Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 28, 2003, 279–302

NONLINEAR HARMONIC MEASURES ON TREES

Robert Kaufman, José G. Llorente, and Jang-Mei Wu

University of Illinois, Department of Mathematics 1409 West Green Street Urbana, Illinois 61801, U.S.A.; rpkaufma@math.uiuc.edu Universitat Autònoma de Barcelona, Department de Matemàtiques ES-08193 Bellaterra, Barcelona, Spain; gonzalez@mat.uab.es University of Illinois, Department of Mathematics 1409 West Green Street, Urbana, Illinois 61801, U.S.A.; wu@math.uiuc.edu

Abstract. We show that nonlinear harmonic measures on trees lack many desirable properties of set functions encountered in classical analysis. Let F be an averaging operator on R^{κ} and ω_{F} be the F -harmonic measure on a κ-regular forward branching tree. Unless F is the usual average, ω_{F} is not a Choquet capacity; union of sets of ω_{F} measure zero can have positive ω_{F} measure when F is permutation invariant; and there exist sets of full ω_{F} measure having “small” dimension. Let A be a monotone operator on R^{κ}, then A-harmonic functions on trees need not obey the strong maximum principle unless the ratio of the ellipticity constants is close to 1.

We show that nonlinear harmonic measures on trees lack many desirable properties of set functions encountered in classical analysis.

Let T be a directed tree with regular κ-branching; in this paper we continue earlier work on p-harmonic functions on trees in [CFPR] and [KW]. We treat the p-Laplacian as a special case of a nonlinear averaging operator F: R^{κ} → R^{1} studied by Alvarez, Rodríguez and Yakubovich in [ARY]. Then the F -potential theory on T is the discrete version of the nonlinear potential theory in the Euclidean space structured on the nonlinear Euler equation of the variational integral ∫ ℱ (x, ∇u) dx with ℱ (x, h) ≈ |h|^{p}; see [GLM] and [HKM].

Each averaging operator F leads to a harmonic measure on the boundary ∂T of the tree. Except when F is the usual average, there exists a number d(κ, F) strictly less than the dimension of ∂T, so that every compact set on ∂T of dimension < d(κ, F) must have zero F -harmonic measure, and there exist sets of dimension ≤ d(κ, F) having full F -harmonic measure. If we could show that every Borel set on ∂T of dimension < d(κ, F) has zero F -harmonic measure, then the statement “the dimension of F -harmonic measure is d(κ, F)” would

2000 Mathematics Subject Classification: Primary 31C05, 31C45; Secondary 31C20, 28A12. The second author was partially supported by a grant from the Ministerio de Educación,

Spain. Part of this research was started when the second author was visiting the University of Illinois in February 2002. He wishes to thank the Department of Mathematics for hospitality. The third author was partially supported by the National Science Foundation DMS-0070312.

follow; here dimension of F -harmonic measure is defined to be the infimum of the dimensions of those Borel sets on ∂T having full F -harmonic measure. When F is the p-Laplacian operator, d(κ, F) is introduced in [KW] and used to study the sizes of Fatou sets and sets of finite radial variations for bounded p-harmonic functions on trees. Here again d(κ, F) is the critical dimension of Fatou sets for bounded F -harmonic functions.

We also prove that when F is permutation invariant and is not the usual average, there exist two sets on ∂T, of zero F -harmonic measure, whose union has positive F -harmonic measure. This answers a question raised by Martio on trees ([ARY], [M]). Our work is motivated by [ARY], in which it is proved that for certain F ’s, there exist congruent sets B_{1}, B_{2}, …, B_{κ} of arbitrarily small positive F -harmonic measure, whose union is ∂T. Our construction starts from the root of the tree and theirs starts from the boundary.

We also show that when F is permutation invariant and is not the usual average, F -harmonic measure is not a Choquet capacity; consequently, it is not left continuous on increasing sequences of sets.

While these results show that F -harmonic measure lacks many desirable properties of set functions in the linear theory, many problems remain: some concern inner approximation of Borel sets by compact sets, others are about the behavior of monotone sequences A_{n} such that ⋃_{1}^{∞} A_{n} = ∂T.

In another direction, we study the analogue of the quasilinear elliptic equation div(A∇u) = 0 ([HKM]) on trees. We examine the notion of A-harmonic functions on trees, when A is a monotone operator on R^{κ}. We find that A-harmonic functions do not always satisfy the strong maximum principle defined in Section 1; this shows that some caution is necessary in arguing from elliptic operators in the Euclidean space to potential theory on trees. When an A-operator is close to the p-Laplacian, the strong maximum principle holds and a Fatou type theorem is valid; our sufficient condition is sharp when p = 2. We also show that the critical dimension d(κ, A) for the A-operator approaches the critical dimension d(κ, p) for the p-Laplacian uniformly as the ellipticity constants of A approach that of the p-Laplacian.

Finally, we comment on the meanings of 1-Laplacian and ∞-Laplacian on trees.

Our operators on trees may be considered as simple analogues of the p-Laplacian div(|∇u|^{p−2}∇u), 1 < p < ∞, on the unit disk D. Martio ([HKM], [M]) asked whether the p-harmonic measure on the unit circle ∂D is subadditive, or whether the union of two null sets must be null when p ≠ 2; for work in this direction, see [AM], [GLM]. The Fatou set ℱ (u) of a function u in D is the set on ∂D where the radial limits exist. The classical theorem of Fatou from 1906 states that ℱ (u) has length 2π for bounded harmonic functions. When p ≠ 2, the Fatou set ℱ (u) of a bounded p-harmonic function u can have length zero; examples are given by Wolff ([W]) for 2 < p < ∞ and by Lewis ([L]) for 1 < p < 2.

It is known that dim ℱ (u) ≥ δ(p) > 0, see ([MW], [FGMS]). The best value of δ(p) is unknown when p ≠ 2, in particular whether δ(p) is equal to 1.

1. Preliminaries

Let κ > 1 be a natural number and T a directed tree with regular κbranching. That is, T consists of the empty set ϕ and all finite sequences (v_{1}, v_{2}, …, v_{r}) of lengths r = 1, 2, 3, …, whose coordinates are chosen from {1, 2, 3, …, κ}. The elements in T are called vertices. Each vertex v has κ successors, obtained by adding another coordinate. These are abbreviated by (v, 1), (v, 2), …, (v, κ) and have length one more than the length of v.

A branch b of T is an infinite sequence (b_{1}, b_{2}, …) with coordinates in {1, 2, …, κ}. And b can be regarded as an infinite sequence of vertices (b_{1}), (b_{1}, b_{2}), …, (b_{1}, b_{2} , …, b_{r}), …, each followed by its immediate successor. The set of all branches forms the boundary ∂T of the tree.

A metric on T ∪∂T is defined as follows. The distance between two sequences (finite or infinite) b = (b_{1}, b_{2}, …) and b′ = (b′_{1}, b′_{2}, …) is κ^{−N+1} when N is the first index n such that b_{n} ≠ b′_{n}; but when b = (b_{1}, b_{2}, …, b_{N}) and b′ = (b_{1}, b_{2}, …, b_{N}, b′_{N+1}, …), the distance is κ^{−N}. Hausdorff measure and Hausdorff dimension are defined using this metric. The tree T and the boundary ∂T then have diameter one, and ∂T has Hausdorff dimension one.

We define a probability measure λ on ∂T through the mapping g(b) = ∑_{1}^{∞} κ^{−r}(b_{r} − 1) onto [0, 1]. The λ-measure of a set E is the Lebesgue measure of g(E); this is the infinite product of uniform distributions on each factor {1, 2, …, κ}. λ is sometimes called Lebesgue measure.

Let F: R^{κ} → R^{1} be a continuous function. We call F an averaging operator if it satisfies the following:

(i) F (0, 0, …, 0) = 0 and F (1, 1, …, 1) = 1;

(ii) F (tx_{1}, tx_{2}, …, tx_{κ}) = tF (x_{1}, x_{2}, …, x_{κ}) for t ∈ R^{1} ;

(iii) F (t + x_{1}, t + x_{2}, …, t + x_{κ}) = t + F (x_{1}, x_{2}, …, x_{κ}) for t ∈ R^{1} ; (iv) F (x_{1}, x_{2}, …, x_{κ}) < max{x_{1}, x_{2}, …, x_{κ}} if not all x_{j} ’s are equal;

(v) F is nondecreasing with respect to each variable.

The definition is adopted from [ARY].

Property (iv) is the strong maximum principle, which implies that if

F (x_{1}, x_{2}, …, x_{κ}) = 0,

unless all x_{j} ’s are zero, there must be a sign change among the entries. Property (v) is the monotonicity property which gives the comparison principle for the Dirichlet problem needed in developing potential theory.

It follows from (i) ∼ (iv) that

(vi) F (1 − x_{1}, 1 − x_{2}, …, 1 − x_{κ}) = 1 − F (x_{1}, x_{2}, …, x_{κ});

and

(vii) there is a number b > 0 such that whenever F (x_{1}, x_{2}, …, x_{κ}) ≥ 0 and max x_{j} ≤ 1, then min x_{j} ≥ −b.

To verify (vii), let S be the set in R^{κ} defined by max x_{j} ≤ 0 and min x_{j} = −1. Then by (ii) and (iv), F (x_{1}, x_{2}, …, x_{κ}) < 0 on S. By continuity, there is an ε > 0 so that F < 0 on the set {(x_{1}, x_{2}, …, x_{κ}) : max x_{j} ≤ ε and min x_{j} = −1}. Hence we can choose b = 1/ε.

Property (vii) is used in the proof of Theorem 1, more precisely, in proving (iii) in Lemma 1. When the strong maximum principle is replaced by

(iv)′ F (x_{1}, x_{2}, …, x_{κ}) ≤ max{x_{1}, x_{2}, …, x_{κ}}, then Lemma 1(iii) fails and the proof of Theorem 1 will be considerably more tedious.

In certain theorems, we require, in addition, F to be permutation invariant: (viii) F (x_{1}, x_{2}, …, x_{κ}) = F (x_{τ(1)}, x_{τ(2)}, …, x_{τ(κ)}) for each permutation τ of

{1, 2, …, κ}.

From now on, we assume F is an averaging operator not necessarily permutation invariant, unless otherwise mentioned. We let

0 = (0, 0, …, 0), 1 = (1, 1, …, 1),

and use X = (x_{1}, x_{2}, …, x_{κ}) to denote vectors in R^{κ}.

Remark. If F is differentiable at 0, then F (X) = Σλ_{j}x_{j} for some λ_{j} ∈ [0, 1] with Σλ_{j} = 1; if F is also permutation invariant then F (X) ≡ Σx_{j}/k is the usual average. To see this, let λ_{j} = (∂/∂x_{j})F (0), and X ≠ 0. Then tF (X) = F (tX) = Σλ_{j}tx_{j} + o(t) as t → 0. This implies that F (X) = Σλ_{j}x_{j}.

We call X ∈ R^{κ} an F -harmonic vector (or F -superharmonic vector) if F (X) = 0 (or F (X) ≤ 0).

Given a function u on T, the gradient of u at a vertex v is

∇u(v) = (u(v, 1) − u(v), u(v, 2) − u(v), …, u(v, κ) − u(v)).

We say u is an F -harmonic function on tree T, if ∇u(v) is F -harmonic at each vertex v, i.e.

F (u(v, 1), u(v, 2), …, u(v, κ)) = u(v) for all v ∈ T.

F -superharmonicity is defined analogously.

For E ⊆ ∂T, let E^{c} = ∂T \E,

U = {u : F -superharmonic on T so that lim inf_{v→b} u(v) ≥ χ_{E}(b) for all b ∈ ∂T }

the upper class of E, and

ω_{F} (v, E) = inf{u(v) : u ∈ U}

the F -harmonic measure function for E; and call ω_{F} (ϕ, E) the F -harmonic measure of E, in short ω_{F} (E). Theorem 3 below shows that ω_{F} is not an outer measure unless F is the usual average.

Following the arguments in [HKM] for the continuous case, we have

(i) 0 ≤ ω_{F} ( · , E) ≤ 1 on T;

(ii) ω_{F} (E) ≤ ω_{F} (G) when E ⊆ G;

(iii) if E is compact, then lim_{v→b} ω_{F} (v, E) = 0 for b ∈ E^{c};

(iv) ω_{F} ( · , E) is F -harmonic on T;

(v) if E and G are disjoint compact sets on ∂T and ω_{F} (E) = ω_{F} (G) = 0, then ω_{F} (E ∪ G) = 0;

(vi) if E is compact, then ω_{F} (E) + ω_{F} (E^{c}) = 1;

(vii) if E_{1} ⊇ E_{2} ⊇ ⋯ ⊇ E_{j} ⊇ ⋯ are compact sets then lim ω_{F} (E_{j}) = ω_{F} (∩E_{j}).

Examples. (1) For 1 < p < ∞, the p-Laplacian Δ_{p} of a vector X in R^{κ} is

∑_{j} x_{j}|x_{j}|^{p−2},

and X is said to be p-harmonic (or p-superharmonic) if Δ_{p}X = 0 (or ≤ 0). The operator p(X) = t from R^{κ} to R^{1} defined implicitly by

Δ_{p}(X − t1) = Σ(x_{j} − t)|x_{j} − t|^{p−2} = 0

is an averaging operator; and Δ_{p}(X −t1) = 0 if and only if ∑_{1}^{κ} |x_{j} −x|^{p} attains its minimum at x = t. Thus p-harmonic functions and F_{p} -harmonic functions are the same, and they are the discrete analogues of the solutions to div(∇u|∇u|^{p−2}) = 0. We use ω_{p} to denote the p-harmonic measure.

(2) The discrete analogue to the quasilinear elliptic equation divA(∇u) = 0, which contains p-Laplacian as a special case, gives rise to another host of averaging operators, see Section 5.

2. The critical dimension d(κ, F)

Given a vector X = (x_{1}, x_{2}, …, x_{κ}), we identify it with a random variable, again called X, with probability P (X = x_{j}) = κ^{−1} for 1 ≤ j ≤ κ. When X contains both positive and negative entries, with ℰ denoting expectation,

β(X) = min{ℰ (e^{tX}) : t ∈ R}

is less than 1 and is attained at some t(X), with t(X) > 0 if Σx_{j} < 0 and t(X) < 0 if Σx_{j} > 0 ([KW]).

Let

m(κ, F) = min{∑_{1}^{κ} e^{x_{j}} : F (X) = 0}

and observe that the minimum is attained. In fact, x_{j} < log κ as soon as the sum is less than κ. It follows from property (vii) of the averaging operators that min x_{j} ≥ −b log κ. The minimum is attained by continuity.

Define

d(κ, F) = log m(κ, F)/ log κ.

Then 0 < d(κ, F) < 1 unless F is the usual average, in which case d(κ, F) = 1, see Lemma 3 below.

The Fatou set ℱ (u) of a function u is the set of branches b = (b_{1}, b_{2}, …) on which lim_{n→∞} u(b_{1}, b_{2}, …, b_{n}) exists and is finite, and BV(u) is the set of branches b on which u has finite variation Σ|u(b_{1}, b_{2}, …, b_{n+1})−u(b_{1}, b_{2}, …, b_{n})|. Recall that F_{p} is the averaging operator associated with the p-Laplacian, and let m(κ, p) = m(κ, F_{p}), d(κ, p) = d(κ, F_{p}). It is proved in [KW] that

min_{ℋ_{p}}dim ℱ (u) = min_{ℋ_{p}}dim BV(u) = d(κ, p)

with ℋ_{p} being the set of bounded p-harmonic functions on T. Values of d(κ, p) can be estimated by Lagrange multipliers, and asymptotics can be found for large p or large κ ([KW]). The same argument gives the following.

Theorem A. Let F be an averaging operator and ℋ_{F} be the set of bounded F -harmonic functions on T. Then

min_{ℋ_{F}}dim ℱ (u) = min_{ℋ_{F}}dim BV(u) = d(κ, F).

Remark. The monotonicity of the averaging operators is not used in the proof. Thus Theorem A is valid for the class of operators satisfying (i) ∼ (iv) only.

Theorem A suggests that dim ω_{F} = d(κ, F) might be true.

Theorem 1. Let F be an averaging operator. Suppose that E is a compact set on ∂T with dim E < d(κ, F) then ω_{F} (E) = 0.

The entropy of a probability measure μ on {1, 2, …, κ} is H(μ) = −Σμ_{j} log μ_{j}, where μ_{j} = μ({j}).

Lemma 1. Let F (x_{1}, x_{2}, …, x_{κ}) = 0. Then there is a probability measure ν on {1, 2, …, κ} such that

(i) Σν_{j}x_{j} = 0,

(ii) H(ν) ≥ log m(κ, F),

(iii) min ν_{j} ≥ c(κ, F) > 0.

Proof. We follow the proof in [KW]. If Σx_{j} = 0, choose ν_{j} = 1/κ for all j. We may assume then Σx_{j} < 0 and recall that (1/κ)Σe^{tx_{j}} attains a minimum β(X) at some τ > 0. Then κβ(X) ≥ m(κ, F) and Σx_{j}e^{τx_{j}} = 0. Define ν_{j} = e^{τx_{j}}/κβ(X), and observe that Σν_{j} = 1, Σν_{j}x_{j} = 0 and H(ν) = log κβ(X) ≥ log m(κ, F). To prove (iii), note that τx_{j} ≤ log κ for all j; then by property (vi) of averaging operators, min τx_{j} ≥ −b log κ. This gives (iii).

In the following, we let S_{r} be the class of subsets of branches of the tree, defined by the first r coordinates. This is a finite field whose atoms are called cylinders of rank r. Now S_{r} contains κ^{r} cylinders 𝒞_{r} of Lebesgue measure κ^{−r} each and the same diameter. The cylinder 𝒞_{0} is ∂T.

Proof of Theorem 1. Let u(v) = ω_{F} (v, E), then 0 ≤ u ≤ 1 on T. Since E is compact, lim_{v→b} u(v) = 0 for all b ∈ ∂T \E. We now apply Lemma 1 to the gradient of u at each vertex v, to find a probability measure μ on ∂T so that u is a martingale with respect to μ. The process resembles the linearization of a solution of a nonlinear operator in the continuous situation, see [CFPR] and [KW]. To define μ, let μ(C_{0}) = 1 and assume μ has been defined on all cylinders of rank ≤ r. Let C_{r} be a cylinder of rank r represented by (b_{1}, b_{2}, …, b_{r}). The gradient (x_{1}, …, x_{κ}) of u at the vertex v = (b_{1}, …, b_{r}) forms an F -harmonic vector. Let ν be the probability measure on {1, 2, …, κ} associated with the present (x_{1}, x_{2}, …, x_{κ}) as in Lemma 1, and define μ on the κ cylinders C_{r+1} of rank r + 1 contained in C_{r} by μ(C_{r+1}) = ν_{j}μ(C_{r}) if C_{r+1} is represented by (b_{1}, b_{2}, …, b_{r}, j). We have defined μ on all cylinders of rank r + 1, and then on ∂T by σ -additivity.

Properties (ii) and (iii) in Lemma 1 yield that μ(C_{r+1}) > c(κ, F)μ(C_{r}) whenever C_{r+1} ⊆ C_{r}, and H(μ | C_{r}) ≥ log m(κ, F). Then by a theorem on entropy and the dimension associated with a measure, known as early as Besicovitch and Eggleston and stated in the form used here in [KW], the measure μ is zero on any subset of ∂T of dimension less than log m(κ, F)/ log κ.

Note from (i) of Lemma 1, u is a bounded martingale with respect to μ, and therefore by the martingale convergence theorem, lim_{v→b} u(v) = u^{∗}(b) exists μ-a.e. on ∂T and

u(ϕ) = ∫_{∂T} u^{∗}(b) dμ(b) = ∫_{E} u^{∗}(b) dμ(b) = 0.

This proves that E has zero F -harmonic measure and thus the theorem.

We need another expression for the quantity m(κ, F) in the next section.

Lemma 2. Let F be an averaging operator, not equal to the usual average. Then there exists Y = (y_{1}, y_{2}, …, y_{κ}) so that F (Y) = 1, ∏ y_{j} > 1 and y_{j} > 0 for all 1 ≤ j ≤ κ.

Proof. Since F is not the usual average, there is some vector X = (x_{1}, …, x_{κ}) such that F (X) = 0 but Σx_{j} > 0. Now we can take Y = 1 + tX for some small positive t.

Let

m^{∗}(κ, F) = inf{min_{t≤0} Σy_{j}^{t} : F (Y) = 1 and y_{j} > 0 for all 1 ≤ j ≤ κ},

and

m^{∗∗}(κ, F) = inf{min_{t≤0} Σy_{j}^{t} : F (Y) = 1, ∏ y_{j} > 1 and y_{j} > 0 for all 1 ≤ j ≤ κ}.

Lemma 3. m(κ, F) = m^{∗}(κ, F) = m^{∗∗}(κ, F). The minimum m(κ, F) is attained. When F is not the usual average, m^{∗}(κ, F) and m^{∗∗}(κ, F) are not attained by vectors defining them, and 0 < m(κ, F) < κ; when F is the usual average, m(κ, F) = κ.

Proof. First suppose F (X) = 0. Then F (1 + tX) = 1 and 1 + tx_{1} , 1 + tx_{2}, …, 1 + tx_{κ} > 0 for small t < 0. Since lim_{t→0^{−}} Σ(1 + tx_{j})^{1/t} = Σe^{x_{j}}, then m^{∗}(κ, F) ≤ m(κ, F).

Conversely, suppose y_{1}, y_{2}, …, y_{κ} > 0 and F (y_{1}, y_{2}, …, y_{κ}) = 1. Hence log y_{j} ≤ y_{j} − 1 and F (y_{1} − 1, y_{2} − 1, …, y_{κ} − 1) = 0. Thus m(κ, F) ≤ Σe^{t(y_{j}−1)} ≤ Σy_{j}^{t} for t ≤ 0. This proves m(κ, F) ≤ m^{∗}(κ, F), and m(κ, F) = m^{∗}(κ, F).

Suppose F is the usual average, then a simple calculation gives m(κ, F) = κ. Otherwise, there exists an F -harmonic vector (x_{1}, x_{2}, …, x_{κ}) with Σx_{j} < 0. Choosing t > 0 and sufficiently small, we have m(κ, F) ≤ Σe^{tx_{j}} < κ. We proved earlier that m(κ, F) is attained and therefore strictly positive. Thus 0 < m(κ, F) < κ.

Suppose ∏_{1}^{κ} y_{j} ≤ 1 and t ≤ 0. By the arithmetic-geometric mean inequality, we have

κ^{−1}Σy_{j}^{t} ≥ exp(κ^{−1}Σt log y_{j}) ≥ 1.

Since 0 < m(κ, F) < κ, m^{∗}(κ, F) = m^{∗∗}(κ, F).

Because m(κ, F) < κ, we only need to consider t < 0 and Y ≠ 1 in the second paragraph of the proof. Since log is strictly concave, m(κ, F) < Σy_{j}^{t}. This shows that m^{∗}(κ, F) and m^{∗∗}(κ, F) are not attained, and completes the proof of Lemma 3.

3. F -harmonic measure—dimension and null sets

When F is not the arithmetic mean, we show in Theorem 2 that there exists a set of dimension at most d(κ, F), having full ω_{F} measure; and in Theorem 3 that the union of two sets of zero ω_{F} measure can have positive ω_{F} measure.

If we were able to prove Theorem 1 for all Borel sets on ∂T, then from Theorem 2,

dimension of ω_{F} = d(κ, F),

here dimension of ω_{F} is inf{dim E : E Borel on ∂T, ω_{F} (E) = 1}. The problem is nontrivial, in view of Theorem 5.

Proposition 1. Let F be an averaging operator not equal to the usual average, and Y be a vector in R^{κ} satisfying F (Y) = 1, ∏ y_{j} > 1 and y_{j} > 0 for all 1 ≤ j ≤ κ. Let X = (log y_{1}, log y_{2}, …, log y_{κ}), then there exists a set E ⊆ ∂T so that ω_{F} (E) = 0, ω_{F} (E^{c}) = 1 and dim(E^{c}) ≤ 1 + log β(X)/ log κ.

Proof. Define an F -harmonic function u on T as follows: let u(ϕ) = 1; suppose u has been defined at a vertex v, define u at its immediate successors (v, 1), (v, 2), …, (v, κ) by y_{1}u(v), y_{2}u(v), …, y_{κ}u(v), respectively. It is clear that u is positive F -harmonic. And suppose that v = (v_{1}, v_{2}, …, v_{n}) with v_{j} ∈ {1, 2, …, κ}, then

u(v) = ∏_{j=1}^{n} y_{v_{j}}.

Let m and n be positive integers, define

E(m, n) = {b : ∏_{j=1}^{n} y_{b_{j}} > m}, E_{m} = ⋃_{n≥1} E(m, n) and E = ⋂_{m=1}^{∞} E_{m}.

It is clear that E_{m} decreases as m increases, and E is exactly the set of b along which u is unbounded.

To calculate dim E^{c}, we note that E^{c} = ⋃_{m=1}^{∞} E_{m}^{c} and E_{m}^{c} ⊆ E(m, n)^{c} for every n ≥ 1. We shall calculate dim E_{m}^{c}. Denote by X also the random variable defined by probability P (X = log y_{j}) = 1/κ for 1 ≤ j ≤ κ; then the expectation ℰ (e^{tX}) = ℰ (Y^{t}) attains its minimum β(X) at t(X) < 0.

Let S_{n} be the sum X_{1} + X_{2} + ⋯ + X_{n} of independent identically distributed random variables with the same law as X. Then the Lebesgue measure

\begin{aligned}λ(E(m, n)^{c}) & = P (S_{n} ≤ log m) = P (e^{t(X)S_{n}} ≥ m^{t(X)}) \\ & ≤ m^{−t(X)}(ℰ (e^{t(X)X}))^{n} = m^{−t(X)}β(X)^{n}. \end{aligned}

Therefore E(m, n)^{c} is contained in β(X)^{n}m^{−t(X)}κ^{n} many balls of diameter κ^{−n}. From this we see that

dim E_{m}^{c} ≤ 1 + log β(X)/ log κ for all m ≥ 1.

Therefore

dim E^{c} ≤ 1 + log β(X)/ log κ,

where

β(X) = min_{t} ∑_{j} e^{t log y_{j}} = min_{t} ∑_{j} y_{j}^{t}.

We now prove ω_{F} (E) = 0. For each m > 1, define u_{m}, an F -harmonic function by stopping time argument. Let u_{m}(ϕ) = u(ϕ) = 1, and suppose u_{m} has been defined at a certain vertex v. If u_{m}(v) < m, then let u_{m}(v, j) = u(v, j) for 1 ≤ j ≤ κ; if u_{m}(v) ≥ m, then u_{m} stops and takes the value u_{m}(v) at all successors (v, j) of v. It is clear that u_{m} is positive F -harmonic, and that lim_{v→b} u_{m}(v) ≥ m for b ∈ E_{m}. It follows that ω_{F} (E) ≤ ω_{F} (E_{m}) ≤ u_{m}(ϕ)/m = 1/m for each m > 1. Therefore ω_{F} (E) = 0.

Recall, from property (vi) stated after the definition of F -harmonic functions, that for compact sets S, ω_{F} (S) + ω_{F} (S^{c}) = 1. Since the identity is unknown for general sets, we need to show ω_{F} (E^{c}) = 1. Note that E^{c} = ⋃_{m} E_{m}^{c} and that 1 − u_{m}/m ≤ 1 on T with boundary values lim_{v→b} 1−u_{m}(v)/m ≤ 0 on E_{m}. It follows from the definition of F -harmonic measure and the monotonicity property of F that ω_{F} (E^{c}) ≥ ω_{F} (E_{m}^{c}) ≥ 1 − u_{m}(ϕ)/m = 1 − 1/m. This says that ω_{F} (E^{c}) = 1.

This completes the proof of Proposition 1.

Theorem 2. Let F be an averaging operator on R^{κ}. Then there exists a set E on ∂T such that ω_{F} (E) = 0, ω_{F} (E^{c}) = 1 and dim E^{c} ≤ d(κ, F).

The fact that m(κ, F) is not attained by min_{t≤0} Σy_{j}^{t} for any vector Y satisfying F (Y) = 1, ∏ y_{j} > 1 and y_{j} > 0 for all 1 ≤ j ≤ κ, complicates the proof of the theorem. For otherwise, we could have used an extremal Y in Proposition 1 and achieved the dimension d(κ, F).

Proof. Assume that F is not the usual average; otherwise take E^{c} = ∂T. Let X be an F -harmonic vector such that Σe^{x_{j}} = m(κ, F), then Σx_{j} < 0.

Let ψ(n) = (log^{+} log^{+} n)^{−1} when n > e^{e} and ψ(n) = 1 otherwise. Let Y^{(n)} = (y_{1}^{(n)}, y_{2}^{(n)} ⋯ y_{κ}^{(n)}) be a sequence of vectors defined by

y_{j}^{(n)} = 1 − ψ(n)x_{j}

provided that n is so large that y_{j}^{(n)} > \frac{1}{2} for all j; and let Y^{(n)} = 1 otherwise. Then F (Y^{(n)}) = 1.

Define now a positive F -harmonic function u on T by the multiplicative process in Proposition 1 with respect to the varying sequence {Y^{(n)}}. That is u(ϕ) = 1, after u(v) has been defined at a vertex v = (v_{1}, v_{2}, …, v_{n}), let u(v, j) = y_{j}^{(n+1)}u(v) for 1 ≤ j ≤ κ. So, for v = (v_{1}, v_{2}, …, v_{n}),

u(v) = ∏_{r=1}^{n}(1 − ψ(r)x_{v_{r}}).

Let η(n) = ψ(n)^{−1} , so η(n) = log log n for n > e^{e}. We want to estimate the average M_{n} of u(v)^{−η(n)} over all vertices of length n. For n large and \sqrt{n} ≤ r ≤ n, we have 1 ≤ ψ(r)η(n) ≤ 1 + 2ψ(n). For these r’s we have a formula for the expected value

\begin{aligned}ℰ (1 − ψ(r)X)^{−η(n)} & = ℰ (e^{−η(n) log(1−ψ(r)X)}) \\ & = ℰ (e^{(1+O(ψ(n)))X}) = κ^{−1}m(κ, F) + 0(ψ(n)). \end{aligned}

Here X is the random variable defined by P (X = x_{j}) = κ^{−1} . For 1 ≤ r < \sqrt{n} , we use y_{j}^{(r)} = 1 − ψ(r)x_{j} > \frac{1}{2} .

Writing γ = log m(κ, F) − log κ, and applying the product formula for u, we can summarize

M_{n} = ℰ (u(v_{1}, v_{2}, …, v_{n})^{−η(n)})

= ℰ ( ∏_{1}^{\sqrt{n} −1}(1 − ψ(r)X)^{−η(n)})ℰ (∏_{\sqrt{n}}^{n}(1 − ψ(r)X)^{−η(n)})

= e^{γn+o(n)}.

Let

E_{n} = {b : u(b_{1}, b_{2}, …, b_{n}) = ∏_{r=1}^{n}(1 − ψ(r)x_{b_{r}}) > n},

and

E = lim sup E_{n}.

Note from the estimate in the last paragraph, the Lebesgue measure

λ(E_{n}^{c}) ≤ M_{n}n^{η(n)} = exp(γn + o(n));

therefore E_{n}^{c} is contained in at most κ^{n} exp(γn + o(n)) many balls of diameter κ^{−n} each. Since E^{c} = lim inf E_{n}^{c}, dim E^{c} ≤ 1 + γ/ log κ = d(κ, F).

Note that u is unbounded at each branch in E; unlike Proposition 1, the set E here does not contain all branches b along which u is unbounded. However arguing as in Proposition 1, we can show that ω_{F} (E) = 0 and ω_{F} (E^{c}) = 1. This proves Theorem 2.

Theorem 3. Let F be a permutation invariant averaging operator, not equal to the usual average. Then there exist congruent sets E_{1}, E_{2}, …, E_{κ} with ∪E_{j} = ∂T such that ω_{F} (E_{j}) = 0 for 1 ≤ j ≤ κ. And there exist sets A and B on ∂T such that ω_{F} (A) = ω_{F} (B) = 0, however ω_{F} (A ∪ B) > 0.

Proof. Let τ be the permutation of 1, 2, 3, …, κ defined by τ1 = 2, τ2 = 3, …, τκ = 1. Then for each state j, the images j, τj, τ^{2}j, τ^{κ−1}j are just the κ states in a different order. The powers of τ operate on the vertices of T in the obvious way, and also on ∂T. Sets A and B in ∂T are congruent if B = τ^{q}A for some q = 0, 1, 2, …, κ − 1. Clearly congruent sets have the same F -harmonic measure. Let E be the set defined in Proposition 1. Then we claim that E ∪ τE ∪ τ^{2}E ∪ ⋯ ∪ τ^{κ−1}E = ∂T. After that the first assertion in the theorem follows. Now let q_{0} = min{q : F − ω(E ∪ τE ∪ τ^{2}E ∪ ⋯ ∪ τ^{q}E) > 0}, A = E ∪ τE ∪ ⋯ τ^{q_{0}−1}E and B = τ^{q_{0}}E, and thus the second assertion follows.

To verify the claim, let Y be the vector in Proposition 1, δ = ∏ y_{j} > 1, and u be the function in the proof of Proposition 1. Note from the definition of u that for a vertex v of length n,

u(v)u(τv)u(τ^{2}v) ⋯ u(τ^{κ−1}v) = δ^{n}.

Hence there is some q = 0, 1, …, κ−1 such that u(τ^{q}v) ≥ δ^{n/κ}. Taking an infinite branch b = (b_{1}, b_{2}, …), we apply this to each segment (b_{1}, b_{2}, …, b_{n}) obtaining a number q(b, n) in {0, 1, 2, …, κ − 1} so that

u(τ^{q(b,n)}(b_{1}, b_{2}, …, b_{n})) ≥ δ^{n/κ}.

For each b, one of the numbers q in {0, 1, 2, …, κ − 1} occurs infinitely often in {q(b, n) : n ≥ 1}. For that number q, u is unbounded on τ^{q}b and therefore b ∈ τ^{κ−q}E. This proves the claim and thus the theorem.

Our next theorem says that there exists E ⊆ ∂T of ω_{p}(E) = 0 and ω_{p}(E^{c}) = 1 for all numbers p in an interval. When κ = 2, p-harmonic functions are 2harmonic; hence we assume κ ≥ 3.

Theorem 4. Given p_{0} > 2, there exists E ⊆ ∂T with dim E^{c} < 1 so that ω_{p}(E) = 0 and ω_{p}(E^{c}) = 1 for all p ≥ p_{0} ; given 1 < p_{0} < 2, there exists E ⊆ ∂T with dim E^{c} < 1 so that ω_{p}(E) = 0 and ω_{p}(E^{c}) = 1 for all 1 < p ≤ p_{0} .

Proof. Fix p_{0} ≠ 2, 1 < p_{0} < ∞; let a = F_{p_{0}}(1, 0, …, 0), i.e.
(1 − a)^{p_{0}−1} − (κ − 1)a^{p_{0}−1} = 0.

Note that 1/κ < a < \frac{1}{2} when p_{0} > 2, and 0 < a < 1/κ when p_{0} < 2. Follow the proof of Lemma 2; let y_{1} = 1 + (1 − a)t, y_{j} = 1 − at(2 ≤ j ≤ k) and Y = (y_{1}, y_{2}, …, y_{κ}) with t fixed (t < 0 when p_{0} > 2 and t > 0 when p_{0} < 2) so that F_{p_{0}}(Y) = 1, ∏ y_{j} > 1 and y_{j} > 0 for 1 ≤ j ≤ κ.

Simple calculations show that Y − 1 is p-superharmonic for p > p_{0} when p_{0} > 2, and p-superharmonic for 1 < p < p_{0} when p < 2.

Assume for now that p_{0} > 2. Follow the construction of E and u in Proposition 1 with F = F_{p_{0}} and the vector Y chosen above. Then u is p_{0} -harmonic and is p-superharmonic for all p > p_{0} . Since F -harmonic measure is defined to be the infimum of an upper class of F -superharmonic functions, the proof in Proposition 1 shows ω_{p}(E) = 0 and ω_{p}(E^{c}) = 1 for all p ≥ p_{0} .

The case p_{0} < 2 is similar.

4. Choquet capacity

Theorem 5. Suppose F is a permutation invariant averaging operator on R^{κ}, not equal to the usual average. Then there exists an increasing sequence of sets B_{1} ⊆ B_{2} ⊆ ⋯ ⊆ B_{j} ⊆ ⋯ ⊆ ∂T so that lim ω_{F} (B_{j}) < ω_{F} (∪B_{j}). In other words, ω_{F} is not a Choquet capacity.

A Choquet capacity 𝒞 on a metric space Ω is a set function defined on all subsets of Ω into [0, ∞] with the following properties:

(i) 𝒞 is monotone: 𝒞 (A) ≤ 𝒞 (B) when A ⊆ B;

(ii) 𝒞 is right continuous on compact sets: if A_{1} ⊇ A_{2} ⊇ ⋯ ⊇ A_{j} ⊇ ⋯ are compact sets then lim 𝒞 (A_{j}) = 𝒞 (∩A_{j});

(iii) 𝒞 is left continuous on arbitrary sets: if B_{1} ⊆ B_{2} ⊆ ⋯ ⊆ B_{j} ⊆ ⋯ are arbitrary sets then lim 𝒞 (B_{j}) = 𝒞 (∪B_{j}).

The Capacitability Theorem of Choquet ([C], [D]) asserts that when Ω is a complete separable metric space then all Borel subsets E of Ω are capacitable:

𝒞 (E) = sup{𝒞 (A) : A compact, A ⊆ E}.

Proof of Theorem 5. Assume 𝒞 is a Choquet capacity on a complete separable metric space Ω. Later, we let Ω = ∂T and 𝒞 = ω_{F} .

Let M be another complete separable metric space and ψ a continuous function of M into Ω. Then the set function 𝒞^{∗}(S) =^{def}𝒞 (ψ(S)) is easily seen to be a Choquet capacity on M.

Let A, B be Borel sets in Ω, then A and B\A are continuous images ψ_{1}(𝒩 ), ψ_{2}(𝒩 ) of the space 𝒩 = N^{N}, the set of sequences of positive integers; ([K, p. 446]). The discrete union 𝒩_{1} ∪𝒩_{2} of two copies of 𝒩 is homeomorphic to 𝒩 and we can map 𝒩_{1} ∪ 𝒩_{2} into Ω by a continuous ψ (induced by ψ_{1} , and ψ_{2}) so that ψ(𝒩_{1}) = A and ψ(𝒩_{2}) = B\A.

Let M = 𝒩_{1} ∪ 𝒩_{2} ; when L is compact in M then L ∩ 𝒩_{1} and L ∩ 𝒩_{2} are both compact. Applying the capacitability theorem to 𝒞^{∗} on M, we see that

𝒞 (A ∪ B) = 𝒞 (ψ(M)) = 𝒞^{∗}(M)
= sup{𝒞^{∗}(L_{1} ∪ L_{2}) : L_{1}, L_{2} compact and L_{1} ⊆ 𝒩_{1}, L_{2} ⊆ 𝒩_{2}}

≤ sup{𝒞 (G_{1} ∪ G_{2}) : G_{1}, G_{2} compact and G_{1} ⊆ A and G_{2} ⊆ B\A}.

Suppose that ω_{F} is a Choquet capacity, and let Ω = ∂T, 𝒞 = ω_{F} , A and B be the sets in Theorem 3 chosen with ω_{F} (A) = ω_{F} (B) = 0 but ω_{F} (A ∪ B) > 0. Note that ω_{F} (G_{1}) = ω_{F} (G_{2}) = 0 for compact G_{1} ⊆ A and G_{2} ⊆ B\A, thus ω_{F} (G_{1} ∪ G_{2}) = 0 by property (v) of F -harmonic measures stated in Section 1. The inequalities in the last paragraph would show that ω_{F} (A ∪ B) = 0. The contradiction says that ω_{F} can not be a Choquet capacity. Since the first two properties of Choquet capacity hold for ω_{F} , property (iii) must fail for ω_{F} . This completes the proof of Theorem 5.

5. Ellipticity and the strong maximum principle

In this section, we study the analogue of the quasilinear elliptic equation divA(∇u) = 0 on a tree T.

Let 1 < p < ∞, q = p/(p − 1), and 0 < α ≤ β. Let A: R^{κ} → R^{κ} be a continuous function satisfying the following structural conditions:

(i) ⟨AX, X⟩ ≥ α∥X∥_{p}^{p},
(ii) ∥AX∥_{q} ≤ β∥X∥_{p}^{p−1} ,
(iii) ⟨AX − AY, X − Y ⟩ > 0 for all X ≠ Y ,
(iv) A(λX) = λ|λ|^{p−2}AX for all λ ∈ R;

A is an example of monotone operator.

Example. Let 1 < p < ∞ and AX = (x_{1}|x_{1}|^{p−2}, x_{2}|x_{2}|^{p−2}, …, x_{κ}|x_{κ}|^{p−2}). Then A satisfies (i) ∼ (iv) with α = β = 1; and ⟨AX, 1⟩ is the p-Laplacian of X.

A vector X ∈ R^{κ} is called an A-harmonic vector if

⟨AX, 1⟩ = 0.

Each monotone operator A defines an operator F_{A} from R^{κ} to R^{1} with F_{A}(X) = t provided that

⟨A(X − t1), 1⟩ = 0,

or X − t1 is A-harmonic.

From (iii), we see that this equation has at most one solution. To prove that there is a solution, we show that lim⟨A(X − t1), 1⟩ = ∓∞ as t → ±∞; in fact A(X + t1) = t|t|^{p−2}A(1 + t^{−1}X) and A is continuous while ⟨A1, 1⟩ > 0.

The solution of ⟨A(X − t1), 1⟩ = 0 is a quasiminimizer for the p-Dirichlet sum ∑_{j} |x_{j} − x|^{p} over all real x. To see this,

\begin{aligned}∥X − t1∥_{p}^{p} & ≤ α^{−1}⟨A(X − t1), X − t1⟩ = α^{−1}⟨A(X − t1), X − x1⟩ \\ & ≤ \frac{β}{α}∥X − t1∥_{p}^{p−1}∥X − x1∥_{p} \end{aligned}

and thus

∥X − t1∥_{p}^{p} ≤ (\frac{β}{α})^{p}∥X − x1∥_{p}^{p}.

It is easy to see that F_{A} has properties (i) ∼ (iii) of the averaging operators. F_{A} satisfies (iv), the strong maximum principle, if and only if every nonzero

A-harmonic vector changes sign. For each p ∈ (1, ∞) and κ > 1, there are A’s for which the strong maximum principle fails, see the second remark below.

When the ratio α/β of the ellipticity constants is close to 1, the strong maximum principle holds. Let

γ(κ, p) = ((κ − 1)^{q−1}[1 + (κ − 1)^{q−1}]^{−1})^{1/q}.

Theorem 6. Under the assumption α/β > γ(κ, p), any nonzero A-harmonic vector changes sign, and the strong maximum principle holds for F_{A}.

From Theorem A and the remark immediately after, we obtain the following.

Corollary. Suppose that α/β > γ(κ, p). Then

min_{ℋ_{A}}dim ℱ (u) = min_{ℋ_{A}}dim BV(u) = d(κ, A),

where ℋ_{A} is the set of bounded A-harmonic functions on T.

Remark. When p = 2, γ(κ, 2) = (1 − 1/κ)^{1/2} , and this is sharp for Theorem 6. To see this, let e = (1, 0, …, 0), then e − (1/κ)1 is its projection on the hyperplane ⟨Y, 1⟩ = 0 and u = (1 − 1/κ)^{−1/2}(e − (1/κ)1) has length 1. Let A be an orthogonal matrix A such that Ae^{T} = u^{T} and ⟨AX, X⟩ ≥ ⟨e, u⟩∥X∥_{2}^{2} for all column vectors X ∈ R^{κ}. Then ⟨e, u⟩ = (1−1/κ)^{1/2} and the structural conditions are satisfied with p = q = 2, α = (1 − 1/κ)^{1/2} and β = 1. Since e is A-harmonic and does not change sign, γ(κ, 2) is sharp for Theorem 6.

Remark. When p ≠ 2, we do not have examples to show the sharpness of γ(κ, p). However for each p ≠ 2, 1 < p < ∞, there exists A satisfying the structural conditions, for which e = (1, 0, …, 0) is A-harmonic. To see this, let

ε_{0}(κ, p) = (κ − 1)^{−1/p}[1 + (κ − 1)^{q−1}]^{−1/p}

and let X = ((1−(κ−1)ε_{o}^{p})^{1/p}, −ε_{0}, −ε_{0}, …, −ε_{0}) (ε_{0} = ε_{0}(p, κ)), then ∥X∥_{p} = 1 and X is p-harmonic. Let P be the p-harmonic operator such that P Y = (y_{1}|y_{1}|^{p−2}, y_{2}|y_{2}|^{p−2}, …, y_{κ}|y_{κ}|^{p−2}) for all Y ∈ R^{κ}, and B be an invertible matrix with Be^{T} = X^{T} and B1^{T} = 1^{T} . Let A = B^{T} P B, then ⟨Ae^{T} , 1^{T} ⟩ = ⟨P Be^{T} , B1^{T} ⟩ = ⟨P X^{T} , 1^{T} ⟩ = 0, so e is A-harmonic. Clearly A has properties (i) ∼ (iv).

The following result controls the oscillation of A-harmonic vectors, from which Theorem 6 follows by taking C = 0.

Proposition 2. Let X ≠ 0 be A-harmonic and C ≥ 0. Suppose that α/β ≥ λγ(κ, p) for some λ ∈ [1, γ(κ, p)^{−1}]. Then max x_{j} ≤ C (or min x_{j} ≥ −C) implies that

∥X∥_{p} ≤ C/ψ^{−1}(λ),

where

ψ(t) = t + [1 − (κ − 1)t^{p}]^{1/p}.

The function ψ is increasing in [0, ε_{0}(κ, p)], ψ(0) = 1 and ψ(ε_{0}(κ, p)) = γ(κ, p)^{−1} .

If X is p-harmonic, then from the a priori bound x_{j} ≤ C it is easy to deduce that |x_{j}| ≤ C(κ − 1)^{1/(p−1)} . Proposition 2 can be considered as a generalization of this fact in the A-harmonic setting. The proof of the proposition is based on the following.

Lemma 4. For 0 ≤ ε ≤ 1, let

ϕ(ε, κ, p) = max{⟨X, Y ⟩ : ∥X∥_{p} = ∥Y ∥_{q} = 1, ⟨Y, 1⟩ = 0, x_{j} ≥ −ε for all 1 ≤ j ≤ κ}.

Then

ϕ(ε, κ, p) = \{ \begin{matrix}γ(κ, p)(ε + [1 − (κ − 1)ε^{p}]^{1/p}), & if 0 ≤ ε ≤ ε_{0}, \\ 1, & if ε_{0} ≤ ε ≤ 1. \end{matrix}

Remark. For 0 ≤ ε ≤ ε_{0}(p, κ), the maximum ϕ(ε, κ, p) is attained when X = ((1 − (κ − 1)ε^{p})^{1/p}, −ε, …, −ε) and Y = (γ, −γ/(κ − 1), …, −γ/(κ − 1)), (γ = γ(κ, p)); and ϕ(ε, p, κ) ≥ γ(p, κ). When ε = 0, X = (1, 0, …, 0) does not change sign and ⟨X, Y ⟩ = γ(p, κ). Observe that ⟨X, Y ⟩ = 1 when ε = ε_{0}(κ, p); Hölder’s inequality implies that ϕ(ε, κ, p) = 1 when ε_{0}(κ, p) ≤ ε ≤ 1.

Proof of Lemma 4. Assume as we may that 0 ≤ ε ≤ ε_{0}(κ, p). Before going to the proof, we need some preliminary computations. Suppose that X, Y ∈ R^{n}, are normalized in the sense that ∥X∥_{p} = ∥Y ∥_{q} = 1 and that ⟨Y, 1⟩ = 0. Assume that y_{1}, …, y_{n} ≥ 0 and y_{n+1}, …, y_{κ} < 0, for some n, 1 ≤ n ≤ κ − 1. Then

\begin{aligned}∑_{j=1}^{n} y_{j}^{q} & ≤ (∑_{j=1}^{n} y_{j})^{q} = ( ∑_{j=n+1}^{κ} |y_{j}|)^{q} \\ & ≤ (κ − 1)^{q−1} ∑_{j=n+1}^{κ} |y_{j}|^{q} = (κ − 1)^{q−1}(1 − ∑_{j=1}^{n} y_{j}^{q}), \end{aligned}

so

∑_{j=1}^{n} y_{j}^{q} ≤ (κ − 1)^{q−1}[1 + (κ − 1)^{q−1}]^{−1},

and, analogously,

∑_{j=n+1}^{κ} |y_{j}|^{q} ≤ (κ − 1)^{q−1}[1 + (κ − 1)^{q−1}]^{−1},

so

∑_{j=1}^{n} y_{j}^{q} ≥ [1 + (κ − 1)^{q−1}]^{−1}.

Now, assume that x_{j} ≥ −ε for all j, and let J = {j : n+1 ≤ j ≤ κ, −ε ≤ x_{j} ≤ 0}. Then

\begin{aligned}⟨X, Y ⟩ & ≤ ∑_{j=1}^{n} x_{j}y_{j} + ∑_{j∈J} |x_{j}| |y_{j}| \\ & ≤ (∑_{j=1}^{n} |x_{j}|^{p})^{1/p}(∑_{j=1}^{n} y_{j}^{q})^{1/q} + (∑_{j∈J} |x_{j}|^{p})^{1/p}(1 − ∑_{j=1}^{n} y_{j}^{q})^{1/q}. \end{aligned}

Now set

a = ∑_{j=1}^{n} |x_{j}|^{p}, b = ∑_{J} |x_{j}|^{p}, c = ∑_{j=1}^{n} y_{j}^{q}.

Observe that, if b_{1} = (κ − 1)ε^{p}, c_{1} = γ(κ, p)^{q}, then a, b, c ≥ 0, a + b ≤ 1, 0 ≤ b ≤ b_{1} ≤ 1 − c_{1} ≤ c ≤ c_{1} < 1, and γ(κ, p)ε + γ(κ, p)[1 − (κ − 1)ε^{p}]^{1/p} = (1 − b_{1})^{1/p}c_{1}^{1/q} + b_{1}^{1/p}(1 − c_{1})^{1/q}. The above considerations show that Lemma 4 is a consequence of the following

Lemma 5. Let f(a, b, c) = a^{1/p}c^{1/q} + b^{1/p}(1 − c)^{1/q} and suppose that 0 ≤ b_{1} ≤ 1 − c_{1} ≤ c_{1} < 1. Then

\begin{matrix}max & \{f(a, b, c) : a, b, c ≥ 0, a + b ≤ 1, 0 ≤ b ≤ b_{1}, 1 − c_{1} ≤ c ≤ c_{1}\} \\ = (1 − b_{1})^{1/p}c_{1}^{1/q} + b_{1}^{1/p}(1 − c_{1})^{1/q}. \end{matrix}

Proof. Note that f is increasing in a and in b, therefore at the maximum, a + b = 1. Let

g(b, c) = (1 − b)^{1/p}c^{1/q} + b^{1/p}(1 − c)^{1/q},

and claim that

max{g(b, c) : 0 ≤ b ≤ b_{1}, 1 − c_{1} ≤ c ≤ c_{1}} = (1 − b_{1})^{1/p}c_{1}^{1/q} + b_{1}^{1/p}(1 − c_{1})^{1/q}.

Fix c ∈ [1 − c_{1}, c_{1}]; then g(b, c) is an increasing function of b over the interval [0, 1 − c]; since b_{1} ≤ 1 − c_{1} ≤ c,

max{g(b, c) : 0 ≤ b ≤ c_{1}} = g(b_{1}, c).

Again g(b_{1}, c) is an increasing function of c over the interval [0, 1 − b_{1}] and c_{1} ≤ 1 − b_{1} , we conclude that

max{g(b_{1}, c) : 1 − c_{1} ≤ c ≤ c_{1}} = g(b_{1}, c_{1}).

This verifies the claim and the lemma.

Proof of Proposition 2. Assume that α/β = λγ(κ, p) with 1 ≤ λ ≤ γ(κ, p)^{−1} . Let X be a nonzero A-harmonic vector, X′ = X/∥X∥_{p} and Y = AX/∥AX∥_{q}, then X′ is also A-harmonic and

∥X′∥_{p} = ∥Y ∥_{q} = 1, ⟨X′, Y ⟩ ≥ α/β.

Suppose now that max x_{j} ≤ C and let ε = C/∥X∥_{p}. We can assume that ε ≤ ε_{0}(κ, p). Then Lemma 4, applied to X′ and Y , gives

λγ(κ, p) ≤ ⟨X′, Y ⟩ ≤ γ(κ, p)ε + γ(κ, p)[1 − (κ − 1)ε^{p}]^{1/p},

so ε = C/∥X∥_{p} ≥ ψ^{−1}(λ) and Proposition 2 follows.

6. Ellipticity and the critical dimension

Fix p, q and let A be a monotone operator on R^{κ} and α, β the constants in (i) and (ii). Assume from now on that α/β > γ(κ, p); then F_{A} has properties (i) ∼ (iv) of the averaging operators, and m(κ, A) =^{def}m(κ, F_{A}) = min{Σe^{x_{j}} : X is A-harmonic} and d(κ, A) = log m(κ, A)/ log κ are attained.

We shall compare the critical dimension d(κ, A) with the critical dimension d(κ, p) when the ratio α/β of the ellipticity constants is close to 1. The estimates obtained are not sharp but are enough to show that if α/β tends to 1, then d(κ, A) tends to d(κ, p) uniformly in A.

Theorem 7. Fix p, q and A as above, and assume that α/β = λγ(p, κ) for some λ ∈ (1, γ^{−1}(p, κ)]. Then

exp{−M(p, 1 − α/β) log κ/ψ^{−1}(λ)} ≤ m(κ, A)/m(κ, p)
≤ exp{M(p, 1 − α/β) log κ/ε_{0}(κ, p)}

where

M(p, δ) = \{ \begin{matrix}δ^{1/p}(p^{1/p} + 1), & if p ≥ 2 and 0 ≤ δ ≤ \frac{q}{2p}, \\ δ^{1/2}((\frac{2q}{p} )^{1/2} + 1), & if 1 < p < 2 and 0 ≤ δ ≤ \frac{p}{2q}, \\ 2, & if δ > min\{\frac{p}{2q} , \frac{q}{2p}\}. \end{matrix}

Corollary. Under the assumptions above, d(κ, A) → d(κ, p) uniformly in A as α/β ↑ 1.

To prove Theorem 7, we associate to any A-harmonic vector X a p-harmonic vector Z (and conversely) in such a way that each z_{j} is close to x_{j} as soon as the ratio α/β is close to 1; we look at the “near equality” case in the arithmeticgeometric inequality and compare Σe^{x_{j}} with Σe^{z_{j}}.

Suppose that 0 ≤ x, z ≤ 1. Then by the arithmetic-geometric inequality,

\frac{x^{p}}{p} + \frac{z^{p}}{q} − xz^{p−1} ≥ 0

with equality if and only if x = z. The following two lemmas give estimates on |x − z| and x + z in the “near equality” situation.

Lemma 6. Let 0 ≤ δ ≤ 1 and

M^{+}(p, δ) = max\{|x − z| : 0 ≤ x, z ≤ 1, \frac{x^{p}}{p} + \frac{z^{p}}{q} − xz^{p−1} = δ\}.

Then

M^{+}(p, δ) ≤ \{ \begin{matrix}(δp)^{1/p}, & when p ≥ 2, \\ (\frac{2δq}{p} )^{1/2}, & when 1 < p < 2. \end{matrix}

Lemma 7. Let 0 ≤ δ ≤ 1 and

M^{−}(p, δ) = max\{x + z : 0 ≤ x, z ≤ 1, \frac{x^{p}}{p} + \frac{z^{p}}{q} + xz^{p−1} = δ\}.

Then

M^{−}(p, δ) ≤ δ^{1/p}(max{p^{1/p}, q^{1/p}} + 1).

Proof of Lemma 6. Case 1: p ≥ 2. Suppose that z ≤ x = z + h. By differentiation on h, it follows that

\frac{(z + h)^{p}}{p} + \frac{z^{p}}{q} − (z + h)z^{p−1} ≥ \frac{h^{p}}{p} .

If x ≤ z = x + h, then also by differentiation we get

\frac{x^{p}}{p} + \frac{(x + h)^{p}}{q} − x(x + h)^{p−1} ≥ \frac{h^{p}}{q} ≥ \frac{h^{p}}{p} ,

so

\frac{x^{p}}{p} + \frac{z^{p}}{q} − xz^{p−1} ≥ \frac{|x − z|^{p}}{p} .

Case 2: 1 < p < 2. Also by differentiation we get now that

\frac{x^{p}}{p} + \frac{z^{p}}{q} − xz^{p−1} ≥ \frac{p − 1}{2}(x − z)^{2}

which proves the lemma.

Proof of Lemma 7. If x ≥ z, then z^{p} ≤ xz^{p−1} ≤ δ and x ≤ (δp)^{1/p} so, x + z ≤ (δ)^{1/p}(1+p^{1/p}). Analogously, if z ≥ x, we get x+z ≤ δ^{1/p}(1+q^{1/p}) and therefore

M^{−}(p, δ) ≤ δ^{1/p}(max{p^{1/p}, q^{1/p}} + 1).

Lemma 8. Let X and Y be vectors in R^{κ} which satisfy ∥X∥_{p} = ∥Y ∥_{q} = 1, ⟨Y, 1⟩ = 0 and ⟨X, Y ⟩ ≥ 1 − δ for some δ ∈ [0, 1]. Let Z ∈ R^{κ} be chosen so that y_{j} = z_{j}|z_{j}|^{p−2} for 1 ≤ j ≤ κ. Then max |x_{j} − z_{j}| ≤ M(p, δ), where M is the function in the statement of Theorem 7.

Proof. Observe that Z is a p-harmonic vector and that

∥Z∥_{p} = 1, 1 − δ ≤ Σx_{j}z_{j}|z_{j}|^{p−2} ≤ 1.

Suppose for now that 0 ≤ δ ≤ min{p/2q, q/2p} and let J^{+} = {j : x_{j}z_{j} > 0} and J^{−} = {1, 2, …, κ}\J^{+} . Suppose i ∈ J^{+} , then

\begin{aligned}1 − δ & ≤ ⟨X, Y ⟩ ≤ ∑_{J^{+}} |x_{j}| |z_{j}|^{p−1} ≤ |x_{i}| |z_{i}|^{p−1} + ∑_{J^{+}\\{i\}}(\frac{|x_{j}|^{p}}{p} + \frac{|z_{j}|^{p}}{q} ) \\ & ≤ |x_{i}| |z_{i}|^{p−1} + \frac{1 − |x_{i}|^{p}}{p} + \frac{1 − |z_{i}|^{p}}{q} , \end{aligned}

so

\frac{|x_{i}|^{p}}{p} + \frac{|z_{i}|^{p}}{q} − |x_{i}| |z_{i}|^{p−1} ≤ δ;

and |x_{i} − z_{i}| ≤ M^{+}(p, δ). On the other hand, if i ∈ J^{−}, then the same argument shows that

\frac{|x_{i}|^{p}}{p} + \frac{|z_{i}|^{p}}{q} + |x_{i}| |z_{i}|^{p−1} ≤ δ

and that |x_{i} − z_{i}| = |x_{i}| + |z_{i}| ≤ M^{−}(p, δ). The lemma follows from the estimates of M^{+} and M^{−} in Lemmas 6 and 7.

Proof of Theorem 7. Let X be a nonzero A-harmonic vector and set X′ = X/∥X∥_{p} and Y′ = AX/∥AX∥_{q}. Choose Z′ ∈ R^{κ} so that y_{j} = z′_{j}|z′_{j}|^{p−2} for all j, and define Z = ∥X∥_{p}Z′. We apply Lemma 8 to X′, Z′, δ = 1 − α/β to get

max |x′_{j} − z′_{j}| ≤ M(p, 1 − α/β),

so

max |x_{j} − z_{j}| ≤ M(p, 1 − α/β)∥X∥_{p};

consequently

exp{−M(p, 1 − α/β)∥X∥_{p}} ≤ Σe^{x_{j}}/Σe^{z_{j}} ≤ exp{M(p, 1 − α/β)∥X∥_{p}}.

Suppose now that α/β = λγ(κ, p) and that X is an extremal vector for m(κ, A). Since 0 is also A-harmonic, we must have Σe^{x_{j}} ≤ κ and max x_{j} ≤ log κ. Apply Proposition 2 to the extremal vector X, we obtain ∥X∥_{p} ≤ log κ/ψ^{−1}(λ). Therefore

exp{−M(p, 1 − α/β) log κ/ψ^{−1}(λ)} ≤ m(κ, A)/m(κ, p).

To reverse the argument, we need some topological properties of the mapping A. First observe that by the monotonicity (iii), A is one to one; and by the homogeneity (iv), the mapping

BX = AX/∥AX∥_{q}

is also one to one from S_{p} = {X : ∥X∥_{p} = 1} to S_{q} = {Y : ∥Y ∥_{q} = 1}. As a consequence of Borsuk’s Antipodal Theorem [Du, Chapter 16], this implies that B is also surjective, and again from homogeneity, it follows that A is surjective too.

Now choose a p-harmonic vector Z, extremal for m(κ, p) and define Y by letting y_{j} = z_{j}|z_{j}|^{p−2}/∥Z∥_{p}^{p−1} for 1 ≤ j ≤ κ. Then ∥Y ∥_{q} = 1 and ⟨Y, 1⟩ = 0. Choose X′ with ∥X′∥_{p} = 1 such that BX′ = Y and set X = ∥Z∥_{p}X′. Then Y = AX/∥AX∥_{q}. Apply Proposition 2 to the p-harmonic case (where α = β = 1), we get ∥X∥_{p} = ∥Z∥_{p} ≤ log κ/ε_{0}(κ, p); therefore

m(κ, A)/m(κ, p) ≤ exp{M(p, 1 − α/β) log κ/ε_{0}(κ, p)}.

This completes the proof of Theorem 7.

7. 1-Laplacian and ∞-Laplacian

Suppose we consider 1-Laplacian Δ_{1} as the limit of Δ_{p} when p → 1^{+} , and define F_{1}(X) = t when

lim_{p→1^{+}} ∑_{j}(x_{j} − t)|x_{j} − t|^{p−2} = 0;

we note immediately that this definition leads to confusion even in the simple situations, e.g. F_{1}(1, 2, 3) and F_{1}(1, 1, 3).

Therefore we define F_{1}(X) through minimizing 1-Dirichlet sums. Given a vector X ∈ R^{κ}, the possible values of the average F_{1}(X) are the numbers x = t that minimize the sum ∑_{j=1}^{κ} |x_{j} − x| over all x ∈ R^{1} .

Suppose that t is the unique minimizer, and let n^{+}, n^{0}, n^{−} be the number of x_{j} ’s that are greater than, equal to, or less than t, respectively. Then, necessarily, n^{−} + n^{0} > n^{+} and n^{+} + n^{0} > n^{−}, or simply n^{0} > |n^{+} − n^{−}|; this is also a sufficient condition that t be the sole minimizer. In this case, define F_{1}(X) = t.

If there is more than one minimizer, then the minimizers occupy an interval [a, b], whose endpoints are among the x_{j} ’s. This occurs if and only if κ is even and there are exactly \frac{1}{2}κ terms x_{j} ≥ b, \frac{1}{2}κ terms x_{j} ≤ a. We then define F_{1}(X) = \frac{1}{2}(a + b).

The operator F_{1} has properties (i) ∼ (iii) for the averaging operators. Since F_{1}(x, 0, 0, …, 0) = 0 for any (x, 0, …, 0) ∈ R^{κ} when κ ≥ 3, F_{1} lacks the strong maximum principle.

We say X is a 1-harmonic vector if F_{1}(X) = 0.

Proposition 3. Let m(κ, 1) = inf{Σe^{x_{i}} : X is 1-harmonic} and d(κ, 1) = log m(κ, 1)/ log κ. Then m(κ, 1) = [\frac{1}{2}κ] + 1 and d(κ, 1) = log([\frac{1}{2}κ] + 1)/ log κ.

In finding the infimum, we make the negative coordinates tend to −∞; the infimum is not attained when κ ≥ 3.

Proof. Let X be 1-harmonic. In the case that 0 is the sole minimizer, we saw that n^{+} + n^{0} > n^{−}. If κ is even, then n^{+} + n^{0} ≥ 1 + \frac{1}{2}κ and Σe^{x_{i}} ≥ 1 + \frac{1}{2}κ. Choosing X with n^{0} = 1+ \frac{1}{2}κ, n^{−} = −1+ \frac{1}{2}κ and n^{+} = 0, we see that the infimum in this special case is 1 + \frac{1}{2}κ. Next if κ is odd, we have n^{+} + n^{0} ≥ \frac{1}{2}(κ + 1) and Σe^{x_{j}} ≥ \frac{1}{2}(κ + 1); choosing vectors X with n^{0} = \frac{1}{2}(κ + 1), n^{−} = \frac{1}{2}(κ − 1) and n^{+} = 0, we see that the infimum is \frac{1}{2}(κ + 1) in this special case.

In the second case, there is a pair of x_{j} ’s equal to a > 0 and −a < 0, and there are in addition exactly \frac{1}{2}(κ − 2) positive x_{i} ’s. Since e^{a} + e^{−a} ≥ 2, Σe^{x_{j}} ≥ 2 + \frac{1}{2}(κ − 2) = 1 + \frac{1}{2}κ. And the infimum of Σe^{x_{j}} in the second case is 1 + \frac{1}{2}κ.

Remark. Alternative definitions of F_{1}(X) in the case there is more than one minimizer will give different values of m(κ, 1) and d(κ, 1).

As for the ∞-Laplacian, first we regard Δ_{∞} as the limit of Δ_{p} when p → ∞, and define F_{∞}(X) = t provided that

lim_{p→∞} ∑_{j}(x_{j} − t)|x_{j} − t|^{p−2} = 0.

This definition leads to problems when the number of coordinates equal to max x_{j} is not the same as the number of coordinates equal to min x_{j}.

Therefore we define F_{∞}(X) = t if t is the minimizer for max_{i} |x_{i} − x| over all x. It is easy to see that

F_{∞}(X) = \frac{1}{2}(max x_{j} + min x_{j}).

Clearly F_{∞} has all properties (i) ∼ (iv) for averaging operators. However F_{∞} is not strictly increasing with respect to each variable, because F (1, −1, 0, …, 0) = F (1, −1, −1, …, −1) = 0 when κ ≥ 3.

We say X is ∞-harmonic if F_{∞}(X) = 0. Simple calculations give the following.

Proposition 4. Let m(κ, ∞) = min{Σe^{x_{i}} : X is ∞-harmonic} and
d(κ, ∞) = log m(κ, ∞)/ log κ.

Then the minimum m(κ, ∞) is attained at

(\frac{1}{2} log(κ − 1), −\frac{1}{2} log(κ − 1), …, −\frac{1}{2} log(κ − 1))

with value 2\sqrt{κ − 1} ; and d(κ, ∞) = log(2\sqrt{κ − 1} )/ log κ.

302 R. Kaufman, J.G. Llorente, and J.-M. Wu

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Received 5 August 2002