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Revista de la Unión Matemática Argentina
Print version ISSN 00416932
Rev. Unión Mat. Argent. vol.50 no.1 Bahía Blanca June 2009
Weak type (1, 1) of maximal operators on metric measure spaces
Marilina Carena
Abstract. A discretization method for the study of the weak type (1, 1) for the maximal of a sequence of convolution operators on has been introduced by Miguel de Guzmán and Teresa Carrillo, by replacing the integrable functions by finite sums of Dirac deltas. Trying to extend the above mentioned result to integral operators defined on metric measure spaces, a general setting containing at once continuous, discrete and mixed contexts, a caveat comes from the result in On restricted weak type (1, 1); the discrete case (Akcoglu M.; Baxter J.; Bellow A.; Jones R., Israel J. Math. 124 (2001), 285297). There a sequence of convolution operators in is constructed such that the maximal operator is of restricted weak type (1, 1), or equivalently of weak type (1, 1) over finite sums of Dirac deltas, but not of weak type (1, 1). The purpose of this note is twofold. First we prove that, in a general metric measure space with a measure that is absolutely continuous with respect to some doubling measure, the weak type (1, 1) of the maximal operator associated to a given sequence of integral operators is equivalent to the weak type (1, 1) over linear combinations of Dirac deltas with positive integer coefficients. Second, for the nonatomic case we obtain as a corollary that any of these weak type properties is equivalent to the weak type (1, 1) over finite sums of Dirac deltas supported at different points.
2000 Mathematics Subject Classification. Primary 42B25
Key words and phrases. Maximal operator; Weak type (1, 1); Dirac delta.
The author was supported by CONICET, CAI+D (UNL) and ANPCyT
The problem of determination of the weak type of maximal operators associated to a sequence of convolution kernels from its behavior on classes of special functions or distributions has as starting point the results of Moon in [14]. There, the weak type of the maximal operator associated to the convolution operators induced by a sequence of integrable kernels in , is proved to be equivalent to the restricted weak type , with . This means that to guarantee the weak type of such operator, is enough to test its action over the collection of all characteristic functions of measurable sets in with finite measure.
The next relevant step was introduced by T. Carrillo y M. de Guzmán (see [8] and [5]), where characteristic functions are substituted by Dirac deltas, again in the Euclidean space. A generalization of these results concerning the structure of the class of special functions providing the weak type () of such a maximal operator on , is proved by F. Chiarenza and A. Villani in [6]. Later on, in [13], T. Menárguez y F. Soria showed how to applied the discrete approach to obtain the best constants for the weak type of maximal operator, which for the HardyLittlewood maximal operator is finally achieved by Melas in [10]. Extensions to weighted inequalities and for non convolution integral operators in are proved by T. Menárguez (see [11] and [12]). Let us also mention some recent results by J. Aldaz and J. Varona in [4], where Dirac deltas are substituted by more general measures for convolution type operators.
A natural question, taking into account the recent developments of real and harmonic analysis on metric spaces, is whether or not these results can be extended to a metric measure space, for example to a space of homogeneous type or even to non doubling settings.
Being the integers with the restriction of the usual distance and with the counting measure a space of homogeneous type, the remarkable example given by Akcoglu, Baxter, Bellow and Jones in [3] gives us the answer to our general aim: no, it is not posible to deduce the weak type of a maximal of a sequence of convolution operators on from its weak type on Dirac deltas.
These facts together leads us to at least two problems. First, if we consider non atomic metric measure spaces and sequences of integral operators with continuous kernels, we ask for the natural extension of the result in [5]. Second, in a general context containing at once discrete, continuous and mixed situations, look for small classes of functions which are enough in order to test the weak type of such a maximal operator. Actually we shall solve the first problem as a corollary of our approach to the second one.
We would like to mention that the main tool for our proof is the dyadic analysis on spaces of homogeneous type started by Christ in [7].
The paper is organized as follows. In Section 2 we introduce the geometric setting and the basic properties of the dyadic families introduced by M. Christ in [7]. In Section 3 we introduce the basic properties of the kernels defining the sequence of integral operators and we state and prove the main results of this paper.
2. Dyadic sets on spaces of homogeneous type
In this section we introduce the geometric setting and we remind some properties of the "dyadic cubes" constructed by Christ. Even when the results hold on quasimetric spaces, a theorem due to Macías an Segovia (see [9]) allows us to work on a metric setting. Let be a metric space and let be a positive Borel measure on . We shall say that is regular on if
for every Borel subset of . The measure satisfies the doubling property on if there exists a constant such that the inequalities
hold for every and every , where . We shall say that a metric measure space is a space of homogeneous type if is a regular measure satisfying the doubling property on . Then if is a space of homogeneous type, the set of all the continuous functions on with compact support is dense in . Notice that if is a complete metric space and is a finite doubling measure on , then is a space of homogeneous type (see [1]).
Given a space of homogeneous type, let us state as a theorem the main properties of the dyadic families constructed by M. Christ in [7]. For , and for each let be a maximal disperse subset of , where is an initial interval of natural numbers that may coincide with , and is finite for every if and only if is bounded. Set .
Theorem 1 (Christ). Let be a space of homogeneous type. Then there exist , , , and a family of subsets of satisfying the following properties.
For the proof see [7] and [2]. Let us write to denote the class of all "dyadic sets" in the above theorem, i.e.
As already mentioned, in a space of homogeneous type the set of all the continuous functions with compact support is dense in . This fact allows to prove that in this case the set of all linear combinations of characteristic functions of dyadic sets is also dense in , which will be essential in the proof of the main result of this paper.
Let us start by introducing some terminology and notation. Let be a metric measure space, where is a finite positive Borel measure on . Let us consider a sequence of kernels, where each is a measurable function such that uniformly in . This means that for each there exists such that
Given we define
Notice that by FubiniTonelli's theorem, for almost every , and then is a measurable function defined on .
If is a continuous function, are different points and , taking
we have that
On the continuity of we have that converges to when tends to zero. By the other hand, in the weak sense when tends to zero, where denotes the Dirac delta concentrated at the point . In this sense we can consider the operator acting over this kind of measures , given by
We shall say that is of weak type (1,1) over finite sums of Dirac deltas (in ) if there exists a constant such that for each the inequality
holds for every , where are different points in .
Notice that is the total variation of the measure .
Also we shall say that the maximal operator is of weak type over finite sums of Dirac deltas if there exists such that for every and every we have
Let us observe that both definitions given above can be written forgetting about Dirac deltas. In particular, for the case of the maximal operator we have that the condition for every and every , is equivalent to say that the inequality
holds for every collection of different points in , for every and for every .
Notice finally that if each is a continuos function with compact support, FubiniTonelli's theorem implies that each is well defined for every and for every , and it is an integrable function. Moreover, is bounded and with compact support. Then is a measurable function defined on every point of , provided that .
With the above definitions we are in position to state and prove the extensions of the above mentioned theorem of Miguel de Guzmán to metric measure spaces. As we already noticed, the characterization of the weak type contained in that theorem is not true in general measure spaces. Actually this is the case of spaces with isolated points, even for convolution operators. In fact, K. H. Moon proves in [14] that the maximal operator associated to a sequence of convolution operators in is of weak type , , if and only if is of restricted weak type , i.e., if the weak type inequality holds for characteristic functions of sets with finite measure. A somehow surprising situation occurs when the extension of Moon's result is considered in such a simple discrete setting as is . In fact, in [3] the authors construct a sequence of convolution operators on whose maximal operator is of restricted weak type but not of weak type . Notice that if is any finite subset of , let us say with different integer numbers, we have that
The above considerations show that a direct extension of the result of M. de Guzmán y T. Carrillo to general metric measure spaces is impossible. Nevertheless the weak type for the maximal of a given sequence of operators is equivalent to its weak type of the class of all linear combinations of Dirac deltas with positive integer coefficients. In fact our result in this direction is the following.
Theorem 2. Let be a metric measure space, where is a measure such that , with and a space of homogeneous type. Let be a sequence of continuous kernels with compact support on . Then is of weak type if and only if there exists a constant such that for every and every finite collection of not necessarily different points, we have
Proof. Let us start by proving that the weak type over linear combinations of Dirac deltas with positive integer coefficients of implies the weak type of on . If for a fixed natural number we call , then it is clear that
and that . Hence it is enough to prove that for each fixed the inequality
holds with independent of . So we take a fix and we will show the weak type of in three steps.
Step 1. We first prove that if with , then for every we have that
If , we write , with , and
where . Then, if we have
Now take and write , with and will be conveniently chosen later, so small as needed. Then taking , for every we have
Since each can be chosen arbitrarily small, we have
for every . The desired inequality follows taking limit for .
Step 2. We want to prove now that is of weak type over linear combinations of characteristic functions of the dyadic sets constructed by Christ (see Section 2 ). Let , with . Notice that we may assume that and that the sets are disjoint. We want to see that for every and for every function as above,
Let us observe first that if is the given simple function, and if is a given positive real number, then we can write, except on a set with measure equal to zero, with disjoint dyadic sets in such that and for every (see properties (3), (4), (6) and (8) in Theorem 1 ). Then we will keep writing and when necessary we shall assume that the diameter of each is as small as we need.
Let where denotes the Dirac delta concentrated at , the "center" of (see properties 2 and 3 in Theorem 1 ). For the fixed and for we write
Then all we have to do is to show that the second term in the last member of the above inequalities can be made arbitrarily small by an adequate choice of the size of the dyadic sets in the definition of the function . In fact, since
we have
where denotes the projection in the first variable of the support of , so it is a bounded set and with finite measure. Since each is a continuous function with compact support, given there exists such that for every , provided that . Since we can take the diameter of each small, we conclude the proof of the Step 2.
Step 3. From the technique of reduction to a dense subspace (see for example [8], Thm. 3.1.1) and the previous step prove we obtain the theorem in one direction.
For the converse, let us assume now that is of weak type . We want to prove that is of weak type over linear combinations of Dirac deltas with positive integer coefficients. In fact, let a set of different points in , and let with a positive integer for every . Defining
we have that when . Fix real numbers and as in Christ's Theorem (Thm. 1 ), and let be a positive integer satisfying . For each , there exists such that . Notice that if then . In fact, let us suppose that there exists . Then
which is absurd if . Let us define the function as
As before, fix , and such that , and write
where means
Hence
As in the Step 2 given we get
by an adequate choice for the diameter of the dyadic sets, since the each kernel is a continuous function and we have a finite number of them. Then we have shown that
as desired.
As we already mentioned, for the nonatomic case we obtain as a corollary that the theorem of de Guzmán and Carrillo can be extended to certain metric measure spaces. More precisely, the following result state that the weak type for the maximal operator of a sequence of integral operators with continuous kernels with compact support, is equivalent to the weak type over finite sums of Dirac deltas supported at different points.
Theorem 3. Let be a metric measure space without isolated points, where is a measure such that , with and a space of homogeneous type. Let be a sequence of continuous kernels with compact support on . Then is of weak type if and only if is of weak type over finite sums of Dirac deltas. In other words, is of weak type if and only if there exists a constant such that for every and every finite set of different points in , we have
Proof. Notice that after Theorem 2 , we only have to prove that if is of weak type over finite sums of Dirac deltas, then it is of weak type over linear combinations of Dirac deltas with positive integer coefficients. In fact, we know that there exists a constant such that for every finite set of different points and for every , we have
We want to see that for every finite set of different points and for every , if with , then
As in the proof of Theorem 2 , it will be sufficient to prove that if is a fixed natural number, then
where is independent of . Then let us fix . Since does not have isolated points, for each we can chose different points in sufficiently close to , and such that the set is also a collection of different points. For each fixed we write
Hence for every such that , we have
We know that
so all we have to do is to show that given and satisfying , we can chose the elements such that . In fact, let
Then
For a fixed , from Chebyshev's inequality we have
where as before denotes the projection in the first variable of the support of , so it is a bounded set and with finite measure. Since each is a continuous function with compact support, given there exists such that for every , provided that . Notice also that only a finite number of kernels are involved, so that becomes small after an appropriate choice of . Hence
and taking we obtain the result.
The next result of this section is devoted to relax the regularity hypothesis on . Its proof is obtained by inspection of the proof of Theorem 2 .
Theorem 4. Let be a metric measure space, where is a measure such that , with and a space of homogeneous type. Let be a sequence of kernels such that each is a measurable function satisfying
Then is of weak type if and only if there exists a constant such that for every and every finite collection of points not necessarily different, we have
It is clear that an analogous extension of Theorem 3 can be proved.
Notice that it is possible to obtain a refined result for spaces wich are neither discrete nor purely continuous. For example, for the set
endowed with the usual distance on and the measure that counts on and measures lengths on , is a space of homogeneous type (see [15]).
Moreover, Macías and Segovia prove in [9] that in spaces of homogeneous type the set of points with positive measure (atoms) is countable and coincides with the set of isolated points. With this characterization for the atoms we have that if of weak type if and only if there exists a constant such that for every finite set of different points in and for every choice of natural numbers satisfying when , we have that
for every .
We shall finally mention that the hypotheses of the above theorems concerning continuity sometimes can be relaxed. For the basic case of HardyLittlewood type operator defined on a space of homogeneous type by
for , where
the continuity required in Theorem 2 does not hold even in Euclidean situations. On the other hand, the continuity required in Theorem 4 does not hold in typical spaces of homogeneous type. In fact,

where denotes the symmetric difference of the sets and , i.e. . The convergence to zero of the last integral when is equivalent to the convergence to zero of for each . The next example shows a nonatomic space of homogeneous type for which this property does not hold. In endowed with the distance , let be the subset defined as
(see Figure 1) with the arc length measure .
Figure 1.
It is not difficult to see that is a space of homogeneous type. Take the sequence in defined as . This sequence converges to the point , and for each (see Figure 2), we have
Figure 2:
Then for each , so that does not tend to zero when tends to infinity.
Nevertheless, the kernels in such a general situation can be controlled by a sequence of continuous kernels. For instance consider
where is the continuous function defined on the nonnegative real numbers by for every in the interval , if , and linear on . It is not difficult to show that each is continuous and that
where denotes the doubling constant for . Then the weak type for the maximal operator associated with the kernels is equivalent to the weak type for the maximal operator associated with the kernels .
The new sequence falls under the scope of Theorem 2 , so that the next result holds even when the kernels are not smooth.
Corollary 5. Let be a space of homogeneous type. Then the HardyLittlewood maximal function is of weak type if and only if there exists a constant such that for every and every finite collection of not necessarily different points,
where
Acknowledgment. The results in this paper are part of my Doctoral Dissertation presented at the Universidad Nacional del Litoral, March 2008. I would like to express deep gratitude to my supervisors Hugo Aimar and Bibiana Iaffei for the constant support.
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M. Carena
Instituto de Matemática Aplicada del Litoral (CONICETUNL),
Departamento de Matemática(FIQUNL),
Universidad Nacional de Litoral,
Santa Fe, Argentina
mcarena@santafeconicet.gov.ar
Recibido: 1 de agosto de 2008
Aceptado: 13 de octubre de 2008