant vector field \(X_{k}\) has an explicit form and satisfies the divergence theorem, see e.g. [8] for the derivation of the exact formula: more precisely, we can formulate
with \(x=(x',x^{(2)},\ldots ,x^{(r)})\), where r is the step of \(\mathbb {G}\) and \(x^{(l)}=(x^{(l)}_1,\ldots ,x^{(l)}_{N_l})\) are the variables in the \(l^{th}\) stratum, see also [8, Section 3.1.5] for a general presentation. The horizontal gradient is given by
and the horizontal divergence is defined by
We now recall the divergence formula in the form of [14, Proposition 3.1]. Let \(f_{k}\in C^{1}(\Omega )\bigcap C({\overline{\Omega }}),\,k=1,\ldots ,N\). Then for each \(k=1,\ldots ,N,\) we have
Consequently, we also have
2 Hardy type inequalities on half-space
2.1 \(L^2\)-Hardy inequality on the half-space of \(\mathbb {G}\)
In this section we present the geometric \(L^2\)-Hardy inequality on the half-space of \(\mathbb {G}\). We define the half-space as follows
where \(\nu :=(\nu _1,\ldots ,\nu _r)\) with \(\nu _j \in \mathbb {R}^{N_j},\, j=1,\ldots ,r,\) is the Riemannian outer unit normal to \(\partial \mathbb {G}^+\) (see [10]) and \(d \in \mathbb {R}\). The Euclidean distance to the boundary \(\partial \mathbb {G}^+\) is denoted by \(dist(x,\partial \mathbb {G}^+)\) and defined as follows
Moreover, there is an angle function on \(\partial \mathbb {G}^+\) which is defined by Garofalo in [10] as
Theorem 2.1
Let \(\mathbb {G}^+\) be a half-space of a stratified group \(\mathbb {G}\). Then for all \(\beta \in \mathbb {R}\) we have
for all \(u \in C^{\infty }_0(\mathbb {G}^+)\) and where \(C_1(\beta ):=-(\beta ^2+ \beta )\).
Remark 2.2
If \(\mathbb {G}\) has step \(r=2\), then for \(i=1,\ldots ,N\) we have the following left-invariant vector fields
where \(a_{m,i}^s\) are the group constants (see, e.g. [7, Formula (2.14)] for the definition). Also we have \(x:=(x',x'')\) with \(x'= (x'_1,\ldots ,x_N')\), \(x''=(x''_1,\ldots ,x''_{N_2})\), and also \(\nu :=(\nu ',\nu '')\) with \(\nu '= (\nu '_1,\ldots ,\nu _N')\) and \(\nu ''=(\nu ''_1,\ldots ,\nu ''_{N_2})\).
Corollary 2.3
Let \(\mathbb {G}^+\) be a half-space of a stratified group \(\mathbb {G}\) of step \(r=2\). For all \(\beta \in \mathbb {R}\) and \(u \in C^{\infty }_0(\mathbb {G}^+)\) we have
where \(C_1(\beta ):=-(\beta ^2+ \beta )\) and \(K(a,\nu ,\beta ):= \beta \sum _{s=1}^{N_2}\sum _{i=1}^{N}a_{i,i}^s\nu _{s}''\).
Proof of Theorem 2.1
To prove inequality (2.3) we use the method of factorization. Thus, for any \(W:=(W_1,\ldots ,W_N),\,\, W_i \in C^1(\mathbb {G}^+)\) real-valued, which will be chosen later, by a simple computation we have
From the above expression we get the inequality
Let us now take \(W_i\) in the form
where
and
Now \(W_i(x)\) can be written as
By a direct computation we have
where
Inserting the expression (2.8) in (2.6) we get
The proof of Theorem 2.1 is finished. \(\square\)
As consequences of Theorem 2.1, we have the geometric Hardy inequalities on the half-space without an angle function, which seems an interesting new result on \(\mathbb {G}\).
Corollary 2.4
Let \(\mathbb {G}^+\) be a half-space of a stratified group \(\mathbb {G}\). Then we have
for all \(u \in C^{\infty }_0(\mathbb {G}^+)\).
Proof of Corollary 2.4
Let \(x:=(x',x^{(2)},\ldots ,x^{(r)}) \in \mathbb {G}\) with \(x'=(x'_1,\ldots ,x'_N)\) and \(x^{(j)} \in \mathbb {R}^{N_j}, \,\, j=2,\ldots ,r\). By taking \(\nu :=(\nu ',0,\ldots ,0)\) with \(\nu '= (\nu '_1,\ldots ,\nu '_N),\) we have that
we have
and
Inserting the above expressions in inequality (2.3) we arrive at
For optimisation we differentiate the right-hand side of integral with respect to \(\beta\), then we have
which implies
We complete the proof. \(\square\)
We also have the geometric uncertainty principle on the half-space of \(\mathbb {G}^+\).
Corollary 2.5
Let \(\mathbb {G}^+\) be a half-space of a stratified group \(\mathbb {G}\). Then we have
for all \(u \in C^{\infty }_0(\mathbb {G}^+)\).
Proof of Corollary 2.5
By using (2.9) and the Cauchy–Schwarz inequality we get
\(\square\)
To demonstrate our general result in a particular case, here we consider the Heisenberg group, which is a well-known example of step \(r=2\) (stratified) group.
Corollary 2.6
Let \(\mathbb {H}^+ = \{(x_1,x_2,x_3)\in \mathbb {H} \, | \, \, x_3>0 \}\) be a half-space of the Heisenberg group \(\mathbb {H}\). Then for any \(u \in C^{\infty }_0(\mathbb {H}^+)\) we have
where \(\nabla _{\mathbb {H}} = \{X_1,X_2\}.\)
Proof of Corollary 2.6
Recall that the left-invariant vector fields on the Heisenberg group are generated by the basis
with the commutator
For \(x= (x_1,x_2,x_3)\), choosing \(\nu =(0,0,1)\) as the unit vector in the direction of \(x_3\) and taking \(d=0\) in inequality (2.3), we get
and
Therefore, with \(\mathcal {W}(x)\) as in (2.2), we have
Substituting these into inequality (2.3) we arrive at
taking \(\beta = -\frac{1}{2}\). \(\square\)
Let us present an example for the step \(r=3\) (stratified) groups. A well-known stratified group with step three is the Engel group, which can be denoted by \(\mathbb {E}\). Topologically \(\mathbb {E}\) is \(\mathbb {R}^4\) with the group law of \(\mathbb {E}\), which is given by
where
The left-invariant vector fields of \(\mathbb {E}\) are generated by the basis
Corollary 2.7
Let \(\mathbb {E}^+ = \{x:=(x_1,x_2,x_3,x_4)\in \mathbb {E} \, | \, \langle x,\nu \rangle >0 \}\) be a half-space of the Engel group \(\mathbb {E}\). Then for all \(\beta \in \mathbb {R}\) and \(u \in C^{\infty }_0(\mathbb {E}^+)\) we have
where \(\nabla _{\mathbb {E}} = \{X_1,X_2\}\), \(\nu := (\nu _1,\nu _2,\nu _3,\nu _4)\), and \(C_1(\beta ) = -(\beta ^2 +\beta )\).
Remark 2.8
If we take \(\nu _4=0\) in (2.12), then we have the following inequality on \(\mathbb {E}\), by taking \(\beta =-\frac{1}{2}\),
Proof of Corollary 2.7
As we mentioned, the Engel group has the following basis of the left-invariant vector fields
with the following two (non-zero) commutators
Thus, we have
A direct calculation gives that
Now substituting these into inequality (2.3) we obtain the desired result. \(\square\)
2.2 \(L^p\)-Hardy inequality on \(\mathbb {G}^+\)
Here we construct an \(L^p\) version of the geometric Hardy inequality on the half-space of \(\mathbb {G}\) as a generalisation of the previous theorem. We define the p-version of the angle function by \(\mathcal {W}_p\), which is given by the formula
Theorem 2.9
Let \(\mathbb {G}^+\) be a half-space of a stratified group \(\mathbb {G}\). Then for all \(\beta \in \mathbb {R}\) we have
for all \(u \in C^{\infty }_0(\mathbb {G}^+)\), \(1<p<\infty\) and \(C_2(\beta ,p):=-(p-1)( |\beta |^{\frac{p}{p-1}} + \beta )\).
Proof of Theorem 2.9
We use the standard method such as the divergence theorem to obtain the inequality (2.14). For \(W \in C^{\infty }(\mathbb {G}^+)\) and \(f \in C^1(\mathbb {G}^+)\), a direct calculation shows that
Here in the last line Hölder’s inequality was applied. For \(p>1\) and \(q>1\) with \(\frac{1}{p}+\frac{1}{q}=1\) recall Young’s inequality
Let us set that
By using Young’s inequality in (2.15) and rearranging the terms, we arrive at
We choose \(W := I_i\), which has the following form \(I_i=(\overset{i}{\overbrace{0,\ldots ,1}},\ldots ,0)\) and set
Now we calculate
and
We also have
Inserting the above calculations in (2.16) and summing over \(i=1,\ldots ,N\), we arrive at
We complete the proof of Theorem 2.9. \(\square\)
Remark 2.10
For \(p\ge 2\), since
we have the following inequality
3 Hardy inequalities on a convex domain of \(\mathbb {G}\)
In this section, we present the geometric Hardy inequalities on the convex domains in stratified groups. The convex domain is understood in the sense of the Euclidean space. Let \(\Omega\) be a convex domain of a stratified group \(\mathbb {G}\) and let \(\partial \Omega\) be its boundary. Below for \(x\in \Omega\) we denote by \(\nu (x)\) the unit normal for \(\partial \Omega\) at a point \(\hat{x}\in \partial \Omega\) such that \(dist(x,\Omega )=dist(x,\hat{x})\). For the half-plane, we have the distance from the boundary \(dist(x,\partial \Omega ) = \langle x, \nu \rangle -d\). As it is introduced in the previous section we also have the generalised angle function
with \(\mathcal {W}(x):=\mathcal {W}_2(x)\).
3.1 Geometric \(L^2\)-Hardy inequality on a convex domain of \(\mathbb {G}\)
Theorem 3.1
Let \(\Omega\) be a convex domain of a stratified group \(\mathbb {G}\). Then for \(\beta <0\) we have
for all \(u \in C_0^{\infty }(\Omega )\), and \(C_1(\beta ):=-(\beta ^2+ \beta )\).
Proof of Theorem 3.1
We follow the approach of Simon Larson [11] by proving inequality (3.1) in the case when \(\Omega\) is a convex polytope. We denote its facets by \(\{ \mathcal {F}_j \}_j\) and unit normals of these facets by \(\{\nu _j\}_j\), which are directed inward. Then \(\Omega\) can be constructed by the union of the disjoint sets \(\Omega _j := \{x \in \Omega : dist(x, \partial \Omega ) = dist(x,\mathcal {F}_j) \}\). Now we apply the same method as in the case of the half-space \(\mathbb {G}^+\) for each element \(\Omega _j\) with one exception that not all the boundary values are zero when we use the partial integration. As in the previous computation we have
where \(n_j\) is the unit normal of \(\partial \Omega _j\) which is directed outward. Since \(\mathcal {F}_j \subset \partial \Omega _j\) we have \(n_j = -\nu _j\).
The boundary terms on \(\partial \Omega\) vanish since u is compactly supported in \(\Omega\). So we only deal with the parts of \(\partial \Omega _j\) in \(\Omega\). Note that for every facet of \(\partial \Omega _j\) there exists some \(\partial \Omega _l\) which shares this facet. We denote by \(\Gamma _{jl}\) the common facet of \(\partial \Omega _j\) and \(\partial \Omega _l\), with \(n_k|_{\Gamma _{jl}}= -n_l|_{\Gamma _{jl}}\). From the above expression we get the following inequality
Now we choose \(W_i\) in the form
and a direct computation shows that
Inserting the expression (3.3) into inequality (3.2) we get
Now we sum over all partition elements \(\Omega _j\) and let \(n_{jl}=n_k|_{{\Gamma _{jl}}}\), i.e. the unit normal of \(\Gamma _{jl}\) pointing from \(\Omega _j\) into \(\Omega _l\). Then we get
Here we used the fact that (by the definition) \(\Gamma _{jl}\) is a set with \(dist(x, \mathcal {F}_j)=dist(x, \mathcal {F}_l)\). From
rearranging \(x \cdot (\nu _j - \nu _l)- d_j + d_l=0\) we see that \(\Gamma _{jl}\) is a hyperplane with a normal \(\nu _j - \nu _l\). Thus, \(\nu _j-\nu _l\) is parallel to \(n_{jl}\) and one only needs to check that \((\nu _j-\nu _l)\cdot n_{jl}>0\). Observe that \(n_{jl}\) points out and \(\nu _j\) points into jth partition element, so \(\nu _j \cdot n_{jl}\) is non-negative. Similarly, we see that \(\nu _l \cdot n_{jl}\) is non-positive. This means we have \((\nu _j-\nu _l)\cdot n_{jl}>0\). In addition, it is easy to see that
which implies that
where \(\alpha _{jl}\) is the angle between \(\nu _j\) and \(\nu _l\). So we obtain
Here with \(\beta <0\) and due to the boundary term signs we verify the inequality for the polytope convex domains.
Let us now consider the general case, that is, when \(\Omega\) is an arbitrary convex domain. For each \(u \in C_0^{\infty }(\Omega )\) one can always choose an increasing sequence of convex polytopes \(\{\Omega _j\}_{j=1}^{\infty }\) such that \(u \in C_0^{\infty }(\Omega _1), \Omega _j \subset \Omega\) and \(\Omega _j \rightarrow \Omega\) as \(j \rightarrow \infty\). Assume that \(\nu _j(x)\) is the above map \(\nu\) (corresponding to \(\Omega _j\)) we compute
Now we obtain the desired result when \(j \rightarrow \infty\). \(\square\)
3.2 \(L^p\)-Hardy’s inequality on a convex domain of \(\mathbb {G}\)
In this section we give the \(L^p\)-version of the previous results.
Theorem 3.2
Let \(\Omega\) be a convex domain of a stratified group \(\mathbb {G}\). Then for \(\beta <0\) we have
for all \(u \in C_0^{\infty }(\Omega )\), and \(C_2(\beta ,p):=-(p-1)( |\beta |^{\frac{p}{p-1}} + \beta )\).
Proof of Theorem 3.2
Let us assume that \(\Omega\) is the convex polytope as in the \(p=2\) case. Thus, we consider the partition \(\Omega _j\) as the previous case. For \(f \in C^1(\Omega _j)\) and \(W \in C^{\infty }(\Omega _j)\), a simple calculation shows that
In the last line Hölder’s inequality was applied. Recall again Young’s inequality for \(p>1\), \(q>1\) and \(\frac{1}{p}+\frac{1}{q}=1\), we have \(ab\le \frac{a^p}{p} + \frac{b^q}{q}, \,\,\, \text {for} \,\, a \ge 0, \, b\ge 0.\) We now take \(q := \frac{p}{p-1}\) and
By using Young’s inequality in (3.6) and rearranging the terms, we arrive at
We choose \(W := I_i\) as a unit vector of the \(i^{th}\) component and let
As before a direct calculation shows that
and
We also have
Inserting the above calculations into (3.7) and summing over \(i=\overline{1,N}\), we arrive at
Now summing up over \(\Omega _j\), and with the interior boundary terms we have
As in the earlier case if the boundary term is positive we can discard it, so we want to show that
Noting the fact that \(n_{jl} = \frac{\nu _j - \nu _l}{\sqrt{2-2\cos (\alpha _{jl})}}\) and \(dist(x,\mathcal {F}_j)=dist(x,\mathcal {F}_l)\) on \(\Gamma _{jl}\), we arrive at
Here we have used the equality \((a-b)(a^{p-1}-b^{p-1})=a^p-a^{p-1}b-b^{p-1}a+b^{p-1}\) with \(a=|\langle X_i(x), \nu _j \rangle |\) and \(b=|\langle X_i(x), \nu _l \rangle |\). From the above expression we note that the boundary term in \(\Omega\) is positive and \(\beta <0\). By discarding the boundary term we complete the proof. \(\square\)
Remark 3.3
For \(p\ge 2\), since
we have the following inequality
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Acknowledgements
The first author was supported by the EPSRC Grant EP/R003025/1, by the Leverhulme Research Grant RPG-2017-151, and by the FWO Odysseus Grant. The second author was supported in parts by the Nazarbayev University Grant N090118FD5342. Third author was supported by the Nazarbayev University program 091019CRP2120 and in parts by the Nazarbayev University Grant 240919FD3901.
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Ruzhansky, M., Sabitbek, B. & Suragan, D. Subelliptic geometric Hardy type inequalities on half-spaces and convex domains. Ann. Funct. Anal. 11, 1042–1061 (2020). https://doi.org/10.1007/s43034-020-00067-9
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DOI: https://doi.org/10.1007/s43034-020-00067-9