Mean Value Property and Harmonicity on Carnot-Carathéodory Groups

Abstract

We study strongly harmonic functions in Carnot–Carathéodory groups defined via the mean value property with respect to the Lebesgue measure. For such functions we show their Sobolev regularity and smoothness. Moreover, we prove that strongly harmonic functions satisfy the sub-Laplace equation for the appropriate gauge norm and that the inclusion is sharp. We observe that appropriate spherical harmonic polynomials in ℍ₁ are both strongly harmonic and satisfy the sub-Laplace equation. Our presentation is illustrated by examples.

Appendix: The Proof of Theorem 4.4

In the Appendix we prove Theorem 4.4, which states that $\mathcal {L}$-harmonic functions satisfy a variant of the mean value property with respect to kernels defined via gradient of pseudonorm given by the fundamental solution of $\mathcal {L}$. We begin with preliminaries regarding the representation of $\mathcal {L}$ in the divergence form.

Fix an orthonormal basis $\{ E_{i} \}_{i = 1}^{N}$ such that $\mathfrak {g}^{1}=\text {span} \{ E_{i} | i = 1,\dots , N_{1} \}$ and

$$\tilde X_{i} u(p) =\frac{d}{dt}u(p\exp(t E_{i} ))|_{t = 0}\quad \text{ for } i = 1,2,\ldots, N. $$

Recall that N = N₁ + … + N_s, cf. Definition 2.1. If x_i, for i = 1, 2, … , N denote the coordinates on G induced by the chosen basis of $\mathfrak {g}$ via the exponential map, then

$$\tilde X_{i} u=\sum\limits_{j = 1}^{N} dx_{j} (\tilde X_{i}) \frac{\partial}{\partial x_{j}}. $$

In coordinates we have

$$\begin{array}{@{}rcl@{}} \mathcal{L}u = \sum\limits_{i=i_{0}}^{N_{1}} \tilde{X}_{i}^{2}u &=& \sum\limits_{j = 1}^{N} \sum\limits_{k = 1}^{N} \sum\limits_{i = 1}^{N_{1}} dx_{j}(\tilde X_{i}) \frac{\partial}{\partial x_{j}} \left( dx_{k}(\tilde X_{i}) \frac{\partial u}{\partial x_{k}}\right) \\ &=& \sum\limits_{j = 1}^{N} \sum\limits_{k = 1}^{N} \sum\limits_{i = 1}^{N_{1}} \frac{\partial}{\partial x_{j}}\left( dx_{j}(\tilde X_{i}) dx_{k}(\tilde X_{i})\frac{\partial u}{\partial x_{k}} \right) \\ &=&\text{div} A\nabla u, \end{array} $$

(31)

where the coefficients of matrix A are given by the formula $A_{j, k} ={\sum }_{i = 1}^{N_{1}} dx_{j} (\tilde X_{i}) dx_{k} (\tilde X_{i})$ for j, k = 1, … , N. Note that the third equality (31) uses the identity $\frac {\partial }{\partial x_{j}} \left (dx_{j}(\tilde X_{i}) \right )= 0$ for each i = 1, … , N₁ which follows from the nilpotency of G.

Expanding in coordinates as above we also get $\langle \nabla _{0} u , \nabla _{0} u \rangle _{G} = \langle A \nabla u \nabla u \rangle _{\mathbb {R}^{N}}$ and note that a mapping (p, q) → 〈∇₀u, ∇₀u〉(p^− 1q) is left invariant and homogeneous of degree 0 with respect to dilation since ∇₀u has degree 0.

Proof of Theorem 4.4

By direct calculation, it is easy to check that the following Green’s identity holds:

$$v \mathcal{L}u -u \mathcal{L}v= \text{div} \left( v A\nabla u - u A \nabla v \right). $$

Since u is $\mathcal {L}$-harmonic in Ω ⊂ G and $v=\mathcal {N}^{2-Q} \circ \tau _{p^{-1}} =-{\Gamma } \circ \tau _{p^{-1}}$, then v is $\mathcal {L}$-harmonic on G ∖{p} and

$${\int}_{B(p,r) \setminus B(p,\epsilon)} \text{div} \left( v A\nabla u - u A \nabla v \right)(q) dq= 0. $$

Applying Stokes theorem gives

$$ {\int}_{\partial B(p,r)} u A \nabla v (\nu) \cdot n(p,\nu) d S(\nu) = {\int}_{\partial B(p,\epsilon)} u A \nabla v (\nu) \cdot n(p,\nu) d S(\nu), $$

(32)

where dS is defined via the Gramm determinant. Note that we have used the following consequences of the choice of u and v:

$$\begin{array}{@{}rcl@{}} {\int}_{\partial B(p,\rho)} v A \nabla u (\nu) \cdot n(p,\nu) d S(\nu) &=&\rho^{2-Q}{\int}_{\partial B(p,\rho)} A \nabla u (\nu) \cdot n(p,\nu) d S(\nu)\\ &=&\rho^{2-Q}{\int}_{\partial B(p,\rho)} \text{div} A \nabla u (q) dq\\ &=&\rho^{2-Q}{\int}_{\partial B(p,\rho)} \mathcal{L}u (q) dq = 0. \end{array} $$

We define the following kernel K(p, q), cf. Definition 5.5.1 and the proof of Theorem 5.5.4 in [3]:

$$K(p,q): = (2-Q)\mathcal{N}(p^{-1}q)^{1-Q} \frac{\langle \nabla_{0} \mathcal{N}(p^{-1} q), \nabla_{0} \mathcal{N}(p^{-1} q) \rangle}{\|\nabla (\mathcal{N} \circ\tau_{p^{-1}})(q)\|} :=\! (2-Q)\mathcal{N}(p^{-1}q)^{1-Q} \mathcal{K} (p,q), $$

where p, q ∈ G and p ≠ q. Next, we define

$$T_{r}(u)(p):={\int}_{\partial B(p,r)} u(q) K(p,q) dS(q)\quad \text{ for any}~p\in G, $$

where r > 0 is such that B(p, r) ⋐ Ω. It turns out that transform T_r satisfies the mean value property. Namely, by Eq. 32 we have T_r(u)(p) = T_𝜖(u)(p) for all 0 < 𝜖 < r. It then follows that

$$\begin{array}{@{}rcl@{}} |T_{r}(u)(p)-u(p)T_{r}(\chi_{\Omega})(p)| &\leq& {\int}_{\partial B(p,\epsilon)} |u(q)-u(p)| |K(p,q)| dS(q)\\ &\leq& \frac{1}{Q-2}\sup\limits_{\partial B(p,\epsilon)} |u(q)-u(p)||T_{r}(\chi_{\Omega})(p)|. \end{array} $$

The continuity of u implies the following:

$$ T_{r}(u)(p) = u(p)T_{r}(\chi_{\Omega})(p). $$

(33)

Furthermore, since T_r(u)(p) = T_t(u)(p) for all t < r, we have

$$\begin{array}{@{}rcl@{}} T_{r}(u)(p)r^{Q}&=&T_{r}(u)(p){{\int}_{0}^{r}}Qt^{Q-1}dt = {{\int}_{0}^{r}} T_{t}(u)(p)Qt^{Q-1}dt\\ &=&{{\int}_{0}^{r}} {\int}_{\partial B(p,t)} u(\nu) K(p,\nu) dS(\nu)Qt^{Q-1}dt\\ &=&Q(2-Q){{\int}_{0}^{r}} {\int}_{\partial B(p,t)} u(\nu) \mathcal{K}(p,\nu) dS(\nu) dt\\ &=&Q(2-Q){\int}_{B(p,r)} u(q) \mathcal{K}(p,q) \|\nabla (\mathcal{N} \circ \tau_{p^{-1}})(q)\| dq \\ &=&Q(2-Q){\int}_{B(p,r)} u(q) \langle \nabla_{0} \mathcal{N}(p^{-1} q), \nabla_{0} \mathcal{N}(p^{-1} q) \rangle dq. \end{array} $$

Finally, from Eq. 33 we get the assertion of Theorem 4.4 for all p ∈ G and all balls B(p, r) ⋐ Ω:

$$u(p) = \frac{{\int}_{B(p,r)} u(q)|\nabla_{0} \mathcal{N}(p^{-1} q)|^{2} dq} {{\int}_{B(p,r)} |\nabla_{0} \mathcal{N}(p^{-1} q)|^{2} dq} =\frac{{\int}_{B(0,r)} u(pq) |\nabla_{0} \mathcal{N}(q)|^{2} dq} {{\int}_{B(0,r)} |\nabla_{0} \mathcal{N}(q)|^{2} dq}. $$

□

As mentioned in Section 4, these computations indicate that $\mathcal {L}$-harmonic functions need not necessarily be strongly harmonic.