Let be the infinitesimal generator of an analytic semigroup on with suitable upper bounds on its heat kernels. Auscher, Duong, and McIntosh defined a Hardy space by means of an area integral function associated with the operator By using a variant of the maximal function associated with the semigroup . a space , of functions of BMO type was defined by Duong and Yan and it generalizes the classical BMO space. In this paper, we show that if has a bounded holomorphic functional calculus on then the dual space of , is where is the adjoint operator of We then obtain a characterization of the space . in terms of the Carleson measure. We also discuss the dimensions of the kernel spaces of BMO when is a second-order elliptic operator of divergence form and when is a Schrödinger operator, and study the inclusion between the classical BMO space and spaces associated with operators.
The introduction and development of Hardy and BMO spaces on Euclidean spaces in the 1960s and 1970s played an important role in modern harmonic analysis and applications in partial differential equations. These spaces were studied extensively in Reference32, Reference22, Reference18, Reference19, Reference31 and many others.
An function on is in the Hardy space if the area integral function of the Poisson integral satisfies
A locally integrable function defined on is said to be in BMO, the space of functions of bounded mean oscillation, if
where the supremum is taken over all balls in and , stands for the mean of over i.e., ,
In Reference19, Fefferman and Stein showed that the space BMO is the dual space of the Hardy space They also obtained a characterization of the BMO space in terms of the Carleson measure, the .- boundedness of convolution operators which satisfy the Hörmander condition, and an interpolation theorem between spaces and the BMO space. From the viewpoint of Calderón-Zygmund operator theory, and BMO spaces are natural substitutes for and spaces, respectively.
Recently, Auscher, McIntosh and the first-named author introduced a class of Hardy spaces associated with an operator by means of the area integral functions in (Equation1.1) in which the Poisson semigroup was replaced by the semigroup (Reference4). They then obtained an characterization for -molecular by using the theory of tent spaces developed by Coifman, Meyer and Stein (Reference7, Reference8 and Reference4). See also Sections 3.2.1 and 4.1 below. In Reference16, we introduced and developed a new function space BMO associated with an operator by using a maximal function introduced by Martell in Reference25. Roughly speaking, if is the infinitesimal generator of an analytic semigroup on with kernel (which decays fast enough), we can view as an average version of (at the scale and use the quantity )
to replace the mean value in the definition (Equation1.2) of the classical BMO space, where is scaled to the radius of the ball We then say that a function (with suitable bounds on growth) is in if
See Section 3.2.2 below. We also studied and established a number of important features of the space such as the John-Nirenberg inequality and complex interpolation (Reference16, Section 3). Note that the spaces and coincide with the classical Hardy and BMO spaces, respectively (Reference16, Section 2).
The main purpose of this paper is to prove a generalization of Fefferman and Stein’s result on the duality of and BMO spaces. We will show that if has a bounded holomorphic functional calculus on and the kernel of the operator in (Equation1.3) satisfies an upper bound of Poisson type, then the space is the dual space of the Hardy space in which denotes the adjoint operator of We also obtain a characterization of functions in in terms of the Carleson measure. See Theorems 3.1 and 3.2 below.
We note that a valid choice of in (Equation1.3) is the Poisson integral which is defined by ,
For this choice of Theorems 3.1 and 3.2 of this article give the classical results of Theorem 2 and the equivalence ,(i) (iii) of Theorem 3 of Reference19, respectively. See also Chapter IV of Reference31.
Note that in our main result, Theorem 3.1, we assume only an upper bound on the kernel of in (Equation1.3) and no regularities on the space variables or Another feature of our result is that we do not assume the conservation property of the semigroup . for This allows our method to be applicable to a large class of operators ..
The paper is organised as follows. In Section 2 we will give some preliminaries on holomorphic functional calculi of operators and on integral operators with kernels satisfying upper bounds of Poisson type. In Section 3 we introduce and describe the assumptions of the operator in this paper, and recall the definitions of and BMO spaces as in Reference4 and Reference16. We then state our main result, Theorem 3.1, which says that the dual space of is In Section 4 we prove a number of important estimates for functions in . and BMO spaces. We then prove Theorem 3.1 in Section 5 by combining the key estimates of Section 4 with certain estimates using the theory of tent spaces and Carleson measures. In Section 6, we study the dimensions of the kernel spaces of BMO when is a second-order elliptic operator of divergence form and when is a Schrödinger operator. We conclude this article with a study of inclusion between the classical BMO space and spaces associated with some differential operators, including a sufficient condition for the classical BMO and spaces to coincide.
Throughout this paper, the letter “ will denote (possibly different) constants that are independent of the essential variables. ”
We first give some preliminary definitions of holomorphic functional calculi as introduced by McIntosh Reference26.
Let We define the closed sector in the complex plane . by
and denote the interior of by .
We employ the following subspaces of the space of all holomorphic functions on :
where and ,
Let A closed operator . in is said to be of type if and for each , there exists a constant such that
If is of type and we define , by
where is the contour parametrized clockwise around and , Clearly, this integral is absolutely convergent in . and it is straightforward to show, using Cauchy’s theorem, that the definition is independent of the choice of , If, in addition, is one-one and has dense range and if then , can be defined by
where It can be shown that . is a well-defined linear operator in We say that . has a bounded calculus on if there exists such that and for ,,
For a detailed study of operators which have holomorphic functional calculi, see Reference6.
In this paper, we will work with a class of integral operators which plays the role of generalized approximations to the identity. We assume that for each , the operator , is defined by its kernel in the sense that
for every function which satisfies the growth condition (Equation3.3) in Section 3.1 below.
We also assume that the kernel of satisfies a Poisson bound of order
in which is a positive, bounded, decreasing function satisfying
for some .
It is easy to check that there exists a constant such that satisfies
uniformly in , See Section 2 of .Reference14.
We recall that the Hardy-Littlewood maximal operator is defined by
where the sup is taken over all balls containing It is well known that the Hardy-Littlewood maximal operator is bounded on . for all Because of the decay of the kernel . in (Equation2.2) and (Equation2.3), one has
There exists a constant such that for any , we have
for all .
In this section, we will give the framework and the main result of this paper.
Let be a linear operator of type on with hence ; generates a holomorphic semigroup , Assume the following two conditions. .
The holomorphic semigroup , is represented by the kernel , which satisfies the upper bound
for , for and is defined on by (Equation2.2).
The operator has a bounded on -calculus That is, there exists . such that and for ,
for any .
We now give some consequences of assumptions (a) and (b) which will be useful in the sequel.
(i) If is a bounded analytic semigroup on whose kernel satisfies the estimate (Equation2.2), then for all the time derivatives of , satisfy
(ii) has a bounded on -calculus if and only if for any non-zero function , satisfies the square function estimate and its reverse
for some where , Note that different choices of . and lead to equivalent quadratic norms of See Reference26.
As noted in Reference26, positive self-adjoint operators satisfy the quadratic estimate (Equation3.2), as do normal operators with spectra in a sector, and maximal accretive operators. For definitions of these classes of operators, we refer the reader to Reference36.
(iii) Under the assumptions (a) and (b), it was proved in Theorem 3.1 of Reference15 and Theorem 6 of Reference14 that the operator has a bounded holomorphic functional calculus on , that is, there exists ; such that and for ,:
for any For . the operator , is of weak-type In .Reference16, it was proved that for the operator , is bounded from into .
We now define the class of functions that the operators act upon. For any a function , is said to be a function of if -type satisfies
We denote by the collection of all functions of If -type. the norm of in is denoted by
It is easy to see that is a Banach space under the norm Note that we use instead of the space as in Reference19 and Reference16 since this gives the appropriate setting for the duality between and BMO. For any given operator we let , and define
Note that if is the Laplacian on then , When . we have ,
For any and we define ,
It follows from the estimate (Equation3.1) that the operators and are well-defined. Moreover, the operator has the following properties:
(i) for any and almost all ,
(ii) the kernel of satisfies
We assume that is an operator which satisfies the assumptions of Section 3.1. will denote the usual upper half-space in The notation . denotes the standard cone (of aperture with vertex ) For any closed subset . , will be the union of all cones with vertices in i.e., , If is an open subset of then the “tent” over , denoted by , is given as ,.
Given a function the area integral function , associated with an operator is defined by
It follows from the assumption (b) of that the area integral function is bounded on (Reference26). It then follows from the assumption (a) of that is bounded on , See Theorem 6 of .Reference4. More specifically, there exist constants such that and
for all See also .Reference35.
By duality, the operator also satisfies the estimate (Equation3.8), where is the adjoint operator of .
The following definition was introduced in Reference4. We say that belongs to a Hardy space associated with an operator denoted by , if , We define its . norm by
Note that if is the Laplacian on then it follows from the area integral characterization of a Hardy space by using convolution that the classical space , coincides with the spaces and and their norms are equivalent. See Reference19 and Reference31.
Following Reference16, we say that is of bounded mean oscillation associated with an operator (abbreviated as if )
where the sup is taken over all balls in and , is the radius of the ball The class of functions of .BMO, modulo where ,
is a Banach space with the norm defined as in (Equation3.9). We refer to Corollary 5.2 in Section 5 for completeness of the space BMO. See also Section 6.1 for a discussion of the kernel space .
We now give the following list of a number of important properties of the spaces BMO. For the proofs, we refer the reader to Sections 2 and 3 of Reference16.
(i) If a function is in the classical space BMO, then it follows from the John-Nirenberg inequality that and See .Reference22. Under the extra condition that satisfies the conservation property of the semigroup for every it can be verified that ,BMO is a subspace of BMO. Moreover, the spaces BMO, and coincide and their norms are equivalent. See also Theorem 6.10 in Section 6.
(ii) If BMO, then for every and every there exists a constant , such that for almost all we have ,
(iii) If BMO, then for any and any there exists a constant , which depends on such that
(iv) A variant of the John-Nirenberg inequality holds for functions in BMO. That is, there exist positive constants and such that for every ball and ,
This and (Equation3.9) imply that for any BMO and the norms ,
with different choices of are all equivalent.
We now state the main result of this paper.
Assume that the operator satisfies the assumptions (a) and (b) in Section Denote by . the adjoint operator of Then, the dual space of the . space is the BMO space, in the following sense.
(i) Suppose BMO. Then the linear functional given by
initially defined on the dense subspace has a unique extension to ,.
(ii) Conversely, every continuous linear functional on the space can be realized as above; i.e., there exists BMO such that Equation3.14 holds and
To state the next theorem, we recall that a measure defined on is said to be a Carleson measure if there is a positive constant such that for each ball on ,
where is the tent over The smallest bound . in (Equation3.15) is defined to be the norm of and is denoted by .
The Carleson measure is closely related to the classical BMO space. We note that for every ,
For the space BMO, we have the following characterization of BMO functions in terms of the Carleson measure.
Assume that the operator satisfies the assumptions (a) and (b) in Section The following conditions are equivalent: .
(i) is a function in BMO;
(ii) and , is a Carleson measure, with .
The proofs of Theorem 3.1 and the implication (ii) (i) of Theorem 3.2 will be given in Section 5. For the proof of the implication (i) (ii) of Theorem 3.2, we refer to Lemma 4.6 of Section 4.
Using Theorems 3.1 and 3.2, we can obtain more information about the Hardy spaces and the spaces. We will discuss the inclusion between the classical BMO space and the spaces associated with some differential operators. See Section 6.
In Reference7, Reference8, Coifman, Meyer and Stein introduced and studied a new family of function spaces, the so-called “tent spaces”. These spaces are useful for the study of a variety of problems in harmonic analysis. In particular, we note that the tent spaces give a natural and simple approach to the atomic decomposition of functions in the classical Hardy space by using the area integral functions and the connection with the theory of Carleson measure. In this paper, we will adopt the same approach of tent spaces.
For any function defined on we will denote
As in Reference8, the “tent space” is defined as the space of functions such that when The resulting equivalence classes are then equipped with the norm When . the space is the class of functions for which and the norm Thus, . if and only if i.e., ,
Next, a function is called a if -atom
the function is supported in (for some ball ;
The following proposition on duality and atomic decomposition for functions in was proved in Reference8.
(a) The following inequality holds, whenever and
(b) The pairing
realizes as equivalent to the Banach space dual of
(c) Every element can be written as where the are atoms, and ,
Proposition 4.1 gives a quick proof of the atomic decomposition for the classical Hardy space Let . For any . we denote by , the Poisson integral and set The atomic decomposition of . in leads to the atomic decomposition of in by using the following identity on :
where for all the function is radial and in with and , for all Note that instead of the condition . we may assume that , for some Then, the operator . maps atoms to appropriate “molecules”. See Lemma 7 of Reference7.
We now give a short discussion of the Hardy space For more details, see .Reference4. First, we need a variant of formula (Equation4.4), which is inspired from the for -calculus We start from the identity: .
which is valid for all in a sector with As a consequence, one has .
where the integral converges strongly in See .Reference26. For any we let , We then have the following identity for all .:
Recall that in Reference4, a function is called an if -molecule
where is a supported in the tent -atom of some ball and , satisfies the condition By using the identity (Equation4.6) in place of (Equation4.4), an decomposition of -molecule in the space is obtained in Theorem 7 of Reference4 as follows.
Let There exist . -molecules and numbers for such that
The sequence satisfies Conversely, the decomposition .Equation4.8 satisfies
The proof of Proposition 4.2 follows from an argument using certain estimates on area integrals and tent spaces. For the details, we refer the reader to Theorem 7 of Reference4.
Let be the set of all with compact support in Consider the operator of (Equation4.6) initially defined on by
Note that for any compact set in
The operator initially defined on , extends to a bounded linear operator from ,
(a) to if ,
(c) to .
The property (b) is contained in the second part of Proposition 4.2. The property (c) will be shown in Section 5.2 as it is a direct result of Theorem 3.1 and the duality of and spaces.
We now verify (a). By using (5.1) of Reference8, we have
for any Hence, we obtain
As a consequence of Lemma 4.3, we have the following corollary.
The space is dense in .
For any by the definition of , we have Define and let
for all This family of functions satisfies
By (a) and (b) of Lemma 4.3, the estimate (i) is straightforward since for each , Moreover, by (b) of Lemma 4.3, .
as This proves property (ii) and completes the proof of Corollary 4.4.
We next prove the following for functions in the space -estimate which will be useful in proving our Theorems 3.1 and 3.2 in Section 5. ,
For any -function supported on a ball with radius there exists a positive constant , such that
Assume that is a ball of radius and centered at One writes .
Note that for any Using Hölder’s inequality and the fact that the area integral function . is bounded on one obtains ,
We now estimate the term First, we will show that there exists a constant . such that for any ,
Let us verify (Equation4.11). Let
and Since we obtain
It follows from the estimate (Equation3.1) that the kernel of the operator satisfies
where is the positive constant in (Equation2.3). Therefore,
We only consider the term since the estimate of the term is even simpler. For and we set , where , For any . and we have ,
which implies hence ; Obviously, for any . and we also have , Note that .
It follows from elementary integration that
The estimate (Equation4.11) then follows readily. Therefore,
Combining the estimates of the terms I and II, we obtain that The proof of Lemma 4.5 is complete.
We now follow Theorem 2.14 of Reference16 to prove the implication (i) (ii) of Theorem 3.2. For the implication (ii) (i) of Theorem 3.2, we will present its proof in Section 5.3.
If BMO, then is a Carleson measure with .
We will prove that there exists a positive constant such that for any ball on ,