Duality of Hardy and BMO spaces associated with operators with heat kernel bounds

By Xuan Thinh Duong and Lixin Yan

Abstract

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.

1. Introduction

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 Reference 32, Reference 22, Reference 18, Reference 19, Reference 31 and many others.

An function on is in the Hardy space if the area integral function of the Poisson integral satisfies

There are a number of equivalent characterizations of functions in the space, including the all-important atomic decomposition (see Reference 21, Reference 31).

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 Reference 19, 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 (Equation 1.1) in which the Poisson semigroup was replaced by the semigroup (Reference 4). They then obtained an -molecular characterization for by using the theory of tent spaces developed by Coifman, Meyer and Stein (Reference 7, Reference 8 and Reference 4). See also Sections 3.2.1 and 4.1 below. In Reference 16, we introduced and developed a new function space BMO associated with an operator by using a maximal function introduced by Martell in Reference 25. 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 (Equation 1.2) 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 (Reference 16, Section 3). Note that the spaces and coincide with the classical Hardy and BMO spaces, respectively (Reference 16, 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 (Equation 1.3) 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 (Equation 1.3) 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 Reference 19, respectively. See also Chapter IV of Reference 31.

Note that in our main result, Theorem 3.1, we assume only an upper bound on the kernel of in (Equation 1.3) 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 Reference 4 and Reference 16. 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.

2. Preliminaries

We first give some preliminary definitions of holomorphic functional calculi as introduced by McIntosh Reference 26.

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 Reference 6.

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 (Equation 3.3) 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 Reference 14.

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 (Equation 2.2) and (Equation 2.3), one has

Proposition 2.1.

There exists a constant such that for any , we have

for all .

Proof.

This is a consequence of the conditions (Equation 2.2), (Equation 2.3) and the definition of See Reference 15, Proposition 2.4.

3. Duality between and spaces

In this section, we will give the framework and the main result of this paper.

3.1. Assumptions and notation

Let be a linear operator of type on with ; hence generates a holomorphic semigroup , . Assume the following two conditions.

Assumption (a).

The holomorphic semigroup , , is represented by the kernel which satisfies the upper bound

for , for and is defined on by (Equation 2.2).

Assumption (b).

The operator has a bounded -calculus on . 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 (Equation 2.2), then for all , the time derivatives of satisfy

for all and almost all . For each , the function might depend on but it always satisfies (Equation 2.3). See Lemma 2.5 of Reference 5.

(ii) has a bounded -calculus on 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 Reference 26.

As noted in Reference 26, positive self-adjoint operators satisfy the quadratic estimate (Equation 3.2), 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 Reference 36.

(iii) Under the assumptions (a) and (b), it was proved in Theorem 3.1 of Reference 15 and Theorem 6 of Reference 14 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 Reference 16, 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 -type if satisfies

We denote by the collection of all functions of -type. If 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 Reference 19 and Reference 16 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