REGULARITY OF MAXIMAL FUNCTIONS ASSOCIATED TO A CRITICAL RADIUS FUNCTION

In this work we obtain boundedness on BMO and Lipschitz type spaces in the context of a critical radius function. We deal with a local maximal operator and a maximal operator of a one parameter family of operators with certain conditions on its kernels that can be applied to the maximal of the semi-group in the context of a Schrödinger operator.


Introduction and preliminaries
In this work we deal with the boundedness of some maximal operators acting on BMO and Lipschitz type spaces that come from the localized analysis considering a critical radius function ρ, i.e., a function that satisfies for all x, y ∈ R d (see [1] and [2]). This analysis appears in the context of the Schrödinger operator L = −∆ + V in R d , d ≥ 3 (see for example [8], [13], and references therein).
For x ∈ R d , a ball of the form B(x, ρ(x)) is called critical and a ball B(x, r) with r < ρ(x) will be called sub-critical. We denote by B ρ the family of all sub-critical balls.
One of the operators we are interested in is the localized maximal operator M ρ defined for f ∈ L 1 loc as In [5] the authors prove that M ρ is bounded on L p (w) for 1 < p < ∞, where w belongs to a suitable class larger than classical A p Muckenhoupt weights. Here we deal with the boundedness of M ρ in a weighted BMO type space that appears in [8] for w = 1, and in [3] with weighted versions.
We also deal with some type of maximal operator of a family of operators presented in Section 5 that is a model to deal with semi-groups appearing in the theory related to the Schrödinger operator L. Some results concerning this operator in a more general context can be found in [14] and [16].
In the rest of this section we present some facts about the critical radius function. Section 2 is devoted to present the classes of weights involved in this work and some properties of them that will be useful. In Section 3 we present some properties of BMO β ρ spaces that will be used later. In Section 4 and Section 5 we state and prove the main results of this work finding the behavior of maximal operators we have already talked about, and finally we present some applications to the context of the Schrödinger operator in Section 6.
As a consequence of (1.1), we have the following result, which can be found in [9], presenting a useful covering of R d by critical balls.

Proposition 1.2.
There exists a sequence of points x j , j ≥ 1, in R d , so that the family Q j = B(x j , ρ(x j )), j ≥ 1, satisfies i) ∪ j Q j = R d . ii) For every σ ≥ 1 there exist constants C and N 1 such that j χ σQj ≤ Cσ N1 .
Given a ball B we shall also need a particular covering by critical balls with centers inside B as the following lemma shows.
where C depends only on the constants in (1.1).

Proof. Consider the family of sets
Observe that if C is a chain in F endowed with the order of inclusion, then V = ∪ S∈C S is an upper bound of C. Therefore, there exists a maximal element S max in F. The set S max must be finite. In fact, due to (1.1), for all x ∈ B, and thus there are no more than N balls in S max with N ≥ r+γδ0 γδ0 d . Denote x 1 , x 2 , . . . , x N the elements of S max . We shall see that B ⊂ ∪ N i=1 B(x i , ρ(x i )) and the overlapping of the balls B(x i , ρ(x i )), i = 1, . . . , N , is finite.

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Suppose there exists y ∈ B such that y / , which is a contradiction by the choice of γ. So S max ∪ {y} belongs to F and this means the contradiction that S max is not a maximal element of F. Now we see that the overlapping {B(x i , ρ(x i ))} N i=1 is finite and depends only on the constants in (1.1).

Weights
Following [5], for 1 < p < ∞, we say that a weight w belongs to the class A ρ,loc for every ball B = B(x, r) with x ∈ R d and r ≤ ρ(x).
In the case p = 1 we define the class A ρ,loc 1 as those weights w satisfying In the rest of this section we will state and prove some facts about weights in the classes defined above which will be useful in what follows and are of interest by themselves.
Proof. In the case p = 1, since w ∈ A ρ,loc 1 = A cρ,loc 1 (see Proposition 2.1) and E ⊂ B, we have for some constant C, For the case p > 1, using the condition w ∈ A cρ,loc p and Hölder's inequality we get, for some constant C, 3) is the constant appearing in (2.1) (or (2.2) when p = 1) for the critical radius function cρ instead of ρ.
Given θ ≥ 0 and p > 1 we introduce the class A ρ,θ p as those weights w such that ˆB w Proof. The proof follows the same lines as that of Lemma 2.2, with the corresponding modifications.

Weighted BMO type spaces
Let β ≥ 0, a weight w, f ∈ L 1 loc and call f B = 1 |B|´B f . Following [3] we say that f belongs to the space BMO where B ρ is the family of sub-critical balls defined in the Introduction. We can give a norm in BMO β ρ (w) as the smallest constant that satisfies (3.1) and (3.2), and we denote it by f BMO β ρ (w) . It is not difficult to see that BMO β ρ (w) ⊂ BMO β (w), where BMO β (w) is the Lipschitz space appearing in [11] in the classical context. On the other hand, if Q is a fixed ball in R d , we call BMO β Q (w) the space of locally integrable functions on Q that satisfy condition (3.1) for all balls B ⊂ Q. From its definition, it is easy to see that Proof. If γ > 0, let us observe that γρ is also a critical radius function.
Without loss of generality, we may suppose γ > 1 (otherwise, we can start with γρ and then we multiply by 1/γ > 1).
Let us start with the inclusion BMO β ρ (w) ⊂ BMO β γρ (w). Given f ∈ BMO β ρ (w), we know that f ∈ BMO(w) and also from (3.3) we have On the other hand, since B ρ ⊂ B γρ , if B / ∈ B γρ then B / ∈ B ρ , and therefore Therefore, it remains to see . Therefore, from Lemma 2.2, we get for some constant C, and the proof is finished.
Proof. Suppose (3.4) holds and consider a ball B = B(x, r), x ∈ R d , and r ≥ ρ(x). In the case that there exists y ∈ B such that ρ(y) > 2r then B ⊂ B(y, ρ(y)). Therefore, by Lemma 2.2, Since x ∈ B(y, ρ(y)), ρ(y) ρ(x) and thus the last quantity is constant. Suppose now that for all y ∈ B, ρ(y) ≤ 2r. From Lemma 1.3, there exist N balls where N and C depend only on the constants in (1.1) and the dimension d. Now for each i = 1, . . . , N consider the ball where C is the constant of Lemma 2.2 and we also have used the bounded overlapping property of the balls B i (see Lemma 1.3).
Following the previous proof, Corollary 1 in [3] may be improved. Actually, instead of w ∈ A ρ,loc p we only need to ask for a doubling condition for the weight w on sub-critical balls.
The following result was proved in [4] for w in the Muckenhoupt class A p . Here we shall prove an extension of that result for w ∈ A ρ,loc Proof. First, we will prove that (3.6) holds. Let us consider the covering {Q k } of critical balls given by Proposition 1.2 and a ball B = B(x, r) with r ≤ ρ(x).
for a constant C independent of x and r. If we have a cube Q and we call BMO β,s Q (w) the space of functions f such that and BMO β,s (w) the space of functions when the supremum (3.7) is considered for all balls B ⊂ R d , according to [4], it follows that BMO β, and thus (3.6) is a consequence of inequality (3.3).

From Proposition 3.2, it is enough to check (3.5) over a critical ballB =
The first term of the right side is bounded following the same argument as before.
For the second term, observe that w 1−s ∈ A ρ,loc s , since w ∈ A ρ,loc p and p ≤ s . Then

The localized maximal operator associated to ρ
In [5] (see Theorem 1 therein) the behavior of M ρ is studied, and it is proved that M ρ is bounded on weighted Lebesgue spaces for localized weights, as is stated in the following theorem.
Now we present one of the main results of this work, which tells us about the behavior of M ρ in the extreme BMO ρ (w).
Proof. Let f ∈ BMO ρ (w). We start by proving condition (3.1) for M ρ f . For B ∈ B ρ , with B = B(x 0 , r), as it is well known it shall be enough to see for some constant c that depends only on f and B. Before starting, observe that if z ∈ B is given and P is a ball such that z ∈ P and P ∈ B ρ , it follows from (1.1) On the other hand, there exist a constantC and a ball Q 0 = B(y 0 , ρ(y 0 )) of the covering given by Proposition 1.2, such that x 0 ∈ Q 0 andB ⊂Q 0 =CQ 0 , with where the maximal operator MQ 0 is defined as Thus, for any constant c, Since for every x ∈ B we have It is not difficult to deduce from (1.1) that if P = B(x P , r P ), such that P ⊂Q 0 and P / ∈ B ρ , then r P ρ(y 0 ). In fact, r P ≤Cρ(y 0 ) and also, r P ≥ ρ(x P ) and ρ(x P ) ρ(y 0 ) (since x P ∈Q 0 = B(y 0 ,Cρ(y 0 ))). Therefore, for every x ∈ B we obtainM

Thus
, As w ∈ A ρ,loc 1 , from Lemma 2.2 and the fact that B ⊂Q 0 we have  αρ(x 0 )) and α = 2 2N0 c 2 ρ +2. We first consider M ρ f 1 . By Hölder's inequality In this way, considering that B * 0 / ∈ B ρ and Lemma 3.3 it follows that the lefthand side of (4.4) is bounded by a constant times f BMOρ(w) . Now, for x ∈ B 0 we will deal with M ρ f 2 (x). It follows from the definition of f 2 that it is enough to take the supremum of the averages over those balls B ∈ B ρ such that x ∈ B and B ∩ (B * 0 ) c = ∅. Let B = B(x B , r B ) be one of those balls. From (1.1), it follows easily that ρ(x 0 ) ρ(x) ρ(x B ). More precisely, we have On the other hand, since B ∩ (B * 0 ) c = ∅, there exists a point z such that z ∈ B and z / ∈ B * 0 ; then In fact, given y ∈ B, it follows that From the bound of M ρ f 2 (x), for every x ∈ B 0 given by (4.5) and Lemma 2.2 we get and this completes the proof.

The maximal operator of a family of operators
Let {T t } t>0 be a family of bounded integral operators on L 2 (R d ) with integrable kernels {T t (x, y)} t>0 . Suppose also that there exist constants C, γ, γ , δ, σ, σ and such that for all t > 0 and x, x 0 , y ∈ R d with |x − x 0 | ≤ t/2 and ρ(x 0 ) ρ(x) the following inequalities hold: For that family of operators we define the maximal operator T * = sup t>0 |T t |.
We present the following technical lemmas that will be used in the proof of Theorem 5.3.
Now we state the main result of this section.
where, for x ∈ R d , Let us start with III. If x ∈ B 0 and 0 < t < ρ(x), then From (5.1) and the definition of M ρ it follows easily that For the second term of (5.4), if k 0 ∈ N 0 is such that 2 k0 t ≤ ρ(x) < 2 k0+1 t and we call B k = B(x, 2 k t), we get In and then the operator M ρ is bounded on L p (w 1−s ) (see Proposition 4.1). Therefore, ifB 0 = c 0 B 0 with c 0 as in (4.1), from Hölder's inequality, Lemma 3.3, and Lemma 2.2, we get Now, we deal with I. Consider x ∈ B 0 and t ≥ ρ(x). Then, Bearing in mind that B(x, t) / ∈ B ρ and B(x, ρ(x)) ⊂ B(x 0 , c 1 ρ(x 0 )) (with c 1 = 1 + c ρ 2 N 0 N 0 +1 ), from (5.1), Lemma 2.5, we havê where in the last inequality we have used the hypothesis σ ≥ β + pθ + d(p − 1). Now, from Lemma 2.2 and the fact that ρ(x) ρ(x 0 ) (see Remark 1.1), we obtain that the last expression is bounded by On the other hand, if we denote B k = B(x, 2 k t), then B k / ∈ B ρ , for any k ∈ N. Hence, from (5.1) and the definition of BMO β ρ (w) we obtain where in the last inequality we have used the hypotheses σ ≥ β + pθ + d(p − 1) and γ > β + pθ + d(p − 1). Therefore, from (5.5) and (5.6) we get and thus we have In order to finish this part, let us see II. Observe that |T t (x, y)||f (y)| dy.

REGULARITY OF MAXIMAL FUNCTIONS 555
If t < ρ(x) andB k = B(x, 2 k ρ(x)), k ∈ N, then from (5.1) From here, we can proceed as in (5.5) and (5.6), replacing B k and t byB k and ρ(x), respectively. Therefore, and this finishes the proof that T * f satisfies (3.4) and then condition (3.2) (see Proposition 3.2).
To estimate the oscillation of T * f we consider a ball B = B(x 0 , r) with 0 < r < ρ(x 0 ). We decompose f = f 1 +f 2 +f 3 , where and f 3 = f B , to deal with each one separately.
We start with f 1 . In this case it is enough to estimate the average sup t>0 |T t f 1 |.
Let us see the case r ≥ t/2. In this case, we estimate the difference by the sum as followŝ We only deal with A; the term B can be estimated analogously. Since in the domain of integration we have |x 0 − y| ≥ 2r ≥ t and also |x − y| ≥ |x 0 − y| − |x − x 0 | > r ≥ t/2, using condition (5.1) and Lemma 5.2 we obtain |x0−y|>2r whenever γ > β + d(p − 1) + pθ. Therefore, from (5.12), (5.14) and (5.15), we obtain for Hence, it follows that , and this finishes the term with f 2 .
To deal with the term with f 3 , we shall find a bound for T * f 3 = T * f B = |f B | T * 1. We will estimate the oscillation of T * f 3 over B subtracting the constant As before, we consider separately the cases t ≥ 2r and t < 2r. We start by assuming t ≥ 2r and then |x − x 0 | ≤ t/2, which allows us to use condition (5.2). We also divide the domain as before aŝ In the case t < 2r, proceeding in the same way as in (5.16) and (5.17) it follows that . (5.19) Having in mind that a < 1 and r/ρ(x 0 ) ≤ 1 we obtain from (5.18) and (5.19) that Finally, from Lemma 5.1 it follows that And this finishes the proof of the theorem. Now will will consider the Poisson maximal operator P * = sup t>0 |P t |, where and {T t } t>0 is a family of integral operators bounded on L 2 (R d ). Hence, we only have to prove condition (5.2), i.e., that for all t > 0 and x, x 0 , y ∈ R d with |x − x 0 | ≤ t/2 and ρ(x 0 ) ρ(x), the following inequality is valid: We start by observing that |x − x 0 | ≤ t/(2s) if and only if s ≤ t/(2|x − x 0 |). Then we see that |P t (x, y) − P t (x 0 , y)| ˆ∞ 0 e −s 2 /4 |T t/s (x, y) − T t/s (x 0 , y)| ds = I + II , β + pθ + d(p − 1), σ 1 > pθ, and 1δ δ+ 1 ≥ d(p − 1) + β, there exists a constant C such that , for every f ∈ BMO β ρ (w).
6. Application to the context of the Schrödinger operator In this section we consider a Schrödinger operator in R d with d ≥ 3, where V ≥ 0, not identically zero, is a function that satisfies for q > d/2 the reverse Hölder inequality for every ball B ⊂ R d . The set of functions with the last property is usually denoted by RH q . For a given potential V ∈ RH q , with q > d/2, as in [13], we consider the auxiliary function ρ defined for x ∈ R d as ρ(x) = sup r > 0 : 1 Under the above conditions on V we have 0 < ρ(x) < ∞. Furthermore, according to [13,Lemma 1.4], if V ∈ RH q/2 the associated function ρ verifies (1.1).
Let k t be the kernel of e −tL , t > 0, where {e −tL } t>0 is called the heat semigroup associated to L. The following estimates for k t are know (see [12] and [10]). Proposition 6.1. Let V ∈ RH q , q > d/2, N > 0, and 0 < λ < min 1, 2 − d q . Then there exist positive constants C,C and C N such that for all t > 0 and x, y, x 0 ∈ R d with |x − x 0 | < √ t we have
We end this section with the following result where we apply Theorem 2 to the maximal operator T * = sup t>0 |e −tL |.