Duality of Hardy and BMO spaces associated with operators with heat kernel bounds
By Xuan Thinh Duong and Lixin Yan
Abstract
Let $L$ be the infinitesimal generator of an analytic semigroup on $L^2({\mathbb{R}}^n)$ with suitable upper bounds on its heat kernels. Auscher, Duong, and McIntosh defined a Hardy space $H_L^1$ by means of an area integral function associated with the operator $L$. By using a variant of the maximal function associated with the semigroup $\{e^{-tL}\}_{t\geq 0}$, a space ${\mathrm{BMO}}_L$ 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 $L$ has a bounded holomorphic functional calculus on $L^2({\mathbb{R}}^n)$, then the dual space of $H_L^1$ is ${\mathrm{BMO}}_{L^{\ast }}$ where $L^{\ast }$ is the adjoint operator of $L$. We then obtain a characterization of the space ${\mathrm{BMO}}_L$ in terms of the Carleson measure. We also discuss the dimensions of the kernel spaces ${\mathcal{K}}_L$ of BMO$_{ L}$ when $L$ is a second-order elliptic operator of divergence form and when $L$ is a Schrödinger operator, and study the inclusion between the classical BMO space and ${\mathrm{BMO}}_L$ spaces associated with operators.
1. Introduction
The introduction and development of Hardy and BMO spaces on Euclidean spaces ${\mathbb{R}}^n$ 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 $L^1$ function $f$ on ${\mathbb{R}}^n$ is in the Hardy space $H^1({\mathbb{R}}^n)$ if the area integral function of the Poisson integral $e^{-t\sqrt {\triangle }}f$ satisfies
There are a number of equivalent characterizations of functions in the $H^1$ space, including the all-important atomic decomposition (see Reference 21, Reference 31).
A locally integrable function $f$ defined on ${\mathbb{R}}^n$ is said to be in BMO, the space of functions of bounded mean oscillation, if
where the supremum is taken over all balls $B$ in ${\mathbb{R}}^n$, and $f_B$ stands for the mean of $f$ over $B$, i.e.,
$$f_B=|B|^{-1}\int _B f(y)dy.$$
In Reference 19, Fefferman and Stein showed that the space BMO is the dual space of the Hardy space $H^1$. They also obtained a characterization of the BMO space in terms of the Carleson measure, the $H^1$-$H^1$ boundedness of convolution operators which satisfy the Hörmander condition, and an interpolation theorem between $L^p$ spaces and the BMO space. From the viewpoint of Calderón-Zygmund operator theory, $H^1$ and BMO spaces are natural substitutes for $L^1$ and $L^{\infty }$ spaces, respectively.
Recently, Auscher, McIntosh and the first-named author introduced a class of Hardy spaces $H^1_L$ associated with an operator $L$ by means of the $L^1$ area integral functions in (Equation 1.1) in which the Poisson semigroup $e^{-t\sqrt {\triangle }}$ was replaced by the semigroup $e^{-tL}$ (Reference 4). They then obtained an $L$-molecular characterization for $H^1_L$ 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$_{ L}$ associated with an operator $L$ by using a maximal function introduced by Martell in Reference 25. Roughly speaking, if $L$ is the infinitesimal generator of an analytic semigroup $\{e^{-tL}\}_{t\geq 0}$ on $L^2$ with kernel $p_t(x,y)$ (which decays fast enough), we can view $P_tf= e^{-tL}f$ as an average version of $f$ (at the scale $t$) and use the quantity
to replace the mean value $f_B$ in the definition (Equation 1.2) of the classical BMO space, where $t_B$ is scaled to the radius of the ball $B.$ We then say that a function $f$ (with suitable bounds on growth) is in ${\mathrm{BMO}}_L$ if
See Section 3.2.2 below. We also studied and established a number of important features of the ${\mathrm{BMO}}_L$ space such as the John-Nirenberg inequality and complex interpolation (Reference 16, Section 3). Note that the spaces $H^1_{\sqrt {\triangle }}$ and ${\mathrm{BMO}}_{\sqrt {\triangle }}$ 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 $H^1$ and BMO spaces. We will show that if $L$ has a bounded holomorphic functional calculus on $L^2$ and the kernel $p_t(x,y)$ of the operator $P_t$ in (Equation 1.3) satisfies an upper bound of Poisson type, then the space ${\mathrm{BMO}}_{L^{\ast }}$ is the dual space of the Hardy space $H^{\mathrm{1}}_{ L}$ in which $L^{\ast }$ denotes the adjoint operator of $L.$ We also obtain a characterization of functions in ${\mathrm{BMO}}_L$ in terms of the Carleson measure. See Theorems 3.1 and 3.2 below.
We note that a valid choice of $P_t$ in (Equation 1.3) is the Poisson integral $P_tf=e^{-t\sqrt {\triangle }}f$, which is defined by
For this choice of $P_t$, Theorems 3.1 and 3.2 of this article give the classical results of Theorem 2 and the equivalence (i)$\Leftrightarrow$ (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 $p_t(x,y)$ of $P_t$ in (Equation 1.3) and no regularities on the space variables $x$ or $y$. Another feature of our result is that we do not assume the conservation property of the semigroup $P_t(1)=1$ for $t>0$. This allows our method to be applicable to a large class of operators $L$.
The paper is organised as follows. In Section 2 we will give some preliminaries on holomorphic functional calculi of operators and on integral operators $P_t$ with kernels $p_t(x,y)$ satisfying upper bounds of Poisson type. In Section 3 we introduce and describe the assumptions of the operator $L$ in this paper, and recall the definitions of $H^{\mathrm{1}}_{ L}$ and BMO$_{ L}$ spaces as in Reference 4 and Reference 16. We then state our main result, Theorem 3.1, which says that the dual space of $H^1_L$ is ${\mathrm{BMO}}_{L^{\ast }}$. In Section 4 we prove a number of important estimates for functions in $H^{\mathrm{1}}_{ L}$ and BMO$_{ L}$ 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 ${\mathcal{K}}_L$ of BMO$_{ L}$ when $L$ is a second-order elliptic operator of divergence form and when $L$ is a Schrödinger operator. We conclude this article with a study of inclusion between the classical BMO space and ${\mathrm{BMO}}_L$ spaces associated with some differential operators, including a sufficient condition for the classical BMO and ${\mathrm{BMO}}_L$ spaces to coincide.
Throughout this paper, the letter “$c$” 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 $0\leq \omega <\nu <\pi$. We define the closed sector in the complex plane ${\mathbb{C}}$ by
Let $0\leq \omega <\pi$. A closed operator $L$ in $L^2({\mathbb{R}}^n)$ is said to be of type $\omega$ if $\sigma (L)\subset S_{\omega }$, and for each $\nu >\omega ,$ there exists a constant $c_{\nu }$ such that
where $\Gamma$ is the contour $\{\xi =re^{\pm i\theta }: r\geq 0\}$ parametrized clockwise around $S_{\omega }$, and $\omega <\theta <\nu$. Clearly, this integral is absolutely convergent in ${\mathcal{L}}(L^2, L^2)$, and it is straightforward to show, using Cauchy’s theorem, that the definition is independent of the choice of $\theta \in (\omega , \nu ).$ If, in addition, $L$ is one-one and has dense range and if $b\in H_{\infty }(S^0_{\nu })$, then $b(L)$ can be defined by
where $\psi (z) =z(1+z)^{-2}$. It can be shown that $b(L)$ is a well-defined linear operator in $L^2({\mathbb{R}}^n)$. We say that $L$ has a bounded $H_{\infty }$ calculus on $L^2$ if there exists $c_{\nu ,2}>0$ such that $b(L)\in {\mathcal{L}}(L^2, L^2)$, and for $b\in H_{\infty }(S^0_{\nu })$,
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 $\{P_t\}_{t>0}$, which plays the role of generalized approximations to the identity. We assume that for each $t>0$, the operator $P_t$ is defined by its kernel $p_t(x,y)$ in the sense that
$$P_tf(x)=\int _{{\mathbb{R}}^n}p_t(x,y)f(y)dy$$
for every function $f$ which satisfies the growth condition (Equation 3.3) in Section 3.1 below.
We also assume that the kernel $p_t(x,y)$ of $P_t$ satisfies a Poisson bound of order $m>0:$
where the sup is taken over all balls containing $x$. It is well known that the Hardy-Littlewood maximal operator is bounded on $L^r$ for all $r\in (1,\infty ]$. Because of the decay of the kernel $p_t(x,y)$ in (Equation 2.2) and (Equation 2.3), one has
Proposition 2.1.
There exists a constant $c>0$ such that for any $f\in L^r$,$1\leq r\leq \infty ,$ we have
3. Duality between $H^1_L$ and ${\mathrm{BMO}}_{L^{\ast }}$ spaces
In this section, we will give the framework and the main result of this paper.
3.1. Assumptions and notation
Let $L$ be a linear operator of type $\omega$ on $L^2({\mathbb{R}}^n)$ with $\omega <\pi /2$; hence $L$ generates a holomorphic semigroup $e^{-zL}$,$0\leq |{\mathrm{Arg}}(z)|<\pi /2-\omega$. Assume the following two conditions.
Assumption (a).
The holomorphic semigroup $e^{-zL}$,$|{\mathrm{Arg}}(z)|<\pi /2-\omega$, is represented by the kernel $p_z(x,y)$ which satisfies the upper bound
$$|p_z(x,y)|\leq c_{\theta }h_{|z|}(x,y)$$
for $x,y\in {\mathbb{R}}^n$,$|{\mathrm{Arg}}(z)|<\pi /2-\theta$ for $\theta >\omega ,$ and $h_t$ is defined on ${\mathbb{R}}^n\times {\mathbb{R}}^n$ by (Equation 2.2).
Assumption (b).
The operator $L$ has a bounded $H_{\infty }$-calculus on $L^2({\mathbb{R}}^n)$. That is, there exists $c_{\nu ,2}>0$ such that $b(L)\in {\mathcal{L}}(L^2, L^2)$, and for $b\in H_{\infty }(S^0_{\nu }):$
We now give some consequences of assumptions (a) and (b) which will be useful in the sequel.
(i) If $\{e^{-tL}\}_{t\geq 0}$ is a bounded analytic semigroup on $L^2({\mathbb{R}}^n)$ whose kernel $p_t(x,y)$ satisfies the estimate (Equation 2.2), then for all $k\in {\mathbb{N}}$, the time derivatives of $p_t$ satisfy
for all $t>0$ and almost all $x,y\in {\mathbb{R}}^n$. For each $k\in {\mathbb{N}}$, the function $s$ might depend on $k$ but it always satisfies (Equation 2.3). See Lemma 2.5 of Reference 5.
(ii) $L$ has a bounded $H_{\infty }$-calculus on $L^2({\mathbb{R}}^n)$ if and only if for any non-zero function $\psi \in \Psi (S^0_{\nu })$,$L$ satisfies the square function estimate and its reverse
for some $0<c_1\leq c_2<\infty$, where $\psi _t(\xi )=\psi (t\xi )$. Note that different choices of $\nu >\omega$ and $\psi \in \Psi (S^0_{\nu })$ lead to equivalent quadratic norms of $f.$ 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 $L$ has a bounded holomorphic functional calculus on $L^p({\mathbb{R}}^n)$,$1<p<\infty$; that is, there exists $c_{\nu , p}>0$ such that $b(L)\in {\mathcal{L}}(L^p, L^p)$, and for $b\in H_{\infty }(S^0_{\nu })$:
for any $f\in L^p({\mathbb{R}}^n)$. For $p=1$, the operator $b(L)$ is of weak-type $(1,1)$. In Reference 16, it was proved that for $p=\infty$, the operator $b(L)$ is bounded from $L^{\infty }$ into ${\mathrm{BMO}}_L$.
We now define the class of functions that the operators $P_t$ act upon. For any $\beta >0$, a function $f\in L^2_{\mathrm{loc}}({\mathbb{R}}^n)$ is said to be a function of $\beta$-type if $f$ satisfies
We denote by ${\mathcal{M}}_{\beta }$ the collection of all functions of $\beta$-type. If $f\in {\mathcal{M}}_{\beta },$ the norm of $f$ in ${\mathcal{M}}_{\beta }$ is denoted by
It is easy to see that ${\mathcal{M}}_{\beta }$ is a Banach space under the norm $\|f\|_{{\mathcal{M}}_{\beta }}.$ Note that we use $L^2_{\mathrm{loc}}({\mathbb{R}}^n)$ instead of the space $L^1_{\mathrm{loc}}({\mathbb{R}}^n)$ as in Reference 19 and Reference 16 since this gives the appropriate setting for the duality between $H^{\mathrm{1}}_{ L}$ and BMO$_{ L}$. For any given operator $L$, we let ${\Theta }(L)=\sup \big \{\epsilon >0: (\xhref[disp-formula]{#texmlid6}{2.3})\ {\mathrm{holds}} \ \big \},$ and define