Annales Academiæ Scientiarum Fennicæ Series A. I. Mathematica Volumen 19, 1994, 393–398

ON SOME PROPERTIES OF HAUSDORFF CONTENT RELATED TO INSTABILITY

Pertti Mattila and Joan Orobitg

University of Jyväskylä, Department of Mathematics P.O. Box 35, SF-40351 Jyväskylä, Finland; pmattila@jylk.jyu.fi Universitat Politècnica de Catalunya, Departament de Matemàtica Aplicada II Colom, 11 , E-08222 Terrassa, Spain; orobitg@ma2.upc.es

Abstract. We shall prove that certain local and global density properties of the αdimensional Hausdorff content in R^{n} are equivalent. Such properties are related to problems in harmonic approximation.

0. Introduction

Let X be a compact subset of R^{n} and α a positive number such that n−2 < α < n. In [M-O] it was established that the following conditions are equivalent:

(1) M_{∗}^{α}(B \ X̊) ≤ C M^{α}(B \ X) for all open balls B and some positive constant C.

(2) lim sup_{r→0} \frac{M^{α}(B(x, r) \ X)}{r^{α}} > 0 for M_{∗}^{α} -a.a. x ∈ ∂X.

Here we denote by X̊ the interior of X, by ∂X the boundary of X, and M^{α} and M_{∗}^{α} are the α-dimensional Hausdorff content and lower Hausdorff content. See later for precise definitions.

It was shown in [M-O] that the above conditions characterize compacts sets X having a Lipschitz harmonic approximation property. Moreover, the proof in [M-O] that (2) implies (1) was based on solving a particular approximation problem. Here we give an easier proof which works for all indexes 0 < α < n and it uses only geometric measure theory. Our method also provides the same result with M_{∗}^{α} replaced by M^{α}. The proof uses, with α = β, the following instability property of the Hausdorff content proved by Fernström [Fe].

If 0 < α ≤ β and A ⊂ R^{n}, then for M^{β} -a.a. x ∈ R^{n} either

lim sup_{r→0} \frac{M^{α}(A ∩ B(x, r))}{r^{α}} ≥ \frac{1}{6^{α}}

1991 Mathematics Subject Classification: Primary 28A12, 28A75.

The second author was supported in part by the grant PB89-0311, DGICYT.

or

lim_{r→0} \frac{M^{α}(A ∩ B(x, r))}{r^{β}} = 0.

The main technique in Fernström’s proof was to define a capacity, equivalent to the Hausdorff content, by means of the fractional maximal function. Here we shall give a shorter proof of the above instability property using only Vitali’s covering lemma and Frostman’s lemma. Actually, we get a slight generalization of this result.

A measure function is a non-decreasing function h(r), r ≥ 0, such that lim_{r→0} h(r) = 0. The Hausdorff content M^{h} related to a measure function h is defined for A ⊂ R^{n} by

M^{h}(A) = inf ∑_{i} h(ϱ_{i}),

where the infimum is taken over all countable coverings of A by open balls B(x_{i}, ϱ_{i}). When h(r) = r^{α}, α > 0, M^{h}(A) = M^{α}(A) is called the α-dimensional Hausdorff content of A. The lower α-dimensional Hausdorff content of A is defined by

M_{∗}^{α}(A) = sup M^{h}(A),

the supremum being taken over all measure functions which satisfy h(r) ≤ r^{α} and lim_{r→0} h(r)r^{−α} = 0.

1. Instability

We shall now prove the following.

Theorem 1. Let A ⊂ R^{n}, 0 < α < n and h(r) = ε(r)r^{α} be a measure function satisfying (i) 0 ≤ ε ≤ 1, (ii) ε is non-decreasing and (iii) h(3r) ≤ C_{1}h(r), where C_{1} is a constant depending on h. Then for M^{h} -a.a. x ∈ R^{n} one of the following conditions holds:

lim sup_{r→0} \frac{M^{α}(A ∩ B(x, r))}{r^{α}} ≥ \frac{C_{2}}{C_{1}} ,
lim_{r→0} \frac{M^{α}(A ∩ B(x, r))}{h(r)} = 0,

where the constant C_{2} depends only on n.

Remark. If h(r) = r^{β}, α ≤ β, we can replace C_{2}/C_{1} by 6^{−α}.

Proof. First we recall a density property of the Hausdorff content (see [F, 2.10.19]). For all subsets F of R^{n}

lim sup_{r→0} \frac{M^{h}(F ∩ B(x, r))}{h(r)} ≥ C_{2} > 0 M^{h} − a.a. x ∈ F,

where C_{2} is a constant which depends only on n.

Let

E = \{x ∈ R^{n} : lim sup_{r→0} \frac{M^{α}(A ∩ B(x, r))}{r^{α}} < \frac{C_{2}}{C_{1}} and

lim sup_{r→0} \frac{M^{α}(A ∩ B(x, r))}{h(r)} > 0\}.

We want to prove that M^{h}(E) = 0. If M^{h}(E) > 0, there would exist y ∈ E and t > 0 such that

(1.1)  M^{α}(A ∩ B(y, t)) < \frac{C_{2}}{C_{1}} t^{α},
(1.2)  M^{h}(E ∩ B(y, t)) > C_{2}h(t).

Condition (1.1) follows from the definition of E and condition (1.2) follows from the above density property.

Let F = A ∩ B(y, t) and G = E ∩ B(y, t). By (1.1) there are balls B_{j} of radii r_{j} ≤ t such that F ⊂ ∪B_{j} and ∑_{j} r_{j}^{α} < C_{2}C_{1}^{−1}t^{α}. On the other hand,

\begin{aligned}M^{h}(G ∩ (∪3B_{j})) & ≤ ∑_{j} h(3r_{j}) ≤ C_{1} ∑ h(r_{j}) ≤ C_{1}ε(t) ∑ r_{j}^{α} \\ & < C_{2}ε(t)t^{α} = C_{2}h(t) < M^{h}(G), \end{aligned}

and consequently M^{h}(G \ ∪3B_{j}) > 0.

Therefore, for some fixed δ > 0 there is a compact set K ⊂ G \ ∪3B_{j} with M^{h}(K) > 0 and satisfying

lim sup_{r→0} \frac{M^{α}(F ∩ B(x, r))}{h(r)} > δ for x ∈ K.

Applying Frostman’s lemma (e.g. [C]), one gets a positive Borel measure μ satisfying (i) spt μ ⊂ K, (ii) μB(x, r) ≤ h(r) and (iii) μ(K) ≥ const M^{h}(K). Take η > 0 such that

\frac{C_{1}}{δ} ∑_{r_{j}<4η} r_{j}^{α} < μ(K).

By Vitali’s covering lemma (e.g. [Z, 1.3.6]), there are disjoint balls B(x_{i}, 3ϱ_{i}) with x_{i} ∈ K and ϱ_{i} < η such that

(a) μ(K \ ∪B(x_{i}, 3ϱ_{i})) = 0.

(b) M^{α}(F ∩ B(x_{i}, ϱ_{i})) > δh(ϱ_{i}). Fixed i, if B_{j} ∩ B(x_{i}, ϱ_{i}) ≠ ∅ then B_{j} ⊂ B(x_{i}, 3ϱ_{i}), because x_{i} ∉ 3B_{j}.

Hence

F ∩ B(x_{i}, ϱ_{i}) ⊂ ⋃_{B_{j}⊂B(x_{i},3ϱ_{i})}B_{j}

and (b) gives

δh(ϱ_{i}) < M^{α}(F ∩ B(x_{i}, ϱ_{i})) ≤ ∑_{B_{j}⊂B(x_{i},3ϱ_{i})} r_{j}^{α}.

Finally, by (a) and using that the balls B(x_{i}, 3ϱ_{i}) are disjoint, one has

μ(K) = ∑_{i} μ(K ∩ B(x_{i}, 3ϱ_{i})) ≤ ∑_{i} h(3ϱ_{i}) ≤ C_{1} ∑_{i} h(ϱ_{i})

≤ \frac{C_{1}}{δ} ∑_{i} ∑_{B_{j}⊂B(x_{i},3ϱ_{i})} r_{j}^{α} ≤ \frac{C_{1}}{δ} ∑_{r_{j}<4η} r_{j}^{α}

< μ(K).

Therefore M^{h}(E) = 0.

In the case h(r) = r^{β}, α ≤ β, we can replace the constant C_{2} in the density property by a better one, 2^{−β}. Moreover, we can use the inequality ∑ r_{j}^{β} ≤

(∑ r_{j}^{α})^{β/α} to get better estimates. Thus, we would prove the same instability result that Fernström proved.

Essentially, the same arguments used in the above proof give the next result. It gives some information only when B \ A has empty interior.

Theorem 2. Let B be a ball of radius δ and A ⊂ B with M^{α}(A) < (δ/6)^{α}, where 0 < α < n. Then there is a subset C ⊂ B with positive Lebesgue measure satisfying

lim_{r→0} \frac{M^{α}(A ∩ B(x, r))}{r^{n}} = 0, for x ∈ C.

2. Equivalent conditions

In this section we show the equivalence of the conditions (1) and (2) of the introduction. More precisely, we prove the following theorems.

Theorem 3. Let 0 < α < n and let X be a compact set in R^{n}. Then the following conditions are equivalent:

(1) M^{α}(B \ X̊) ≤ C M^{α}(B \ X) for all open balls B and some positive constant C.

(2) lim sup_{r→0} \frac{M^{α}(B(x, r) \ X)}{r^{α}} > 0 for M^{α} -a.a. x ∈ ∂X.

Theorem 4. Let 0 < α < n and let X be a compact set in R^{n}. Then the following conditions are equivalent:

(1) M_{∗}^{α}(B \ X̊) ≤ C M^{α}(B \ X) for all open balls B and some positive constant C.

(2) lim sup_{r→0} \frac{M^{α}(B(x, r) \ X)}{r^{α}} > 0 for M_{∗}^{α} -a.a. x ∈ ∂X.

In Theorem 3, (1) implies (2) follows easily from the density property stated at the beginning of the proof of Theorem 1. An analogous density result for the lower Hausdorff content (see [M-O, Lemma 3.5]) also gives that (1) implies (2) in Theorem 4. Thus, one only has to show (2) ⟹ (1) in both theorems. Since the proofs are similar we only give a complete description of (2) ⟹ (1) in Theorem 3.

First note that the instability property gives that condition (2) is equivalent to

lim sup_{r→0} \frac{M^{α}(B(x, r) \ X)}{r^{α}} ≥ 6^{−α} for M^{α} − a.a. x ∈ ∂X.

Let B be an open ball. We split B \ X̊ = (B \ X) ∪ (B ∩ ∂X), and then it is enough to see that

M^{α}(B ∩ ∂X) ≤ CM^{α}(B \ X).

Without loss of generality we can assume that for each x ∈ B ∩ ∂X

lim sup_{r→0} (M^{α}(B(x, r) \ X))/r^{α} ≥ 7^{−α}.

Then (see [Z, 1.3.1]) there is a family of disjoint balls B_{j} = B(x_{j}, r_{j}) with x_{j} ∈ B ∩ ∂X such that each B_{j} is contained in B, r_{j}^{α} ≤ 6^{α}M^{α}(B_{j} \ X) and B ∩ ∂X ⊂ ∪5B_{j}. By Melnikov’s covering lemma [O, p. 72] there is a subfamily B_{jk} satisfying (a) M^{α}(∪5B_{j}) ≤ C ∑_{k} r_{jk}^{α}.

(b) ∑_{B_{jk}⊂D} r_{jk} ≤ Cr^{α} for all open balls D of radius r. Therefore, with C denoting different constants,

M^{α}(B ∩ ∂X) ≤ M^{α}(∪5B_{j})
by (a) ≤ C ∑_{k} r_{jk}^{α}
by construction ≤ C ∑_{k} M^{α}(B_{jk} \ X)
by (b) ≤ CM^{α}(∪B_{jk} \ X)
≤ CM^{α}(B \ X).

We finish this note with an example of a planar compact set X which satisfies the equivalent conditions of Theorem 4 but it does not realize the conditions of Theorem 3.

Let Q be the rectangle [0, 1] × [−1, 1]. Let {B_{j}} be the sequence of open balls with center ((2k − 1)/(2^{n+1}), 1/2^{n}), 1 ≤ k ≤ 2^{n}, n = 1, 2, …, and radius 4^{−n}, that is,

⋃_{j}B_{j} = ⋃_{n=1}^{∞}⋃_{k=1}^{2^{n}}B((\frac{2k − 1}{2^{n+1}}, \frac{1}{2^{n}} ), 4^{−n}).

Consider the compact set

X = Q \ ∪B_{j}.

Then

\begin{aligned}X̊ & = Q̊ \ (∪\overline{B_{j}} ∪ [0, 1]), \\ ∂X & = ∂Q ∪ (∪∂B_{j}) ∪ [0, 1]. \end{aligned}

Take an open ball B of radius r with center lying on [0, 1]. We have B \X̊ = B ∩ ([0, 1] ∪ (∪\overline{B_{j}})) and B \ X = B ∩ (∪B_{j}). Then, since M_{∗}^{1}([0, 1]) = 0,

M_{∗}^{1}(B \ X̊) = M_{∗}^{1}(B ∩ (∪\overline{B_{j}})) = M^{1}(B \ X).

Moreover, given any ε > 0 there is such an open ball B of radius r such that M^{1}(B \ X) < εr. On the other hand, we get M^{1}(B \ X̊) ≥ r/2 because B \ X̊ contains B ∩ [0, 1].

Thus, the compact set X satisfies condition (1) of Theorem 4 but it does not have property (1) of Theorem 3.

Acknowledgement. This paper was made while the first author was visiting the Centre de Recerca Matemàtica of the Institut d’Estudis Catalans. He wishes to express his gratitude to this Institution for its hospitality.

References

[C] Carleson, L.: Selected problems on exceptionals sets. - Van Nostrand Math. Studies 13, Van Nostrand, Princeton, N.J., 1967.

[F] Federer, H.: Geometric measure theory. - Springer-Verlag, Berlin–Heidelberg–New York, 1969.

[Fe] Fernström, C.: On the instability of Hausdorff content. - Math. Scand. 60, 1987, 19–30. [M-O] Mateu, J., and J. Orobitg: Lipschitz approximation by harmonic functions and some

applications to spectral synthesis. - Indiana Univ. Math. J. 39, 1990, 703–736.

[O] O’Farrell, A.G.: Instability pairs. - Proc. Roy. Irish Acad. Sect. A 86, 1986, 67–80.

[Z] Ziemer, W.P.: Weakly differentiable functions. - Springer-Verlag, New York, 1989.

Received 5 May 1993