The theorem and the local oscillation decomposition for Banach space valued functions
Abstract.
We prove that the operator norm of every Banach space valued CalderónZygmund operator on the weighted LebesgueBochner space depends linearly on the Muckenhoupt characteristic of the weight. In parallel with the proof of the realvalued case, the proof is based on pointwise dominating every Banach space valued CalderónZygmund operator by a series of positive dyadic shifts. In common with the realvalued case, the pointwise dyadic domination relies on Lerner’s local oscillation decomposition formula, which we extend from the realvalued case to the Banach space valued case. The extension of Lerner’s local oscillation decomposition formula is based on a Banach space valued generalization of the notion of median.
Key words and phrases:
Banach space, vectorvalued, CalderonZygmund operator, Bochner space, local oscillation decomposition, Lerner’s formula, Muckenhoupt weight, median, dyadic domination, A_2, A22010 Mathematics Subject Classification:
42B20, 46E401. Introduction
In this paper we introduce a Banach space valued generalization of a median. Using the generalization of a median, we extend Lerner’s local oscillation decomposition formula from realvalued functions to Banach space valued functions. As an application of the extension of Lerner’s local oscillation decomposition formula, we prove that, in common with the realvalued case, every Banach space valued CalderónZygmund operator is pointwise dominated by a series of positive dyadic shifts. As an immediate consequence of the pointwise dyadic domination, we obtain the theorem for Banach space valued CalderónZygmund operators.
Let denote the class of all the weights with a finite Muckenhoupt characteristic . The theorem states that for each CalderónZygmund operator with the Hölder exponent we have
The theorem in full generality was first proven by Hytönen [12]. The result was preceded by many intermediate results by others. See [16] for a list of contributions to the theorem. The proof in [12] consists of two steps: The first step is to pointwise represent every CalderónZygmund operator as a series (over complexity ) of dyadic shift operators (with complexity ) averaged over an infinite number of randomized dyadic grids. The second step is to obtain the estimate for the dyadic shift operators (with such a decay in complexity that the series converges).
Hytönen, Lacey, and Pérez [14], and Lerner [16] showed that every CalderónZygmund operator is pointwise dominated by a series (over complexity ) of simple positive dyadic shift operators (with complexity ) summed over a finite number of translated dyadic grids (parameterized by ),
(1.1) 
This result simplifies the first proof of the theorem [12], because the pointwise domination (1.1) is simpler than the representation theorem in [12] and because the estimate is obtained more simply for the operator than for a general dyadic shift operator.
Moreover, Lerner [15] proved that the formal adjoint of each operator is pointwise dominated (linearly in complexity ) by the operator . Hence, by duality and the selfadjointness of , the estimate for the operator follows from the estimate for the operator , as shown in [15]. This result simplifies further the proof of the theorem, because the estimate for the operator is simple to obtain, as shown in [5, The proof of Theorem 1]. See [16] for a selfcontained proof of the theorem based on the simplifications mentioned. Both of the results on domination [14, 16] and [15] are based on Lerner’s local oscillation decomposition formula [17, 15].
In our paper we extend the results discussed in the preceding paragraphs from realvalued functions to Banach space valued functions. In what follows we summarize the results in an informal manner. The results together with the definitions are stated formally in Section 2. Let be a Banach space. Suppose that is an valued CalderónZygmund operator with the Hölder exponent on the LebesgueBochner space . Assume that is a Bochner measurable function.
In this paper we prove that, in common with the realvalued case, for each valued CalderónZygmund operator we have the pointwise dyadic domination theorem
and, as a corollary, the theorem
Once we have an valued generalization of Lerner’s local oscillation decomposition formula, the proof of the valued dyadic domination theorem proceeds in parallel with the proof of the realvalued dyadic domination theorem. The difficulty in extending Lerner’s formula from realvalued functions to valued functions is that the formula is derived using the notion of a median, notion which is based on the ordering of the real line. We circumvent the difficulty by introducing an valued generalization of a median, which we call a quasioptimal center of oscillation and denote by . By using the notion of a quasioptimal center of oscillation, we extend Lerner’s local oscillation formula to valued functions,
We note that twoweight norm inequalities of the form
were studied in [6, Section 8] for a valued maximal operator and in [18] for another valued operator. The setting in [6, Section 8] and [18] differs from ours, because neither the operator studied in [6, Section 8] nor the operator studied in [18] is (albeit each one is similar to) a valued CalderónZygmund operator and because instead of valued functions we study valued functions for an abstract Banach space .
Our paper is organized as follows. In Section 2 we first introduce the setting along with the notation. Then we state the pointwise dyadic domination theorem, Theorem 2.9, and the theorem, Corollary 2.10, for Banach space valued CalderónZygmund operators. We conclude Section 2 by defining the notion of a quasioptimal center of oscillation and by stating the Banach space valued generalization of Lerner’s local oscillation decomposition formula, Theorem 2.13. In Section 3 we prove, assuming the generalization of Lerner’s formula, the pointwise dyadic domination theorem and the theorem for Banach space valued CalderónZygmund operators. In Section 4 we prove the generalization of Lerner’s formula.
2. Vectorvalued setting and the main theorems
The material from Definition 2.1 to Definition 2.6 consists of defining the notions of a vectorvalued LebesgueBochner space, of a vectorvalued CalderónZygmund operator, and of a Muckenhoupt weight. The reader familiar with these notions may prefer to move on to Definition 2.7.
Notation.
Let be a Banach space. Denote by the space of bounded linear operators from to , and denote by the usual operator norm. Let denote the Lebesgue measure space. Denote by the closed ball with center and radius in . Let denote the function .
Suppose that and are sets. Let and be functions. The notation ”” and the notation ” for all ” both mean that for each there exists a constant such that for all .
Definition 2.1 (Bochner measurability).
A function is called (Lebesgue) measurable, if and only if for every Borel set of .
A function is called essentially separably valued (with respect to the Lebesgue measure space), if and only if there exist a set of measure zero such that the image of the complement of is separable.
A function is called strongly measurable (with respect to the Lebesgue measure space) or Bochner measurable (with respect to the Lebesgue measure space), if and only if it is both essentially separably valued and Lebesgue measurable.
Definition 2.2 (Weight function and weight measure).
A locally integrable function is called a weight function. A weight function gives rise to the weight measure by setting
Definition 2.3 (Weighted and unweighted LebesgueBochner space).
The LebesgueBochner space, denoted by , is defined as
We denote .
Let be a weight. The weighted LebesgueBochner space, denoted by , is defined as
We denote .
Definition 2.4 (Muckenhoupt weights).
Let denote the HardyLittlewood maximal function. Suppose that is a weight function. Define the dual weight function of , denoted by , by setting for each . We define the auxiliary quantities
and
For we define the Muckenhoupt characteristic, denoted by , of a weight by setting
and we define the Muckenhoupt’s class, denoted by , as
Definition 2.5 (Vectorvalued singular kernel).
A function is called a singular kernel, if and only if

The function obeys the decay estimate

The function obeys the Höldertype estimates
and
for some Hölder exponent .
Definition 2.6 (Vectorvalued CalderónZygmund operator).
Let . A linear operator is called a vectorvalued CalderónZygmund operator, if and only if

is bounded.

There exists a singular kernel such that
for every strongly measurable, bounded, and compactly supported function and for every that lies outside the support of .
Remark.
We include the condition (i) as a part of the definition of an valued CalderónZygmund operator. In case of many classes of operators the condition (i) is checked by using theorems such as [3, Theorem 5], an valued theorem [8], an valued theorem [10], or an operatorvalued Fourier multiplier theorem [19]. These theorems presume that has the UMDproperty, which means that valued martingale difference sequences are unconditional in . Moreover, in the case of the Hilberttransform, which is a prototype of a singular integral operator, for the fulfilment of the condition (i) it is not only sufficient [4] but also necessary [2] that the Banachspace has the UMDproperty.
Next we define the dyadic model operators that dominate each vectorvalued CalderónZygmund operator. The operators are precisely the same dyadic model operators that dominate each CalderónZygmund operator in the realvalued case [14].
Definition 2.7 (Pairwise nearly disjoint collection).
Let . A collection of measurable sets is called pairwise nearly disjoint (with the parameter ), if and only if

For every there exists a measurable subset such that .

For every and such that we have .
Definition 2.8 (Dyadic model operator ).
Let be a collection of dyadic cubes. Let denote the :th ancestor of a dyadic cube . We define the dyadic model operator by
Recall that for each translation parameter we have the translated dyadic system
Next we state our main theorem.
Theorem 2.9 (Pointwise dyadic domination theorem for vectorvalued CalderónZygmund operators).
Suppose that is a strongly measurable, bounded, and compactly supported function. Let be a cube that contains the support of . Suppose that is a vectorvalued CalderónZygmund operator with the Hölder exponent .
Then for each translated dyadic system and for each there exists a collection of dyadic cubes such that the collection is pairwise nearly disjoint and for almost every we have
The implicit constant in the inequality depends only on the dimension and on the constants that are implicit in the definition of a vectorvalued CalderónZygmund operator. Each collection depends on and .
The proof of the theorem is deferred to Section 3. We can use the theorem as a method to transfer results about realvalued model operators into results about vectorvalued CalderónZygmund operators. In regard to estimating the norm, note that by definition if , then and
Hence by Theorem 2.9 for each we have that
Now by using a known estimate for we obtain the following corollary, the proof of which is deferred to Section 3.
Corollary 2.10 ( theorem for vectorvalued CalderónZygmund operators).
Let be a Banach space. Suppose that is a vectorvalued CalderónZygmund operator with the Hölder exponent . Then we have that
(2.1) 
for all and for all . The implicit constant in the inequality depends only on the dimension and on the constants that are implicit in the definition of a vectorvalued CalderónZygmund operator.
Remark.
By the sharp version of Rubio de Francia’s extrapolation theorem [7], version which extends with the same proof for Banach space valued functions, we have that the weighted norm estimate (2.1) for implies the weighted norm estimate (2.1) for every . However, we prove the weighted norm estimate (2.1) for every directly, without using the sharp version of Rubio de Francia’s extrapolation theorem.
The dyadic domination theorem for realvalued CalderónZygmund operators [14] is based on Lerner’s local oscillation decomposition formula [17, 15]. Similarly, the dyadic domination theorem for vectorvalued CalderónZygmund operators is based on a vectorvalued generalization of Lerner’s local oscillation decomposition formula, to which we turn next. First we recall the notion of a local oscillation.
Definition 2.11 (Local oscillation ).
Let be a Lebesgue measurable function. Suppose that is a Lebesgue measurable set. Let . The local oscillation or the optimal oscillatory bound of on with a portion disregarded, denoted by , is defined as
Remark.
Recall that the decreasing rearrangement of , denoted by , is defined as . Note that .
Next we formulate the definition of a quasioptimal center of oscillation, which is a vectorvalued counterpart of a median. In Section 4 we show that there always exists a quasioptimal center of oscillation and discuss the idea behind the notion.
Definition 2.12 (Quasioptimal center of oscillation ).
Let be a Lebesgue measurable function. Suppose that is a Lebesgue measurable set. Let A quasioptimal center of oscillation of on with a portion disregarded or a pseudomedian of on , denoted by , is defined as any vector such that
Endowed with the vectorvalued generalization of a median, we now state the vectorvalued generalization of Lerner’s local oscillation decomposition formula.
Theorem 2.13 (Vectorvalued generalization of Lerner’s local oscillation decomposition formula).
Suppose that is a Banach space. Let . Suppose that is a strongly measurable function. Let be a cube.
Then there exists a collection of dyadic subcubes of such that the collection is pairwise nearly disjoint with the parameter and for almost every and for every quasioptimal center of oscillation we have
3. Pointwise dyadic domination theorem and theorem for vectorvalued CalderónZygmund operators
Notation.
Let be a cube. We denote by the side length and by the center of the cube . Let denote the cube that has the same center as but that has the side length . We denote by the :th dyadic ancestor of a cube . We denote by the standard dyadic system,
We use the notations
and
Theorem 3.1 (Pointwise dyadic domination theorem for vectorvalued CalderónZygmund operators).
Suppose that is a strongly measurable, bounded, and compactly supported function. Let be a cube that contains the support of . Suppose that is a vectorvalued CalderónZygmund operator with the Hölder exponent .
Then for each translated dyadic system and for each there exists a collection of dyadic cubes such that the collection is pairwise nearly disjoint and for almost every we have
The implicit constant in the inequality depends only on the dimension and on the constants that are implicit in the definition of a vectorvalued CalderónZygmund operator. Each collection depends on and .
Proof.
The proof of the dyadic domination theorem in the vectorvalued case proceeds parallel to the proof in the realvalued case [14]. By using the vectorvalued generalization of Lerner’s local oscillation decomposition formula, Theorem 2.13, we dominate by a series of optimal oscillatory bounds summed over all the cubes of a pairwise nearly disjoint (with the parameter ) collection of dyadic cubes,
Let . We claim that for each cube the optimal oscillatory bound is bounded by a series of the nondyadic integral averages ,
Moreover, we claim that for each cube that contains the support of we have
Assuming for the moment these claims, which are stated and proven as Lemma 3.2, we complete the proof.
Recall that for each translation parameter there is the translated dyadic system
and that for each cube and each we can find a cube that is dyadic in the translated dyadic systems for some and that satiesfies , and [14, Proposition 2.5.]. Hence we can dominate each nondyadic integral average by the dyadic integral average ,
Let . By the definition of a pairwise nearly disjoint collection with the parameter , for each there exists a measurable subset such that and for each and such that we have . Since and are of comparable size, we have
Hence the collection is pairwise nearly disjoint with the parameter . Moreover, observe that since the sidelengths and are both powers of and since , we have in fact that . Since a cube with sidelength can contain at most cubes of side lenth and since , we have
Observe that we can decompose the pairwise nearly disjoint collection , which contains dyadic cubes of various translated dyadic systems, into pairwise nearly disjoint collections , each of which contains dyadic cubes of at most one translated dyadic system,
Lemma 3.2.
Suppose that Let denote the set of all strongly measurable, bounded, and compactly supported functions from to . Suppose that is a vectorvalued CalderónZygmund operator with the Hölder exponent .
Then we have that
for all and all cubes . Furthermore, we have that
for all and all cubes that contain the support of .
Proof.
We shall first proof the estimate for . Recall that the Euclidean distance and the distance are equivalent in the sense that . From the equivalence of the distances it follows that if a singular kernel satisfies the decay and the Hölder estimates of its definition for the Euclidean distance , then satisfies the same estimates (with the implicit constants depending on ) for the distance . For this proof we shall work with the distance in order to slightly simplify the use of the kernel estimates. Let be a strongly measurable, bounded, and compactly supported function. Suppose that is a cube. Let . Since
we have
(3.1) 
Observe that for the last term on the righthand side we can use the integral presentation with a singular kernel, because lies outside the support of , and we can use the Hölder estimate of a singular kernel, because for we have . Therefore
(3.2)  
Let be a Lebesgue measurable function. Recall that the decreasing rearrangement of g, denoted by , is defined as
In order to estimate the optimal oscillatory bound by exploiting the properties of a decreasing rearrangement, we write the optimal oscillatory bound in terms of the decreasing rearrangement
We shall use the following properties, which are all wellknown, of a decreasing rearrangement.


for every constant vector .

If for some real constant we have that for almost every point , then for every .
By the properties (ii) and (iii) of a decreasing rearrangement and the inequalities and we can estimate the optimal oscillatory bound as follows.
Recall the weaktype (1,1) inequality for ,
which is wellknown and proven in [1, The proof of Theorem 1]. By the property (i) of a decreasing rearrangement together with the weaktype (1,1) inequality for we have
Hence altogether we have