Densities with the Mean Value Property for Sub-Laplacians: An Inverse Problem

Abstract

Inverse problem results, related to densities with the mean value property for the harmonic functions, were recently proved by the authors. In the present paper we improve and extend them to the sub-Laplacians on stratified Lie groups.

Dedicated to Richard L. Wheeden

Appendix: \(\mathcal{L}\)-Superharmonic Functions

In this section we recall the definition and list some properties of the $\mathcal{L}$-superharmonic functions, as presented in [5, Chap. 8].

Let $\Omega \subseteq \mathbb{G}$ be open and let $u: \Omega \rightarrow ] -\infty,\infty ]$ be lower semicontinuous. We say that u is $\mathcal{L}$-superharmonic in $\Omega$ if

• (a) $u \in L_{\mathrm{loc}}^{1}(\Omega )$ and $\mathcal{L}(u) \leq 0$ in $\Omega$ in the weak sense of distributions,

• (b) u is M _r -continuous; i.e.,

  $$\displaystyle{u(x) =\lim _{r\rightarrow 0^{+}}M_{r}(u)(x)\qquad \forall x \in \Omega.}$$

Here M _r denotes the average operator in (4).

A function $v: \Omega \rightarrow [-\infty,\infty [$ is $\mathcal{L}$-subharmonic if − v is $\mathcal{L}$-superharmonic. We say that v is $\mathcal{L}$-harmonic if v is smooth and $\mathcal{L}v = 0$.

Let $\Gamma$ be the fundamental solution of $\mathcal{L}$ and let μ be a nonnegative Radon measure in $\mathbb{G}$. The $\Gamma$-potential of μ is defined as follows

$$\displaystyle{\Gamma _{\mu }(x):=\int _{\mathbb{G}}\Gamma (x^{-1} \circ y)\,d\mu (y),\qquad x \in \mathbb{G}.}$$

Obviously, if $\Omega$ is an open set such that $\mu (\Omega ^{c}) = 0$,

$$\displaystyle{\Gamma _{\mu }(x) =\int _{\Omega }\Gamma (x^{-1} \circ y)\,d\mu (y),\qquad x \in \Omega.}$$

The function $\Gamma _{\mu }$ is nonnegative and lower semicontinuous; it is $\mathcal{L}$-superharmonic in $\mathbb{G}$ if and only if there exists $z \in \mathbb{G}$ such that $\Gamma _{\mu }(z) <\infty$, see [5, Theorem 9.3.2].

In this case, see [5, Theorem 9.3.5],

$$\displaystyle{\mathcal{L}\Gamma _{\mu } = -\mu \quad \text{in the sense of distributions}}$$

and

$$\displaystyle{\Gamma _{\mu }\ \text{is }\mathcal{L}\text{-harmonic in }\mathbb{G}\setminus \mathop{ \mathrm{supp}}\nolimits \mu.}$$

For our purposes, the following remark is crucial.

Remark

Let $(\Omega,\mu,x_{0})$ be a $\Gamma$-triple (see Definition 2.2) and let $A \subseteq \Omega$ be a Borel set. Then the function

$$\displaystyle{\mathbb{G} \ni x\mapsto \Gamma _{\mu _{A}}(x):=\int _{A}\Gamma (x^{-1} \circ y)\,d\mu (y)}$$

is the $\Gamma$-potential of $\mu _{A}:=\mu \llcorner A$ and satisfies

$$\displaystyle{\Gamma _{\mu _{A}}(x) \leq \Gamma _{\mu }(x) = \Gamma (x^{-1} \circ x_{ 0}) <\infty \quad \forall x \in \Omega ^{c}.}$$

Moreover, $\Gamma _{\mu _{A}}$ is $\mathcal{L}$-superharmonic in $\mathbb{G}$ and

$$\displaystyle{\Gamma _{\mu _{A}}\ \text{is }\mathcal{L}\text{-harmonic in }O}$$

for every open set O ⊆ A ^c. Indeed O ⊆ A ^c implies $O \subseteq \overline{A}^{c} \subseteq (\mathop{\mathrm{supp}}\nolimits \mu _{A})^{c}$.