(Mean Value Theorem). [7] Let Ω be a stratified set whose free strata have the same dimension, let u be a harmonic function on ${\mathsf{\Omega }}^{\circ }$, and let $S_r\left(X_0\right)$, $X_0\in {\mathsf{\Omega }}^{\circ }$, be an admissible sphere. Then

(Harnack’s Inequality). [8] Let Ω be a stratified set and let K be an arbitrary compact set in ${\mathsf{\Omega }}^{\circ }$. Then the inequality holds for every nonnegative harmonic function u on ${\mathsf{\Omega }}^{\circ }$ with a constant $C=C(K,{\mathsf{\Omega }}^{\circ })$ independent of u.

(Removable Singularity Theorem). Let Ω be a strongly sturdy stratified set, $S\subset {\mathsf{\Omega }}^{\circ }$ a relatively closed set whose intersection with the closure of any free stratum has harmonic capacity zero in the affine space including the stratum, and $u:{\mathsf{\Omega }}^{\circ }\setminus S\to \mathbb{R}$ a bounded harmonic function. Then u has a harmonic extension to all of ${\mathsf{\Omega }}^{\circ }$.

Let Ω be a strongly sturdy stratified set of dimension n and $u:{\mathsf{\Omega }}^{\circ }\setminus {\mathsf{\Sigma }}_{n-2}\to \mathbb{R}$ a bounded harmonic function. Then u has a harmonic extension to all of ${\mathsf{\Omega }}^{\circ }$.

It suffices to prove that, for every point $X_0\in {\mathsf{\Sigma }}^n\cup {\mathsf{\Sigma }}^{n-1}$ and any admissible ball $B_r\left(X_0\right)$, the restriction ${u|}_{B_r\left(X_0\right)\setminus S}$ admits an extension to a bounded harmonic function on $B_r\left(X_0\right)$.

If $X_0\in {\mathsf{\Sigma }}^n$ then the existence of a sought extension follows from the removability of a relatively closed set of capacity zero for bounded harmonic functions on subdomains of ${\mathbb{R}}^n$. Therefore, suppose that $X_0$ belongs to an $(n-1)$-dimensional stratum ${\sigma }_{n-1}$ and let $B_r\left(X_0\right)$ be an admissible ball. Without loss of generality, we assume $\mathsf{\Omega }$ to be the closed stratified ball $B_r\left(X_0\right)\cup S_r\left(X_0\right)$ in which $B_r\left(X_0\right)$ serves as the interior and $S_r\left(X_0\right)$ as the boundary (see Figure 4).

Let ${\sigma }_{n1},\cdots ,{\sigma }_{nm}$ be all strata contiguous to ${\sigma }_{n-1}$.

If we have exactly two strata, ${\sigma }_{n1}$ and ${\sigma }_{n2}$, then $B_r\left(X_0\right)$ (with its interior metric) is isometric to the usual Euclidean ball B of radius r in ${\mathbb{R}}^n$; moreover, ${\sigma }_{n1}$ and ${\sigma }_{n2}$ go under isometry into half-balls separated by the $(n-1)$-dimensional disk that is the image of ${\sigma }_{n-1}$. The function $\tilde{u}$ corresponding to u under the isometry is an ordinary harmonic function on $B\setminus \tilde{S}$, where $\tilde{S}$ is the image of $S\cap B_r\left(X_0\right)$ under the isometry. Applying the standard removable singularity theorem to $\tilde{u}$, we obtain a harmonic extension of $\tilde{u}$ to B. Executing the inverse isometry, we get a harmonic extension of u to $B_r\left(X_0\right)$.

If we have an odd number of strata then we double their number by slightly rotating the ball $B_r\left(X_0\right)$ around ${\sigma }_{n-1}$ and extend u to the new strata by using this rotation.

Thus, we may assume that there are $2l$ strata in total, ${\sigma }_{nj}$, $j=1,\cdots ,2l$, contiguous to ${\sigma }_{n-1}$. Again we take a Euclidean ball B of radius r in ${\mathbb{R}}^n$ divided into two half-balls $B^1$ and $B^2$ resting on an equatorial $(n-1)$-dimensional disk D. Let $J\subset \{1,2,\cdots ,2l\}$ be an arbitrary collection of l distinct numbers and let $\overline{J}$ be the complementary collection. Now, map ${\sigma }_{nj}\cup {\sigma }_{n-1}$, $j\in J$, onto $B^1\cup D$ and map ${\sigma }_{n\overline{j}}\cup {\sigma }_{n-1}$, $\overline{j}\in \overline{J}$, onto $B^2\cup D$ isometrically so that the stratum ${\sigma }_{n-1}$ go onto D. Let $\tilde{S}$ be the union of the images of $({\sigma }_{nj}\cup {\sigma }_{n-1})\cap S$ over all j under the isometry. Translating u by isometry from ${\sigma }_{nj}\cup {\sigma }_{n-1}$ to $B^1\cup D$, we receive functions $u_j$, $j\in J$. Proceeding similarly, we also receive functions $u_{\overline{j}}$, $\overline{j}\in \overline{J}$. It is easily seen that the function $u_J$, equal to ${\sum }_{j\in J}{u}_j$ on $(B^1\cup D)\setminus \tilde{S}$ and equal to ${\sum }_{\overline{j}\in \overline{J}}{u}_{\overline{j}}$ on $(B^2\cup D)\setminus \tilde{S}$ (notice that both the functions agree on the intersection of their domains), is an ordinary harmonic function on $B\setminus \tilde{S}$. Applying to it the standard removable singularity theorem, we get a harmonic extension of $u_J$ to the whole ball B. Since this is true for every collection J, an easy combinatorics shows that each individual function $u_j$ gets an extension rather than their combination. Returning to u, we obtain an extension of u to the whole set ${\sigma }_{nj}\cup {\sigma }_{n-1}$ for every j; furthermore, the resultant function is harmonic on the whole ball $B_r\left(X_0\right)$. □

Equation (3) imposes no conditions on the strata of dimension $\le n-2$. Thus, the lemma asserts in fact that u extends continuously to ${\mathsf{\Omega }}^{\circ }$.

We extend u from ${\mathsf{\Omega }}^{\circ }\setminus {\mathsf{\Sigma }}_{n-2}$ to ${\mathsf{\Omega }}^{\circ }$ by using spherical means. To this end, we state the following lemma whose proof is postponed to the next section.

Leaning on this lemma, we put where r is any admissible radius.

Note that Let us prove by induction on k decreasing from $k=n-1$ to 0 that $M\left(X\right)$ is continuous on ${\mathsf{\Omega }}^{\circ }\setminus {\mathsf{\Sigma }}_{k-1}$. Then $M\left(X\right)$ will give us the sought extension of u.

The induction base is trivial. Suppose that $M\left(X\right)$ is continuous on ${\mathsf{\Omega }}^{\circ }\setminus {\mathsf{\Sigma }}_k$, $k<n-1$, and hence harmonic on ${\mathsf{\Omega }}^{\circ }\setminus {\mathsf{\Sigma }}_k$. Take $X_0\in {\sigma }_k$ for a k-dimensional stratum ${\sigma }_k\subset {\mathsf{\Omega }}^{\circ }$. We are to prove that This will justify the induction step and, therefore, complete the proof of Lemma 2.

For a proof, it suffices to apply Lemma 5 to the function $-u$, leaning upon the equality $lim\; inf(-u)=-lim\; supu$.

Lemmas 5 and 6 yield ${lim}_{X\to X_0}M\left(X\right)=M\left(X_0\right)$. This and Lemma 4 imply (4), thus finishing the proof of Lemma 2.

(Gradient Estimate). Let Ω be a stratified set whose free strata have the same dimension n and let $u:{\mathsf{\Omega }}^{\circ }\setminus {\mathsf{\Sigma }}_{n-2}\to \mathbb{R}$ be a bounded harmonic function. Then for every $X\in {\mathsf{\Omega }}^{\circ }\setminus {\mathsf{\Sigma }}_{n-2}$ the estimate holds: where $C=C\left(\mathsf{\Omega }\right)$ is a constant depending only on the structure of the stratified set Ω, and $\rho \left(X\right)=\mathrm{dist}(X,{\mathsf{\Sigma }}_{n-2}\cup 𝜕{\mathsf{\Omega }}^{\circ })$.

For every $X_0\in {\mathsf{\Omega }}^{\circ }\setminus {\mathsf{\Sigma }}_{n-2}$ and any admissible ball $B_r\left(X_0\right)$ the estimate holds: where $C=C\left(\mathsf{\Omega }\right)$ is a constant depending only on the structure of Ω.

We use the same trick as in the proof of Lemma 1.

If $X_0$ belongs to an n-dimensional stratum, then the required estimate follows from the above-mentioned classical result. We therefore assume that $X_0$ belongs to some $(n-1)$-dimensional stratum ${\sigma }_{n-1}$ and let $B_r\left(X_0\right)$ be an admissible ball. Without loss of generality, we consider $\mathsf{\Omega }$ to be the closed stratified ball $B_r\left(X_0\right)\cup S_r\left(X_0\right)$ in which $B_r\left(X_0\right)$ serves as interior and $S_r\left(X_0\right)$ as boundary (Figure ??).

Let ${\sigma }_{n1},\cdots ,{\sigma }_{nm}$ be all strata contiguous to ${\sigma }_{n-1}$.

If we have only two strata, ${\sigma }_{n1}$ and ${\sigma }_{n2}$, then $B_r\left(X_0\right)$ (with its interior metric) is isometric to the ordinary Euclidean ball of radius r in ${\mathbb{R}}^n$; moreover, under the isometry the strata ${\sigma }_{n1}$ and ${\sigma }_{n2}$ go onto two half-balls separated by an $(n-1)$-dimensional disk which is the image of ${\sigma }_{n-1}$. The function $\tilde{u}$ corresponding to u under the isometry is an ordinary harmonic function on the Euclidean ball and the required estimate is valid for it. By isometry, the required estimate holds for u, too.

Proceeding as in the proof of Lemma 1, we may assume that we have $2l$ strata ${\sigma }_{nj}$, $j=1,\cdots ,2l$, contiguous to ${\sigma }_{n-1}$. Repeating the construction from the proof of Lemma 1, we take a Euclidean ball B of radius r in ${\mathbb{R}}^n$ divided into two half-balls $B^1$ and $B^2$ resting on an $(n-1)$-dimensional disk D. Next, for an arbitrary collection $J\subset \{1,2,\cdots ,2l\}$ of l distinct numbers and the complementary collection $\overline{J}$, we obtain the functions $u_j$ on $B^1\cup D$, $j\in J$, and the functions $u_{\overline{j}}$ on $B^2\cup D$, $\overline{j}\in \overline{J}$, that correspond to the functions ${u|}_{{\sigma }_{nj}\cup {\sigma }_{n-1}}$, $j\in J$, and the functions ${u|}_{{\sigma }_{n\overline{j}}\cup {\sigma }_{n-1}}$, $\overline{j}\in \overline{J}$, by isometry. Then the function $u_J$, equal to ${\sum }_{j\in J}{u}_j$ on $B^1\cup D$ and equal to ${\sum }_{\overline{j}\in \overline{J}}{u}_{\overline{j}}$ on $B^2\cup D$, is an ordinary harmonic function on B and for it we have the required estimate. Since this is true for any J, an easy combinatorics shows that the sought estimate is valid for each function $u_j$, $j=1,\cdots ,2l$, and by isometry for the function u. □

If $X\in {\mathsf{\Sigma }}^n$ then $\mathrm{dist}(X,{\mathsf{\Sigma }}_{n-1}\cup 𝜕{\mathsf{\Omega }}^{\circ })$ is an admissible radius for X. There exists $\alpha \in (0,1]$ (depending on Ω) such that if $X\in {\mathsf{\Sigma }}^{n-1}$ then $\alpha \mathrm{dist}(X,{\mathsf{\Sigma }}_{n-2}\cup 𝜕{\mathsf{\Omega }}^{\circ })$ is an admissible radius for X.

Take $X\in {\mathsf{\Omega }}^{\circ }\setminus {\mathsf{\Sigma }}_{n-2}$ and

If $\mathrm{dist}(X,{\mathsf{\Sigma }}^{n-1})<\alpha \rho /4$ then there is an $(n-1)$-dimensional stratum ${\sigma }_{n-1}$ and a point $X_0\in {\sigma }_{n-1}$ such that $X\in B_{\alpha \rho /4}\left(X_0\right)$. Observe that $\mathrm{dist}(X_0,{\mathsf{\Sigma }}_{n-2}\cup 𝜕{\mathsf{\Omega }}^{\circ })>\rho /2$. By Lemma 8, $\alpha \rho /2$ is an admissible radius for $X_0$ and by Lemma 7

If $\mathrm{dist}(X,{\mathsf{\Sigma }}^{n-1})\ge \alpha \rho /4$ then, by Lemma 8, $\alpha \rho /4$ is an admissible radius for X and by Lemma 7 Anyway we get the required estimate. □

Let Ω be a stratified set whose free strata have the same dimension n and let $u:{\mathsf{\Omega }}^{\circ }\setminus {\mathsf{\Sigma }}_{n-2}\to \mathbb{R}$ be a bounded harmonic function. Then $\nabla u$ has zero flux through any admissible sphere $S_r\left(X\right)$:

If $B_r\left(X\right)\cap {\mathsf{\Sigma }}_{n-2}=⌀$ then the conclusion of the lemma follows from the application of the divergence theorem, which is expressed by (1), to the vector field $p\nabla u\in {\overrightarrow{C}}^1\left(B_r\left(X\right)\right)$, with the equality $\nabla ·(p\nabla u)=0$ taken into account.

Therefore, assume $B_r\left(X\right)\cap {\mathsf{\Sigma }}_{n-2}\ne ⌀$. Let be all strata whose closures contain X and dimensions do not exceed $n-2$. Consider the dimensions to increase: Note that there is exactly one stratum, ${\sigma }_{d_{1}1}$, of dimension $d_1$, and $X\in {\sigma }_{d_{1}1}$.

Let $0<{\rho }_1<r$ and

Choose ${\rho }_2$, $0<{\rho }_2<{\rho }_1$, so that the sets be pairwise disjoint for $j=1,\cdots ,j_2$ and do not intersect the strata ${\sigma }_{d_{i}j}$, $i>2$, that are not contiguous to ${\sigma }_{d_{2}j}$.

Proceeding by induction, finally choose ${\rho }_k$, $0<{\rho }_k<{\rho }_{k-1}$, so that the sets be pairwise disjoint for $j=1,\cdots ,j_k$.

Put for $l=1,\cdots ,k$ and consider the stratified set with boundary where

Figure 6 shows what remains after erasing the “cylinders” ${\sigma }_{d_{k}j}\left({\rho }_k\right)$ from the ball; in three-dimensional space the process of constructing ${\mathcal{P}}_l$ is not very diverse and might involve only one or two steps in dependence of the dimension of the stratum containing the center of the ball. In this figure the invisible part of the sphere is also included into the boundary $𝜕{\mathcal{P}}_l^{\circ }$.

Inducting on l decreasing from $l=k$ to 0, let us prove that where $d\mu$ is the stratified measure on $𝜕{\mathcal{P}}_l^{\circ }$.

Take $l=k$. The set $\mathsf{\Sigma }({\rho }_1,\cdots ,{\rho }_k)$ represents a neighborhood of the set ${\mathsf{\Sigma }}_{n-2}\cap B_r\left(X\right)$. Then $p\nabla u\in {\overrightarrow{C}}^1\left({\mathcal{P}}_k^{\circ }\right)$, and the divergence theorem (see (1)) gives which justifies the induction base.

Assuming (11) true, let us validate the equality for $l-1$. We have Hence, where the signs of the integrals on the right-hand side are chosen in accord with the fact that ${\mathsf{\Gamma }}_{d_{l}s}^l$ as part of the boundary of ${\sigma }_{d_{l}s}\left({\rho }_l\right)$.

The next relations will be justified below:

Applying them, we get from (12): completing validation of the induction step.

Since $𝜕{\mathcal{P}}_0^{\circ }=S_r\left(X\right)$, the equality (11) for $l=0$ coincides with (10).

To finish the proof of the theorem, it remains to justify (13) and (14).

Since $p=1$ on the strata of dimension n and $p=0$ on the other strata, we have where the summation is over all n-dimensional strata whose closures contain X, and $d{\mu }^{n-1}$ is the surface $(n-1)$-measure.

By Theorem 4, the estimate holds:

The following lemma is rather obvious and we leave it without proof.

Note that for $Z\in {\sigma }_{nj}$ where $L_{d_{i}s}$ is the affine subspace including ${\sigma }_{d_{i}s}$, and $\mathrm{dist}(Z,L_{d_{i}s})$ is the distance from Z to $L_{d_{i}s}$ inside $L_{nj}$. From (15)–(16) and Lemma 9 we then conclude that ${(p\nabla u)}_{\nu }$ is $\mu$-integrable on $𝜕{\mathcal{P}}_{l-1}^{\circ }$. This proves (13).

By analogy to (15), we have with the summation taken over the n-dimensional strata contiguous to ${\sigma }_{d_{l}s}$.

Fix a stratum ${\sigma }_{nj}$ contiguous to ${\sigma }_{d_{l}s}$ and consider

First, suppose that $d_l\le n-3$. In view of (16), The following assertion is proven by straightforward calculations.