Outer measures and weak type (1,1) estimates of Hardy-Littlewood maximal operators

. We will introduce the k times modi(cid:12)ed centered and uncentered Hardy-Littlewood maximal operators on nonhomogeneous spaces for k > 0. We will prove the k times modi(cid:12)ed centered Hardy-Littlewood maximal operator is weak type (1 ; 1) bounded with constant 1 when k (cid:21) 2 if the Radon measure of the space has \con-tinuitiy" in some sense. In the proof, we will use the outer measure associated with the Radon measure. We will also prove other results of Hardy-Littlewood maximal operators on homogeneous spaces and on the real line by using outer measures.


INTRODUCTION
Hardy-Littlewood maximal operators were first introduced by G.H.Hardy and J.E.Littlewood ( [8]) in one dimensional case for the purpose of the application to Complex Analysis. Then N. Wiener ([19]) introduced this operator in higher dimensional Eucledian spaces for the purpose of the application to Ergodic Theory. Later, R.Coifman and G.Weiss ( [5]) defined Hardy-Littlewood maximal operators on quasi-metric measure spaces satisfying doubling conditions (which we call homogeneous spaces). More recently, F.Nazarov, S.Treil and A.Volberg ( [14]) defined modified Hardy-Littlewood maximal operators on quasi-metric measure spaces possesing a Radon measure that does not satisfy a doubling condition (which we call nonhomogeneous spaces), which are used in harmonic analysis on nonhomogeneous spaces. In this paper, we will treat weak type (1, 1) inequalities satisfied by several types of Hardy-Littlewood maximal operators. As is well known, weak type (1, 1) inequalities satisfied by Hardy-Littlewood maximal operators are keys to prove their strong type (p, p) boundedness via Marcinkiewicz's interpolation theorem. To prove their weak type (1, 1) inequalities, the unification of our approach is the use of outer measures. The advantage of the use of outer measures over usual measures is that they could measure any subsets of a total space, even when they are nonmeasurable.
Let (X, µ) be a metric space possesing a nondegenerate Radon measure such that µ(B(x, r)) is continuous with respect to the variable r > 0 when the variable x ∈ X is fixed, where B(x, r) denotes a ball centered at x and of radius r. We will define the k times modified centered Hardy-Littlewood maximal operator as follows: We will prove that the k times modified centered Hardy-Littlewood maximal operator M k is weak-(1, 1) bounded when k is larger than or equal to 2, and that their weak-(1, 1) constant (which is the infimum (concequently the minimum)) of the constant appearing in the weak type (1, 1) inequality) is less than or equal to 1. We will state the main idea of the proof of this fact. Let R > 0 be fixed. Let k > 0. We consider the k times modified centered Hardy-Littlewood maximal operator with bounded radius : The set A λ is easily seen to be an open set. From the continuity of the measure, we can asume that k > 2. Let J ⊂ A λ be an arbitrary compact set. For each x ∈ J, we choose r x such that The set J n is not necessarily measurable. So we use the outer measure associated with µ to estimate the "size" of the set J n . Take 0 < θ < 1 such that 1 < (k −1)θ.
Then there exists x 1 ∈ J n such that θR 1 < r x 1 . And it holds that Here, µ * is the outer measure associated with µ, i.e., If B(x 1 , kr x 1 ) ̸ ⊃ J n , we set We proceed in the same way. This process ends in finite times, because of the compactness of J and the lower uniform boundness of r x . Thus we obtain the proof.
Furthermore, we will treat the weighted weak-(1,1) inequality of the centerd Hardy-Littlewood maximal operator on a metric space possesing a doubling Radon measure. We will get some upper bound of the weak-(1, 1) constant of the weighted weak-(1, 1) inequality of the centerd Hardy-Littlewood maximal operator. We should remark that the method of this proof resembles to that of the above mentioned result on nonhomogeneous spaces.
The following is the constitution of our paper.
In Section 2, we will prove weak-(1, 1) boundedness of the k times modified centered Hardy-Littlewood maixmal operators on nonhomogeneous spaces with measures which have "continuities" in some sense when k is larger than 2. (We will state what is meant by the word "continuities" later. ) After our result, Yoshihiro Sawano ( [15]) proved a result of the same type in the setting of a separable metric space without this continuity assumption.
In Section 3, we will prove weak-(1, 1) boundedness of centered Hardy-Littlewood maximal operators under A 1 -weights ( the definition of which we will state later) with better constants than are previously known (as far as we know). The weak-(1, 1) norm of the centered Hardy-Littlewood maximal operator on the real line is recently determined by A.D. Melas ([10]). Our result may be regarded as some upper bound estimates of the weak-(1, 1) norms of the centerd Hardy-Littlewood maximal operator on homogeneous sapaces under general A 1 -weights.
In Section 4, we will prove weak-(1, 1) norms of one-sided Hardy-Littlewood maximal operators on the real line with absolutely continuous measure are less than or equal to 1. A.Bernal ([1]) proved more general results under only assumptions that the measures on the real line are Borel. We will give a different proof of special cases of A.Bernal's result. In fact, this kind of proof of the result is already known (cf. [11], [12], [19]). However, we include this proof here since this kind of proof of the result may be regarded as the easiest example of our method.

MODIFIED HARDY-LITTLEWOOD MAXIMLAL OPERATORS ON NONHOMOGENEOUS SPACES
To fix the terminology, we will include here the definition of the Hardy-Littlewood maximal operators on a metric measure space. We will consider Hardy-Littlewood maximal operators on a metric measure space X possesing a nondegenerate Radon measure µ which we will denote as (X, µ). Here a Radon measure means a measure which is defined on a σ-algebra on X including all Borel sets and which is inner regular on open sets and outer regular on Borel sets. A nondegenerate Radon measure is a Radon measure such that the measure of balls which have positive radious are positive. We will also assume here that the measures of balls which have finite radius are finite.
There are two types of the Hardy-Littlewood maximal operators, namely the centered one and the uncentered one. We will recall the definition of these here. Definition 2.1. Let (X, µ) be a metric space possesing a nondegenerate Radon measure. Let f be a locally integrable function on (X, µ). The centered Hardy-Littlewood maximal function M f of f is defined as follows.
We call the operator M associating f to M f the centered Hardy-Littlewood maximal operator. Next, we define the uncentered Hardy-Littlewood maximal operator. The uncentered Hardy-Littlewood maximal function M uc f of the locally integrable function f is defined as We call the operator M uc associating f to M uc f the uncentered Hardy-Littlewood maximal operator.
Let us assume that (X, µ) satisfies a doubling conditon, and let C be their doubling constant. Then, for any locally integrable function on (X, µ), the inequalities M f ≤ M uc f and M uc f ≤ C 2 · M f holds pointwise. The centered Hardy-Littlewood maximal operaotor M and the uncentered one M uc are both weak type (1, 1) and strong type (p, p) (1 < p ≤ +∞). We can prove that the operaotors M and M uc are both strong type (p, p) (1 < p < +∞) from the fact that they are weak type (1, 1) and strong type (+∞, +∞) by using Marcinkiewicz's interpolation theorem. It is trivial that M and M uc are both strong type (+∞, +∞), so the problem is to prove that they are weak type (1,1) .
, it suffices to prove that M uc is weak type (1, 1) . We can prove that M uc is weak type (1, 1) by using the following (finite type) Vitali's covering lemma.

Theorem 2.2. Let X be a metric spce and let a finite collection of balls
be given. Then we can find a subcollection of balls F.Nazarov, S.Treil and A.Volberg introduced a type of modified Hardy-Littlewood maximal operators on nonhomogeneous spaces. We will introduce the k times modified centered Hardy-Littlewood maximal operators and k times modified uncenterd Hardy-Littlewood maximal operators. Definition 2.3. Let f be a locally integrable function on a metric measue space (X, µ). Then the k times modified centered Hardy-Littlewood maximal function We call the operator M k the k times modified centered Hardy-Littlewood maximal operator. The k times modified uncentered Hardy-Littlewood maximal function M k f of f is defined as follows.
We call the operator M k,uc the k times modified uncentered Hardy-Littlewood maximal operaotor.
As is easily seen, the pointwise inequalities is lower semicontinuous for any locally integrable function f. We can easily prove that M 3,uc is weak-(1,1) bounded by using Vitali's covering lemma. Note that modified Hardy-Littlewood maximal operators introduced by F.Nazarov, S.Treil and A.Volberg are M 3 f in our notations and that they proved M 3 is weak-(1, 1) bounded.
Let X be a metric space possesing a nondegenerate Radon measure µ such that the measure is "continuous" in the sense that µ (B(x, r)) is continuous with the variable r > 0 when x ∈ X is fixed. Then we can show that M k f (x) = sup r>0 B(x,r) |f (y)| dµ(y) is weak-(1, 1) bounded with constant 1 when k is larger than 2. In the course of the proof, we will meet subsets of X which are not necessarily measurable. So we cannot use measures to estimate "sizes" of these sets. So, we use instead an outer measure to estimate "sizes" of these sets. Namely, Proof. Let R > 0 be fixed. Let k > 0. We consider the centered Hardy-Littlewood maximal operator with bounded radius : By the absolute continuity of the integral, there exists a compact set K ⊂ B(x 0 , r) such that 1 µ (B(x 0 , kr)) If we take δ sufficiently small, then for any y satisfying |y − x 0 | < δ, it holds that K ⊂ B(y, r) and that

Since µ(B(x, r)) is continuous with the variable r > 0 when
So we have only to prove the theorem in the case k > 2. Let J ⊂ A λ be an arbitrary compact set. For each Take 0 < θ < 1 such that 1 < (k − 1)θ. Set Then there exists x 1 ∈ J n such that θR 1 < r x 1 . And it holds that Here, µ * is the outer measure associated with µ, i.e., Then there exists x 2 ∈ J n \ B(x 1 , kr x 1 ) such that θR 2 < r x 2 . And it holds that |f (y)|dy > µ(B(x 2 , kr x 2 )).
We should remark that Using this , we can show that B( Then there exists We can show that B( We repeat this process. Then, finally, we have For, if not, we can take an infinite sequence . This, however, contradicts the compactness of J. Thus we have Here we use the fact that (For the proof of (1) , see Lemma 2.8. at the end of this chapter.). Since J is an arbitrary compact set contained in A λ , we have µ(A λ ) ≤ 1 λ ∥f ∥ 1 by the inner regularity of µ. Since the righthand side is independent of R > 0, we have Remark 2.5. After our result, Yoshihiro Sawano ( [15]) proved the following theorem: Theorem 2.6. Let X be a separable metric space with nondegenerate Radon measure. Then the two times modified centered modified Hardy-Littlewood maximal operators M 2 as is defined above is weak-(1, 1) bounded with constant 1. Namely, the following inequality holds.
for any f ∈ L 1 (X, µ). Furthermore, this result is sharp in the following sense. There exists a separable metric space with nondegenerate Radon measure such that M k is not weak-(1, 1) bounded for all positive k smaller than 2.
He proved this theorem by some variant of Vitali's covering lemma and Lindelöf's covering lemma. He did not use outer measure which we used to prove this theorem. He showed the sharpness of the result by using Kolmogorov's extension law in measure theory. Furthermore, using this theorem, he proved some type of vector-valued inequalities of singular integral operators and Fefferman-Stein's vector-valued version of Hardy-Littlewood maximal inequality on nonhomogeneous spaces. For details, the reader should refer to [15]. Remark 2.7. For completeness, we will include the proof of the following lemma. The following lemma is from [6]. Proof. From the definition of µ * , for any A ⊂ X, there exists a µ-measurable set C such that A ⊂ C and µ * (A) = µ(C). Therefore for each J k , there exists a µ- On the other hand, since µ * (J k ) ≤ µ(J), we have lim k→+∞ µ * (J k ) ≤ µ(J).

WEIGHTED WEAK-(1, 1) ESTIMATES OF HARDY-LITTLEWOOD MAXIMAL OPERATORS ON HOMOGENEOUS SPACES
We will prove in this section the weighted weak-(1, 1) inequality of the centered Hardy-Littlewood maximal operator on a metric space possesing a doubling Radon measure. We must emphasize that this type of inequality is well known. It is proved by A.P. Calderón ([2]). In this paper, we will prove the weighted weak-(1, 1) inequality with better constant than previously known (as far as we know). When w ≡ 1, we have the ordinary unweighted weak-(1, 1) inequality of the centered Hardy-Littlewood maximal operator, and even in this case, the proof of the main theorem gives a new proof the weak-(1, 1) boundness of the centered Hardy-Littlewood maximal operator. The author got some hints of this proof from H.Carlsson's paper ( [3]) and D.Termini and C.Vitanza's paper ( [18]). The reader should also notice that the method of the proof resembles to Theorem 2.4. Theorem 3.1. Let X be a metric space possseing a doubling Radon measure µ. Let w be an A 1 -weight. Namely, there exists a positive number c > 0 such that holds for any ball B(x, r). Let d be an A 1 -constant of w, and set e λ = inf{e | w(B(x, λr)) ≤ e · w(B(x, r)), ∀x ∈ X, ∀r > 0}.
Proof. Let R > 0 be fixed. We will show that holds for any f ∈ L 1 (X, µ). Let o ∈ X be a fixed point. Let r > 0 be a positive number. Set We choose r x ≤ R for each x ∈ J such that wdµ. wdµ.
We will take x i in the same way. Then, wdµ.
Then, we finally have Adding the previous inequalities, we have Here, w * is the outer measure associated with the weighted measure w. Letting n → +∞, we have Since we can choose r > 0 arbitrary in the definition of J, we have in the set {x | M + f (x) > λ}. We can choose for each x ∈ K, h x > 0 such that the inequality Here, [·] is a Gauss symbol. Let ϵ > 0 be an arbitrary positive number. Then, by the absolute continuity of the measure µ, there exists a positive number δ > 0 such that if |E| < δ, then µ(E) < ϵ. (Here, |E| denotes the Lebesgue measure of the set E.) There exists a point x 1 ∈ K n such that x 1 < inf K n + δ m+1 . By the definition of K n , the inequality holds. If K n ⊂ (−∞, x 1 + h x 1 ), we stop here. If not, there exists a point x 2 such that x 2 < inf(K n \ (−∞, x 1 + h x 1 )) + δ m+1 . And the inequality holds. If K n ⊂ (−∞, x 2 + h x 2 ), we stop here. If not, there exists a point x 3 such that x 3 < inf(K n \ (−∞, x 2 + h x 2 )) + δ m+1 . And the inequality holds. We will proceed in the same way. Then finally, for x k ∈ K n we have K n ⊂ (−∞, x k + h x k ). And we have the inequality Adding the inequalities about the integral, we have Here, µ * is the outer measure associated with measure µ. Thus Letting n → +∞,we have Since K is an arbitrary compact set contained in the set {x | M µ,+ f (x) > λ} and since µ is a Radon measure, we have