American Mathematical Society

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

By Xuan Thinh Duong and Lixin Yan

Abstract

Let upper L be the infinitesimal generator of an analytic semigroup on upper L squared left-parenthesis double-struck upper R Superscript n Baseline right-parenthesis with suitable upper bounds on its heat kernels. Auscher, Duong, and McIntosh defined a Hardy space upper H Subscript upper L Superscript 1 by means of an area integral function associated with the operator upper L . By using a variant of the maximal function associated with the semigroup StartSet e Superscript minus t upper L Baseline EndSet Subscript t greater-than-or-equal-to 0 , a space normal upper B normal upper M normal upper O Subscript upper 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 upper L has a bounded holomorphic functional calculus on upper L squared left-parenthesis double-struck upper R Superscript n Baseline right-parenthesis , then the dual space of upper H Subscript upper L Superscript 1 is normal upper B normal upper M normal upper O Subscript upper L Sub Superscript asterisk where upper L Superscript asterisk is the adjoint operator of upper L . We then obtain a characterization of the space normal upper B normal upper M normal upper O Subscript upper L in terms of the Carleson measure. We also discuss the dimensions of the kernel spaces script upper K Subscript upper L of BMO Subscript upper L when upper L is a second-order elliptic operator of divergence form and when upper L is a Schrödinger operator, and study the inclusion between the classical BMO space and normal upper B normal upper M normal upper O Subscript upper L spaces associated with operators.

1. Introduction

The introduction and development of Hardy and BMO spaces on Euclidean spaces double-struck upper R Superscript 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 Reference32, Reference22, Reference18, Reference19, Reference31 and many others.

An upper L Superscript 1 function f on double-struck upper R Superscript n is in the Hardy space upper H Superscript 1 Baseline left-parenthesis double-struck upper R Superscript n Baseline right-parenthesis if the area integral function of the Poisson integral e Superscript minus t StartRoot white up pointing triangle EndRoot Baseline f satisfies

StartLayout 1st Row with Label left-parenthesis 1.1 right-parenthesis EndLabel script upper S left-parenthesis f right-parenthesis left-parenthesis x right-parenthesis equals left-parenthesis integral Subscript 0 Superscript normal infinity Baseline integral Underscript StartAbsoluteValue y minus x EndAbsoluteValue less-than t Endscripts StartAbsoluteValue StartFraction partial-differential Over partial-differential t EndFraction e Superscript minus t StartRoot white up pointing triangle EndRoot Baseline f left-parenthesis y right-parenthesis EndAbsoluteValue squared t Superscript 1 minus n Baseline d y d t right-parenthesis Superscript 1 slash 2 Baseline element-of upper L Superscript 1 Baseline left-parenthesis double-struck upper R Superscript n Baseline right-parenthesis period EndLayout

There are a number of equivalent characterizations of functions in the upper H Superscript 1 space, including the all-important atomic decomposition (see Reference21, Reference31).

A locally integrable function f defined on double-struck upper R Superscript n is said to be in BMO, the space of functions of bounded mean oscillation, if

StartLayout 1st Row with Label left-parenthesis 1.2 right-parenthesis EndLabel double-vertical-bar f double-vertical-bar Subscript normal upper B normal upper M normal upper O Baseline equals sup Underscript upper B Endscripts StartFraction 1 Over StartAbsoluteValue upper B EndAbsoluteValue EndFraction integral Underscript upper B Endscripts StartAbsoluteValue f left-parenthesis y right-parenthesis minus f Subscript upper B Baseline EndAbsoluteValue d y less-than normal infinity comma EndLayout

where the supremum is taken over all balls upper B in double-struck upper R Superscript n , and f Subscript upper B stands for the mean of f over upper B , i.e.,

f Subscript upper B Baseline equals StartAbsoluteValue upper B EndAbsoluteValue Superscript negative 1 Baseline integral Underscript upper B Endscripts f left-parenthesis y right-parenthesis d y period

In Reference19, Fefferman and Stein showed that the space BMO is the dual space of the Hardy space upper H Superscript 1 . They also obtained a characterization of the BMO space in terms of the Carleson measure, the upper H Superscript 1 - upper H Superscript 1 boundedness of convolution operators which satisfy the Hörmander condition, and an interpolation theorem between upper L Superscript p spaces and the BMO space. From the viewpoint of Calderón-Zygmund operator theory, upper H Superscript 1 and BMO spaces are natural substitutes for upper L Superscript 1 and upper L Superscript normal infinity spaces, respectively.

Recently, Auscher, McIntosh and the first-named author introduced a class of Hardy spaces upper H Subscript upper L Superscript 1 associated with an operator upper L by means of the upper L Superscript 1 area integral functions in (Equation1.1) in which the Poisson semigroup e Superscript minus t StartRoot white up pointing triangle EndRoot was replaced by the semigroup e Superscript minus t upper L (Reference4). They then obtained an upper L -molecular characterization for upper H Subscript upper L Superscript 1 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 Subscript upper L associated with an operator upper L by using a maximal function introduced by Martell in Reference25. Roughly speaking, if upper L is the infinitesimal generator of an analytic semigroup StartSet e Superscript minus t upper L Baseline EndSet Subscript t greater-than-or-equal-to 0 on upper L squared with kernel p Subscript t Baseline left-parenthesis x comma y right-parenthesis (which decays fast enough), we can view upper P Subscript t Baseline f equals e Superscript minus t upper L Baseline f as an average version of f (at the scale t ) and use the quantity

StartLayout 1st Row with Label left-parenthesis 1.3 right-parenthesis EndLabel upper P Subscript t Sub Subscript upper B Baseline f left-parenthesis x right-parenthesis equals integral Underscript double-struck upper R Superscript n Baseline Endscripts p Subscript t Sub Subscript upper B Baseline left-parenthesis x comma y right-parenthesis f left-parenthesis y right-parenthesis d y EndLayout

to replace the mean value f Subscript upper B in the definition (Equation1.2) of the classical BMO space, where t Subscript upper B is scaled to the radius of the ball upper B period We then say that a function f (with suitable bounds on growth) is in normal upper B normal upper M normal upper O Subscript upper L if

StartLayout 1st Row sup Underscript upper B Endscripts StartFraction 1 Over StartAbsoluteValue upper B EndAbsoluteValue EndFraction integral Underscript upper B Endscripts StartAbsoluteValue f left-parenthesis x right-parenthesis minus upper P Subscript t Sub Subscript upper B Subscript Baseline f left-parenthesis x right-parenthesis EndAbsoluteValue d x less-than normal infinity period EndLayout

See Section 3.2.2 below. We also studied and established a number of important features of the normal upper B normal upper M normal upper O Subscript upper L space such as the John-Nirenberg inequality and complex interpolation (Reference16, Section 3). Note that the spaces upper H Subscript StartRoot white up pointing triangle EndRoot Superscript 1 and normal upper B normal upper M normal upper O Subscript StartRoot white up pointing triangle EndRoot 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 upper H Superscript 1 and BMO spaces. We will show that if upper L has a bounded holomorphic functional calculus on upper L squared and the kernel p Subscript t Baseline left-parenthesis x comma y right-parenthesis of the operator upper P Subscript t in (Equation1.3) satisfies an upper bound of Poisson type, then the space normal upper B normal upper M normal upper O Subscript upper L Sub Superscript asterisk is the dual space of the Hardy space upper H Subscript upper L Superscript 1 in which upper L Superscript asterisk denotes the adjoint operator of upper L period We also obtain a characterization of functions in normal upper B normal upper M normal upper O Subscript upper L in terms of the Carleson measure. See Theorems 3.1 and 3.2 below.

We note that a valid choice of upper P Subscript t in (Equation1.3) is the Poisson integral upper P Subscript t Baseline f equals e Superscript minus t StartRoot white up pointing triangle EndRoot Baseline f , which is defined by

upper P Subscript t Baseline f left-parenthesis x right-parenthesis equals integral Underscript double-struck upper R Superscript n Baseline Endscripts p Subscript t Baseline left-parenthesis x minus y right-parenthesis f left-parenthesis y right-parenthesis d y comma t greater-than 0 comma normal w normal h normal e normal r normal e p Subscript t Baseline left-parenthesis x right-parenthesis equals StartFraction c Subscript n Baseline t Over left-parenthesis t squared plus StartAbsoluteValue x EndAbsoluteValue squared right-parenthesis Superscript left-parenthesis n plus 1 right-parenthesis slash 2 Baseline EndFraction period

For this choice of upper P Subscript t , Theorems 3.1 and 3.2 of this article give the classical results of Theorem 2 and the equivalence (i) left right double arrow (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 p Subscript t Baseline left-parenthesis x comma y right-parenthesis of upper P Subscript t in (Equation1.3) 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 upper P Subscript t Baseline left-parenthesis 1 right-parenthesis equals 1 for t greater-than 0 . This allows our method to be applicable to a large class of operators upper 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 upper P Subscript t with kernels p Subscript t Baseline left-parenthesis x comma y right-parenthesis satisfying upper bounds of Poisson type. In Section 3 we introduce and describe the assumptions of the operator upper L in this paper, and recall the definitions of upper H Subscript upper L Superscript 1 and BMO Subscript upper L spaces as in Reference4 and Reference16. We then state our main result, Theorem 3.1, which says that the dual space of upper H Subscript upper L Superscript 1 is normal upper B normal upper M normal upper O Subscript upper L Sub Superscript asterisk . In Section 4 we prove a number of important estimates for functions in upper H Subscript upper L Superscript 1 and BMO Subscript upper 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 script upper K Subscript upper L of BMO Subscript upper L when upper L is a second-order elliptic operator of divergence form and when upper L is a Schrödinger operator. We conclude this article with a study of inclusion between the classical BMO space and normal upper B normal upper M normal upper O Subscript upper L spaces associated with some differential operators, including a sufficient condition for the classical BMO and normal upper B normal upper M normal upper O Subscript upper 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 Reference26.

Let 0 less-than-or-equal-to omega less-than nu less-than pi . We define the closed sector in the complex plane double-struck upper C by

upper S Subscript omega Baseline equals StartSet z element-of double-struck upper C colon StartAbsoluteValue normal a normal r normal g z EndAbsoluteValue less-than-or-equal-to omega EndSet union StartSet 0 EndSet

and denote the interior of upper S Subscript omega by upper S Subscript omega Superscript 0 .

We employ the following subspaces of the space upper H left-parenthesis upper S Subscript nu Superscript 0 Baseline right-parenthesis of all holomorphic functions on upper S Subscript nu Superscript 0 :

upper H Subscript normal infinity Baseline left-parenthesis upper S Subscript nu Superscript 0 Baseline right-parenthesis equals StartSet b element-of upper H left-parenthesis upper S Subscript nu Superscript 0 Baseline right-parenthesis colon StartAbsoluteValue EndAbsoluteValue b StartAbsoluteValue EndAbsoluteValue Subscript normal infinity Baseline less-than normal infinity EndSet comma

where StartAbsoluteValue EndAbsoluteValue b StartAbsoluteValue EndAbsoluteValue Subscript normal infinity Baseline equals normal s normal u normal p StartSet StartAbsoluteValue b left-parenthesis z right-parenthesis EndAbsoluteValue colon z element-of upper S Subscript nu Superscript 0 Baseline EndSet , and

normal upper Psi left-parenthesis upper S Subscript nu Superscript 0 Baseline right-parenthesis equals StartSet psi element-of upper H left-parenthesis upper S Subscript nu Superscript 0 Baseline right-parenthesis colon there-exists s greater-than 0 comma StartAbsoluteValue psi left-parenthesis z right-parenthesis EndAbsoluteValue less-than-or-equal-to c StartAbsoluteValue z EndAbsoluteValue Superscript s Baseline left-parenthesis 1 plus StartAbsoluteValue z EndAbsoluteValue Superscript 2 s Baseline right-parenthesis Superscript negative 1 Baseline EndSet period

Let 0 less-than-or-equal-to omega less-than pi . A closed operator upper L in upper L squared left-parenthesis double-struck upper R Superscript n Baseline right-parenthesis is said to be of type omega if sigma left-parenthesis upper L right-parenthesis subset-of upper S Subscript omega , and for each nu greater-than omega comma there exists a constant c Subscript nu such that

double-vertical-bar left-parenthesis upper L minus lamda script upper I right-parenthesis Superscript negative 1 Baseline double-vertical-bar less-than-or-equal-to c Subscript nu Baseline StartAbsoluteValue lamda EndAbsoluteValue Superscript negative 1 Baseline comma lamda not-an-element-of upper S Subscript nu Baseline period

If upper L is of type omega and psi element-of normal upper Psi left-parenthesis upper S Subscript nu Superscript 0 Baseline right-parenthesis , we define psi left-parenthesis upper L right-parenthesis element-of script upper L left-parenthesis upper L squared comma upper L squared right-parenthesis by

StartLayout 1st Row with Label left-parenthesis 2.1 right-parenthesis EndLabel psi left-parenthesis upper L right-parenthesis equals StartFraction 1 Over 2 pi i EndFraction integral Underscript normal upper Gamma Endscripts left-parenthesis upper L minus lamda script upper I right-parenthesis Superscript negative 1 Baseline psi left-parenthesis lamda right-parenthesis d lamda comma EndLayout

where normal upper Gamma is the contour StartSet xi equals r e Superscript plus-or-minus i theta Baseline colon r greater-than-or-equal-to 0 EndSet parametrized clockwise around upper S Subscript omega , and omega less-than theta less-than nu . Clearly, this integral is absolutely convergent in script upper L left-parenthesis upper L squared comma upper L squared right-parenthesis , and it is straightforward to show, using Cauchy’s theorem, that the definition is independent of the choice of theta element-of left-parenthesis omega comma nu right-parenthesis period If, in addition, upper L is one-one and has dense range and if b element-of upper H Subscript normal infinity Baseline left-parenthesis upper S Subscript nu Superscript 0 Baseline right-parenthesis , then b left-parenthesis upper L right-parenthesis can be defined by

StartLayout 1st Row b left-parenthesis upper L right-parenthesis equals left-bracket psi left-parenthesis upper L right-parenthesis right-bracket Superscript negative 1 Baseline left-parenthesis b psi right-parenthesis left-parenthesis upper L right-parenthesis comma EndLayout

where psi left-parenthesis z right-parenthesis equals z left-parenthesis 1 plus z right-parenthesis Superscript negative 2 . It can be shown that b left-parenthesis upper L right-parenthesis is a well-defined linear operator in upper L squared left-parenthesis double-struck upper R Superscript n Baseline right-parenthesis . We say that upper L has a bounded upper H Subscript normal infinity calculus on upper L squared if there exists c Subscript nu comma 2 Baseline greater-than 0 such that b left-parenthesis upper L right-parenthesis element-of script upper L left-parenthesis upper L squared comma upper L squared right-parenthesis , and for b element-of upper H Subscript normal infinity Baseline left-parenthesis upper S Subscript nu Superscript 0 Baseline right-parenthesis ,

StartLayout 1st Row StartAbsoluteValue EndAbsoluteValue b left-parenthesis upper L right-parenthesis StartAbsoluteValue EndAbsoluteValue less-than-or-equal-to c Subscript nu comma 2 Baseline StartAbsoluteValue EndAbsoluteValue b StartAbsoluteValue EndAbsoluteValue Subscript normal infinity Baseline period EndLayout

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 StartSet upper P Subscript t Baseline EndSet Subscript t greater-than 0 , which plays the role of generalized approximations to the identity. We assume that for each t greater-than 0 , the operator upper P Subscript t is defined by its kernel p Subscript t Baseline left-parenthesis x comma y right-parenthesis in the sense that

upper P Subscript t Baseline f left-parenthesis x right-parenthesis equals integral Underscript double-struck upper R Superscript n Baseline Endscripts p Subscript t Baseline left-parenthesis x comma y right-parenthesis f left-parenthesis y right-parenthesis d y

for every function f which satisfies the growth condition (Equation3.3) in Section 3.1 below.

We also assume that the kernel p Subscript t Baseline left-parenthesis x comma y right-parenthesis of upper P Subscript t satisfies a Poisson bound of order m greater-than 0 colon

StartLayout 1st Row with Label left-parenthesis 2.2 right-parenthesis EndLabel StartAbsoluteValue p Subscript t Baseline left-parenthesis x comma y right-parenthesis EndAbsoluteValue less-than-or-equal-to h Subscript t Baseline left-parenthesis x comma y right-parenthesis equals t Superscript negative n slash m Baseline s left-parenthesis StartFraction StartAbsoluteValue x minus y EndAbsoluteValue Over t Superscript 1 slash m Baseline EndFraction right-parenthesis comma EndLayout

in which s is a positive, bounded, decreasing function satisfying

StartLayout 1st Row with Label left-parenthesis 2.3 right-parenthesis EndLabel limit Underscript r right-arrow normal infinity Endscripts r Superscript n plus epsilon Baseline s left-parenthesis r right-parenthesis equals 0 EndLayout

for some epsilon greater-than 0 .

It is easy to check that there exists a constant c greater-than 0 such that h Subscript t Baseline left-parenthesis x comma y right-parenthesis satisfies

c Superscript negative 1 Baseline less-than-or-equal-to integral Underscript double-struck upper R Superscript n Baseline Endscripts h Subscript t Baseline left-parenthesis x comma y right-parenthesis d x less-than-or-equal-to c normal a normal n normal d c Superscript negative 1 Baseline less-than-or-equal-to integral Underscript double-struck upper R Superscript n Baseline Endscripts h Subscript t Baseline left-parenthesis y comma x right-parenthesis d x less-than-or-equal-to c

uniformly in y element-of double-struck upper R Superscript n , t greater-than 0 . See Section 2 of Reference14.

We recall that the Hardy-Littlewood maximal operator upper M f is defined by

upper M f left-parenthesis x right-parenthesis equals sup Underscript x element-of upper B Endscripts StartFraction 1 Over StartAbsoluteValue upper B EndAbsoluteValue EndFraction integral Underscript upper B Endscripts StartAbsoluteValue f left-parenthesis y right-parenthesis EndAbsoluteValue d y comma

where the sup is taken over all balls containing x . It is well known that the Hardy-Littlewood maximal operator is bounded on upper L Superscript r for all r element-of left-parenthesis 1 comma normal infinity right-bracket . Because of the decay of the kernel p Subscript t Baseline left-parenthesis x comma y right-parenthesis in (Equation2.2) and (Equation2.3), one has

Proposition 2.1

There exists a constant c greater-than 0 such that for any f element-of upper L Superscript r , 1 less-than-or-equal-to r less-than-or-equal-to normal infinity comma we have

StartAbsoluteValue upper P Subscript t Baseline f left-parenthesis x right-parenthesis EndAbsoluteValue less-than-or-equal-to integral Underscript double-struck upper R Superscript n Baseline Endscripts h Subscript t Baseline left-parenthesis x comma y right-parenthesis StartAbsoluteValue f left-parenthesis y right-parenthesis EndAbsoluteValue d y less-than-or-equal-to c upper M f left-parenthesis x right-parenthesis

for all t greater-than 0 .

Proof.

This is a consequence of the conditions (Equation2.2), (Equation2.3) and the definition of upper M f period See Reference15, Proposition 2.4.

3. Duality between upper H Subscript upper L Superscript 1 and normal upper B normal upper M normal upper O Subscript upper L Sub Superscript asterisk spaces

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

3.1. Assumptions and notation

Let upper L be a linear operator of type omega on upper L squared left-parenthesis double-struck upper R Superscript n Baseline right-parenthesis with omega less-than pi slash 2 ; hence upper L generates a holomorphic semigroup e Superscript minus z upper L , 0 less-than-or-equal-to StartAbsoluteValue normal upper A normal r normal g left-parenthesis z right-parenthesis EndAbsoluteValue less-than pi slash 2 minus omega . Assume the following two conditions.

Assumption (a)

The holomorphic semigroup e Superscript minus z upper L , StartAbsoluteValue normal upper A normal r normal g left-parenthesis z right-parenthesis EndAbsoluteValue less-than pi slash 2 minus omega , is represented by the kernel p Subscript z Baseline left-parenthesis x comma y right-parenthesis which satisfies the upper bound

StartAbsoluteValue p Subscript z Baseline left-parenthesis x comma y right-parenthesis EndAbsoluteValue less-than-or-equal-to c Subscript theta Baseline h Subscript StartAbsoluteValue z EndAbsoluteValue Baseline left-parenthesis x comma y right-parenthesis

for x comma y element-of double-struck upper R Superscript n , StartAbsoluteValue normal upper A normal r normal g left-parenthesis z right-parenthesis EndAbsoluteValue less-than pi slash 2 minus theta for theta greater-than omega comma and h Subscript t is defined on double-struck upper R Superscript n Baseline times double-struck upper R Superscript n by (Equation2.2).

Assumption (b)

The operator upper L has a bounded upper H Subscript normal infinity -calculus on upper L squared left-parenthesis double-struck upper R Superscript n Baseline right-parenthesis . That is, there exists c Subscript nu comma 2 Baseline greater-than 0 such that b left-parenthesis upper L right-parenthesis element-of script upper L left-parenthesis upper L squared comma upper L squared right-parenthesis , and for b element-of upper H Subscript normal infinity Baseline left-parenthesis upper S Subscript nu Superscript 0 Baseline right-parenthesis colon

StartLayout 1st Row StartAbsoluteValue EndAbsoluteValue b left-parenthesis upper L right-parenthesis f StartAbsoluteValue EndAbsoluteValue Subscript 2 Baseline less-than-or-equal-to c Subscript nu comma 2 Baseline StartAbsoluteValue EndAbsoluteValue b StartAbsoluteValue EndAbsoluteValue Subscript normal infinity Baseline double-vertical-bar f double-vertical-bar Subscript 2 EndLayout

for any f element-of upper L squared left-parenthesis double-struck upper R Superscript n Baseline right-parenthesis .

We now give some consequences of assumptions (a) and (b) which will be useful in the sequel.

(i) If StartSet e Superscript minus t upper L Baseline EndSet Subscript t greater-than-or-equal-to 0 is a bounded analytic semigroup on upper L squared left-parenthesis double-struck upper R Superscript n Baseline right-parenthesis whose kernel p Subscript t Baseline left-parenthesis x comma y right-parenthesis satisfies the estimate (Equation2.2), then for all k element-of double-struck upper N , the time derivatives of p Subscript t satisfy

StartLayout 1st Row with Label left-parenthesis 3.1 right-parenthesis EndLabel StartAbsoluteValue StartFraction partial-differential Superscript k Baseline p Subscript t Baseline Over partial-differential t Superscript k Baseline EndFraction left-parenthesis x comma y right-parenthesis EndAbsoluteValue less-than-or-equal-to c t Superscript minus StartFraction n plus k m Over m EndFraction Baseline s left-parenthesis StartFraction StartAbsoluteValue x minus y EndAbsoluteValue Over t Superscript 1 slash m Baseline EndFraction right-parenthesis EndLayout

for all t greater-than 0 and almost all x comma y element-of double-struck upper R Superscript n . For each k element-of double-struck upper N , the function s might depend on k but it always satisfies (Equation2.3). See Lemma 2.5 of Reference5.

(ii) upper L has a bounded upper H Subscript normal infinity -calculus on upper L squared left-parenthesis double-struck upper R Superscript n Baseline right-parenthesis if and only if for any non-zero function psi element-of normal upper Psi left-parenthesis upper S Subscript nu Superscript 0 Baseline right-parenthesis , upper L satisfies the square function estimate and its reverse

StartLayout 1st Row with Label left-parenthesis 3.2 right-parenthesis EndLabel c 1 double-vertical-bar f double-vertical-bar Subscript 2 Baseline less-than-or-equal-to left-parenthesis integral Subscript 0 Superscript normal infinity Baseline double-vertical-bar psi Subscript t Baseline left-parenthesis upper L right-parenthesis f double-vertical-bar Subscript 2 Superscript 2 Baseline StartFraction d t Over t EndFraction right-parenthesis Superscript 1 slash 2 Baseline less-than-or-equal-to c 2 double-vertical-bar f double-vertical-bar Subscript 2 EndLayout

for some 0 less-than c 1 less-than-or-equal-to c 2 less-than normal infinity , where psi Subscript t Baseline left-parenthesis xi right-parenthesis equals psi left-parenthesis t xi right-parenthesis . Note that different choices of nu greater-than omega and psi element-of normal upper Psi left-parenthesis upper S Subscript nu Superscript 0 Baseline right-parenthesis lead to equivalent quadratic norms of f period 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 upper L has a bounded holomorphic functional calculus on upper L Superscript p Baseline left-parenthesis double-struck upper R Superscript n Baseline right-parenthesis , 1 less-than p less-than normal infinity ; that is, there exists c Subscript nu comma p Baseline greater-than 0 such that b left-parenthesis upper L right-parenthesis element-of script upper L left-parenthesis upper L Superscript p Baseline comma upper L Superscript p Baseline right-parenthesis , and for b element-of upper H Subscript normal infinity Baseline left-parenthesis upper S Subscript nu Superscript 0 Baseline right-parenthesis :

double-vertical-bar b left-parenthesis upper L right-parenthesis f double-vertical-bar Subscript p Baseline less-than-or-equal-to c Subscript nu comma p Baseline StartAbsoluteValue EndAbsoluteValue b StartAbsoluteValue EndAbsoluteValue Subscript normal infinity Baseline double-vertical-bar f double-vertical-bar Subscript p

for any f element-of upper L Superscript p Baseline left-parenthesis double-struck upper R Superscript n Baseline right-parenthesis . For p equals 1 , the operator b left-parenthesis upper L right-parenthesis is of weak-type left-parenthesis 1 comma 1 right-parenthesis . In Reference16, it was proved that for p equals normal infinity , the operator b left-parenthesis upper L right-parenthesis is bounded from upper L Superscript normal infinity into normal upper B normal upper M normal upper O Subscript upper L .

We now define the class of functions that the operators upper P Subscript t act upon. For any beta greater-than 0 , a function f element-of upper L Subscript normal l normal o normal c Superscript 2 Baseline left-parenthesis double-struck upper R Superscript n Baseline right-parenthesis is said to be a function of beta -type if f satisfies

StartLayout 1st Row with Label left-parenthesis 3.3 right-parenthesis EndLabel left-parenthesis integral Underscript double-struck upper R Superscript n Baseline Endscripts StartFraction StartAbsoluteValue f left-parenthesis x right-parenthesis EndAbsoluteValue squared Over 1 plus StartAbsoluteValue x EndAbsoluteValue Superscript n plus beta Baseline EndFraction d x right-parenthesis Superscript 1 slash 2 Baseline less-than-or-equal-to c less-than normal infinity period EndLayout

We denote by script upper M Subscript beta the collection of all functions of beta -type. If f element-of script upper M Subscript beta Baseline comma the norm of f in script upper M Subscript beta is denoted by

double-vertical-bar f double-vertical-bar Subscript script upper M Sub Subscript beta Subscript Baseline equals inf left-brace c greater-than-or-equal-to 0 colon left-parenthesis 3.3 right-parenthesis normal h normal o normal l normal d normal s right-brace period

It is easy to see that script upper M Subscript beta is a Banach space under the norm double-vertical-bar f double-vertical-bar Subscript script upper M Sub Subscript beta Subscript Baseline period Note that we use upper L Subscript normal l normal o normal c Superscript 2 Baseline left-parenthesis double-struck upper R Superscript n Baseline right-parenthesis instead of the space upper L Subscript normal l normal o normal c Superscript 1 Baseline left-parenthesis double-struck upper R Superscript n Baseline right-parenthesis as in Reference19 and Reference16 since this gives the appropriate setting for the duality between upper H Subscript upper L Superscript 1 and BMO Subscript upper L . For any given operator upper L , we let normal upper Theta left-parenthesis upper L right-parenthesis equals sup left-brace epsilon greater-than 0 colon left-parenthesis 2.3 right-parenthesis normal h normal o normal l normal d normal s right-brace comma and define

StartLayout 1st Row script upper M equals StartLayout Enlarged left-brace 1st Row 1st Column script upper M Subscript normal upper Theta left-parenthesis upper L right-parenthesis Baseline 2nd Column normal i normal f normal upper Theta left-parenthesis upper L right-parenthesis less-than normal infinity semicolon 2nd Row 1st Column Blank 3rd Row 1st Column union Underscript beta colon 0 less-than beta less-than normal infinity Endscripts script upper M Subscript beta Baseline 2nd Column normal i normal f normal upper Theta left-parenthesis upper L right-parenthesis equals normal infinity period EndLayout EndLayout

Note that if upper L is the Laplacian white up pointing triangle on double-struck upper R Superscript n , then normal upper Theta left-parenthesis white up pointing triangle right-parenthesis equals normal infinity . When upper L equals StartRoot white up pointing triangle EndRoot , we have normal upper Theta left-parenthesis StartRoot white up pointing triangle EndRoot right-parenthesis equals 1 period

For any left-parenthesis x comma t right-parenthesis element-of double-struck upper R Superscript n times left-parenthesis 0 comma plus normal infinity right-parenthesis and f element-of script upper M , we define

StartLayout 1st Row with Label left-parenthesis 3.4 right-parenthesis EndLabel upper P Subscript t Baseline f left-parenthesis x right-parenthesis equals e Superscript minus t upper L Baseline f left-parenthesis x right-parenthesis equals integral Underscript double-struck upper R Superscript n Baseline Endscripts p Subscript t Baseline left-parenthesis x comma y right-parenthesis f left-parenthesis y right-parenthesis d y EndLayout

and

StartLayout 1st Row with Label left-parenthesis 3.5 right-parenthesis EndLabel upper Q Subscript t Baseline f left-parenthesis x right-parenthesis equals t upper L e Superscript minus t upper L Baseline f left-parenthesis x right-parenthesis equals integral Underscript double-struck upper R Superscript n Baseline Endscripts minus t left-parenthesis StartFraction d Over d t EndFraction p Subscript t Baseline left-parenthesis x comma y right-parenthesis right-parenthesis f left-parenthesis y right-parenthesis d y period EndLayout

It follows from the estimate (Equation3.1) that the operators upper P Subscript t Baseline f and upper Q Subscript t Baseline f are well-defined. Moreover, the operator upper Q Subscript t has the following properties:

(i) for any t 1 comma t 2 greater-than 0 and almost all x element-of double-struck upper R Superscript n ,

upper Q Subscript t 1 Baseline upper Q Subscript t 2 Baseline f left-parenthesis x right-parenthesis equals t 1 t 2 left-parenthesis StartFraction d squared upper P Subscript t Baseline Over d t squared EndFraction vertical-bar Subscript t equals t 1 plus t 2 Baseline f right-parenthesis left-parenthesis x right-parenthesis semicolon

(ii) the kernel q Subscript t Sub Superscript m Baseline left-parenthesis x comma y right-parenthesis of upper Q Subscript t Sub Superscript m satisfies

StartLayout 1st Row with Label left-parenthesis 3.6 right-parenthesis EndLabel StartAbsoluteValue q Subscript t Sub Superscript m Subscript Baseline left-parenthesis x comma y right-parenthesis EndAbsoluteValue less-than-or-equal-to c t Superscript negative n Baseline s left-parenthesis StartFraction StartAbsoluteValue x minus y EndAbsoluteValue Over t EndFraction right-parenthesis comma EndLayout

where the function s satisfies the condition (Equation2.3). This property is the same as the estimate (Equation3.1).

3.2. Hardy spaces and BMO spaces associated with operators

3.2.1. Hardy space upper H Subscript upper L Superscript 1

We assume that upper L is an operator which satisfies the assumptions of Section 3.1. double-struck upper R Subscript plus Superscript n plus 1 will denote the usual upper half-space in double-struck upper R Superscript n plus 1 . The notation normal upper Gamma left-parenthesis x right-parenthesis equals StartSet left-parenthesis y comma t right-parenthesis element-of double-struck upper R Subscript plus Superscript n plus 1 Baseline colon StartAbsoluteValue x minus y EndAbsoluteValue less-than t EndSet denotes the standard cone (of aperture 1 ) with vertex x element-of double-struck upper R Superscript n . For any closed subset upper F subset-of double-struck upper R Superscript n , script upper R left-parenthesis upper F right-parenthesis will be the union of all cones with vertices in upper F , i.e., script upper R left-parenthesis upper F right-parenthesis equals union Underscript x element-of upper F Endscripts normal upper Gamma left-parenthesis x right-parenthesis period If upper O is an open subset of double-struck upper R Superscript n , then the “tent” over upper O , denoted by ModifyingAbove upper O With caret , is given as ModifyingAbove upper O With caret equals left-bracket script upper R left-parenthesis upper O Superscript c Baseline right-parenthesis right-bracket Superscript c .

Given a function f element-of upper L Superscript 1 Baseline left-parenthesis double-struck upper R Superscript n Baseline right-parenthesis , the area integral function script upper S Subscript upper L Baseline left-parenthesis f right-parenthesis associated with an operator upper L is defined by

StartLayout 1st Row with Label left-parenthesis 3.7 right-parenthesis EndLabel script upper S Subscript upper L Baseline left-parenthesis f right-parenthesis left-parenthesis x right-parenthesis equals left-parenthesis integral Underscript normal upper Gamma left-parenthesis x right-parenthesis Endscripts StartAbsoluteValue upper Q Subscript t Sub Superscript m Subscript Baseline f left-parenthesis y right-parenthesis EndAbsoluteValue squared StartFraction d y d t Over t Superscript n plus 1 Baseline EndFraction right-parenthesis Superscript 1 slash 2 Baseline period EndLayout

It follows from the assumption (b) of upper L that the area integral function script upper S Subscript upper L Baseline left-parenthesis f right-parenthesis is bounded on upper L squared left-parenthesis double-struck upper R Superscript n Baseline right-parenthesis (Reference26). It then follows from the assumption (a) of upper L that script upper S Subscript upper L Baseline left-parenthesis f right-parenthesis is bounded on upper L Superscript p , 1 less-than p less-than normal infinity . See Theorem 6 of Reference4. More specifically, there exist constants c 1 comma c 2 such that 0 less-than c 1 less-than-or-equal-to c 2 less-than normal infinity and

StartLayout 1st Row with Label left-parenthesis 3.8 right-parenthesis EndLabel c 1 double-vertical-bar f double-vertical-bar Subscript p Baseline less-than-or-equal-to double-vertical-bar script upper S Subscript upper L Baseline left-parenthesis f right-parenthesis double-vertical-bar Subscript p Baseline less-than-or-equal-to c 2 double-vertical-bar f double-vertical-bar Subscript p EndLayout

for all f element-of upper L Superscript p Baseline comma 1 less-than p less-than normal infinity . See also Reference35.

By duality, the operator upper S Subscript upper L Sub Superscript asterisk Baseline left-parenthesis f right-parenthesis also satisfies the estimate (Equation3.8), where upper L Superscript asterisk is the adjoint operator of upper L .

The following definition was introduced in Reference4. We say that f element-of upper L Superscript 1 belongs to a Hardy space associated with an operator upper L , denoted by upper H Subscript upper L Superscript 1 , if upper S Subscript upper L Baseline left-parenthesis f right-parenthesis element-of upper L Superscript 1 . We define its upper H Subscript upper L Superscript 1 norm by

double-vertical-bar f double-vertical-bar Subscript upper H Sub Subscript normal upper L Sub Superscript 1 Subscript Baseline equals double-vertical-bar script upper S Subscript upper L Baseline left-parenthesis f right-parenthesis double-vertical-bar Subscript upper L Sub Superscript 1 Subscript Baseline period

Note that if upper L is the Laplacian white up pointing triangle on double-struck upper R Superscript n , then it follows from the area integral characterization of a Hardy space by using convolution that the classical space upper H Superscript 1 Baseline left-parenthesis double-struck upper R Superscript n Baseline right-parenthesis coincides with the spaces upper H Subscript white up pointing triangle Superscript 1 Baseline left-parenthesis double-struck upper R Superscript n Baseline right-parenthesis and upper H Subscript StartRoot white up pointing triangle EndRoot Superscript 1 Baseline left-parenthesis double-struck upper R Superscript n Baseline right-parenthesis and their norms are equivalent. See Reference19 and Reference31.

3.2.2. The function space normal upper B normal upper M normal upper O Subscript upper L

Following Reference16, we say that f element-of script upper M is of bounded mean oscillation associated with an operator upper L (abbreviated as normal upper B normal upper M normal upper O Subscript upper L ) if

StartLayout 1st Row with Label left-parenthesis 3.9 right-parenthesis EndLabel sup Underscript upper B Endscripts StartFraction 1 Over StartAbsoluteValue upper B EndAbsoluteValue EndFraction integral Underscript upper B Endscripts StartAbsoluteValue f left-parenthesis x right-parenthesis minus upper P Subscript r Sub Subscript upper B Sub Superscript m Subscript Baseline f left-parenthesis x right-parenthesis EndAbsoluteValue d x equals double-vertical-bar f double-vertical-bar Subscript normal upper B normal upper M normal upper O Sub Subscript upper L Subscript Baseline less-than normal infinity comma EndLayout

where the sup is taken over all balls in double-struck upper R Superscript n , and r Subscript upper B is the radius of the ball upper B . The class of functions of BMO Subscript upper L , modulo script upper K Subscript upper L , where

StartLayout 1st Row with Label left-parenthesis 3.10 right-parenthesis EndLabel script upper K Subscript upper L Baseline equals StartSet f element-of script upper M colon upper P Subscript t Baseline f left-parenthesis x right-parenthesis equals f left-parenthesis x right-parenthesis normal f normal o normal r normal a normal l normal m normal o normal s normal t normal a normal l normal l x element-of double-struck upper R Superscript n Baseline normal a normal n normal d normal a normal l normal l t greater-than 0 EndSet comma EndLayout

is a Banach space with the norm double-vertical-bar f double-vertical-bar Subscript normal upper B normal upper M normal upper O Sub Subscript upper L defined as in (Equation3.9). We refer to Corollary 5.2 in Section 5 for completeness of the space BMO Subscript upper L . See also Section 6.1 for a discussion of the kernel space script upper K Subscript upper L .

We now give the following list of a number of important properties of the spaces BMO Subscript upper L . For the proofs, we refer the reader to Sections 2 and 3 of Reference16.

(i) If a function f is in the classical space BMO, then it follows from the John-Nirenberg inequality that f element-of upper L Subscript normal l normal o normal c Superscript 2 Baseline left-parenthesis double-struck upper R Superscript n Baseline right-parenthesis and f element-of script upper M . See Reference22. Under the extra condition that upper L satisfies the conservation property of the semigroup upper P Subscript t Baseline left-parenthesis 1 right-parenthesis equals 1 for every t greater-than 0 , it can be verified that BMO is a subspace of BMO Subscript upper L . Moreover, the spaces BMO, normal upper B normal upper M normal upper O Subscript white up pointing triangle and normal upper B normal upper M normal upper O Subscript StartRoot white up pointing triangle EndRoot coincide and their norms are equivalent. See also Theorem 6.10 in Section 6.

(ii) If f element-of BMO Subscript upper L , then for every t greater-than 0 and every upper K greater-than 1 , there exists a constant c greater-than 0 such that for almost all x element-of double-struck upper R Superscript n , we have

StartLayout 1st Row with Label left-parenthesis 3.11 right-parenthesis EndLabel StartAbsoluteValue upper P Subscript t Baseline f left-parenthesis x right-parenthesis minus upper P Subscript upper K t Baseline f left-parenthesis x right-parenthesis EndAbsoluteValue less-than-or-equal-to c left-parenthesis 1 plus normal l normal o normal g upper K right-parenthesis double-vertical-bar f double-vertical-bar Subscript normal upper B normal upper M normal upper O Sub Subscript upper L Subscript Baseline period EndLayout

(iii) If f element-of BMO Subscript upper L , then for any delta greater-than 0 and any x 0 element-of double-struck upper R Superscript n , there exists a constant c Subscript delta which depends on delta such that

StartLayout 1st Row with Label left-parenthesis 3.12 right-parenthesis EndLabel integral Underscript double-struck upper R Superscript n Baseline Endscripts StartFraction StartAbsoluteValue f left-parenthesis x right-parenthesis minus upper P Subscript t Baseline f left-parenthesis x right-parenthesis EndAbsoluteValue Over left-parenthesis t Superscript 1 slash m Baseline plus StartAbsoluteValue x minus x 0 EndAbsoluteValue right-parenthesis Superscript n plus delta Baseline EndFraction d x less-than-or-equal-to StartFraction c Subscript delta Baseline Over t Superscript delta slash m Baseline EndFraction double-vertical-bar f double-vertical-bar Subscript normal upper B normal upper M normal upper O Sub Subscript upper L Subscript Baseline period EndLayout

(iv) A variant of the John-Nirenberg inequality holds for functions in BMO Subscript upper L . That is, there exist positive constants c 1 and c 2 such that for every ball upper B and alpha greater-than 0 ,

StartLayout 1st Row StartAbsoluteValue StartSet x element-of upper B colon StartAbsoluteValue f left-parenthesis x right-parenthesis minus upper P Subscript r Sub Subscript upper B Sub Superscript m Subscript Baseline f left-parenthesis x right-parenthesis EndAbsoluteValue greater-than alpha EndSet EndAbsoluteValue less-than-or-equal-to c 1 StartAbsoluteValue upper B EndAbsoluteValue normal e normal x normal p StartSet minus StartFraction c 2 alpha Over double-vertical-bar f double-vertical-bar Subscript normal upper B normal upper M normal upper O Sub Subscript upper L Subscript Baseline EndFraction EndSet period EndLayout

This and (Equation3.9) imply that for any f element-of BMO Subscript upper L and 1 less-than-or-equal-to p less-than normal infinity , the norms

StartLayout 1st Row with Label left-parenthesis 3.13 right-parenthesis EndLabel double-vertical-bar f double-vertical-bar Subscript p comma normal upper B normal upper M normal upper O Sub Subscript upper L Subscript Baseline equals sup Underscript upper B Endscripts left-parenthesis StartFraction 1 Over StartAbsoluteValue upper B EndAbsoluteValue EndFraction integral Underscript upper B Endscripts StartAbsoluteValue f left-parenthesis x right-parenthesis minus upper P Subscript r Sub Subscript upper B Sub Superscript m Subscript Baseline f left-parenthesis x right-parenthesis EndAbsoluteValue Superscript p Baseline d x right-parenthesis Superscript 1 slash p EndLayout

with different choices of p are all equivalent.

3.3. Main theorems

We now state the main result of this paper.

Theorem 3.1

Assume that the operator upper L satisfies the assumptions (a) and (b) in Section 3.1 . Denote by upper L Superscript asterisk the adjoint operator of upper L . Then, the dual space of the upper H Subscript upper L Superscript 1 space is the BMO Subscript upper L Sub Superscript asterisk space, in the following sense.

(i) Suppose f element-of BMO Subscript upper L Sub Superscript asterisk . Then the linear functional script l given by

StartLayout 1st Row with Label left-parenthesis 3.14 right-parenthesis EndLabel script l left-parenthesis g right-parenthesis equals integral Underscript double-struck upper R Superscript n Baseline Endscripts f left-parenthesis x right-parenthesis g left-parenthesis x right-parenthesis d x comma EndLayout

initially defined on the dense subspace upper H Subscript upper L Superscript 1 Baseline intersection upper L squared , has a unique extension to upper H Subscript upper L Superscript 1 .

(ii) Conversely, every continuous linear functional script l on the upper H Subscript upper L Superscript 1 space can be realized as above; i.e., there exists f element-of BMO Subscript upper L Sub Superscript asterisk such that Equation3.14 holds and double-vertical-bar f double-vertical-bar Subscript normal upper B normal upper M normal upper O Sub Subscript upper L Sub Sub Superscript asterisk Sub Subscript Subscript Baseline less-than-or-equal-to c double-vertical-bar script l double-vertical-bar period

To state the next theorem, we recall that a measure mu defined on double-struck upper R Subscript plus Superscript n plus 1 is said to be a Carleson measure if there is a positive constant c such that for each ball upper B on double-struck upper R Superscript n ,

StartLayout 1st Row with Label left-parenthesis 3.15 right-parenthesis EndLabel mu left-parenthesis ModifyingAbove upper B With caret right-parenthesis less-than-or-equal-to c StartAbsoluteValue upper B EndAbsoluteValue comma EndLayout

where ModifyingAbove upper B With caret is the tent over upper B . The smallest bound c in (Equation3.15) is defined to be the norm of mu and is denoted by StartAbsoluteValue EndAbsoluteValue StartAbsoluteValue mu EndAbsoluteValue StartAbsoluteValue EndAbsoluteValue Subscript c .

The Carleson measure is closely related to the classical BMO space. We note that for every f element-of normal upper B normal upper M normal upper O ,

mu Subscript f Baseline left-parenthesis x comma t right-parenthesis equals StartAbsoluteValue t StartFraction partial-differential Over partial-differential t EndFraction e Superscript minus t StartRoot white up pointing triangle EndRoot Baseline f left-parenthesis x right-parenthesis EndAbsoluteValue squared StartFraction d x d t Over t EndFraction

is a Carleson measure on double-struck upper R Subscript plus Superscript n plus 1 . See Reference19 and Chapter 4 of Reference21.

For the space BMO Subscript upper L , we have the following characterization of BMO Subscript upper L functions in terms of the Carleson measure.

Theorem 3.2

Assume that the operator upper L satisfies the assumptions (a) and (b) in Section 3.1 . The following conditions are equivalent:

(i) f is a function in BMO Subscript upper L Baseline left-parenthesis double-struck upper R Superscript n Baseline right-parenthesis ;

(ii) f element-of script upper M , and mu Subscript f Baseline left-parenthesis x comma t right-parenthesis equals StartAbsoluteValue upper Q Subscript t Sub Superscript m Subscript Baseline left-parenthesis script upper I minus upper P Subscript t Sub Superscript m Subscript Baseline right-parenthesis f left-parenthesis x right-parenthesis EndAbsoluteValue squared StartFraction d x d t Over t EndFraction is a Carleson measure, with double-vertical-bar vertical-bar mu Subscript f Baseline double-vertical-bar vertical-bar Subscript c Baseline tilde double-vertical-bar f double-vertical-bar Subscript normal upper B normal upper M normal upper O Sub Subscript upper L Subscript Superscript 2 .

The proofs of Theorem 3.1 and the implication (ii) right double arrow (i) of Theorem 3.2 will be given in Section 5. For the proof of the implication (i) right double arrow (ii) of Theorem 3.2, we refer to Lemma 4.6 of Section 4.

Remark

Using Theorems 3.1 and 3.2, we can obtain more information about the Hardy spaces upper H Subscript upper L Superscript 1 and the normal upper B normal upper M normal upper O Subscript upper L spaces. We will discuss the inclusion between the classical BMO space and the normal upper B normal upper M normal upper O Subscript upper L spaces associated with some differential operators. See Section 6.

4. Properties of upper H Subscript upper L Superscript 1 and BMO Subscript upper L spaces

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.

4.1. Tent spaces and applications

For any function f left-parenthesis y comma t right-parenthesis defined on double-struck upper R Subscript plus Superscript n plus 1 we will denote

StartLayout 1st Row with Label left-parenthesis 4.1 right-parenthesis EndLabel script upper A left-parenthesis f right-parenthesis left-parenthesis x right-parenthesis equals left-parenthesis integral Underscript normal upper Gamma left-parenthesis x right-parenthesis Endscripts StartAbsoluteValue f left-parenthesis y comma t right-parenthesis EndAbsoluteValue squared StartFraction d y d t Over t Superscript n plus 1 Baseline EndFraction right-parenthesis Superscript 1 slash 2 EndLayout

and

StartLayout 1st Row with Label left-parenthesis 4.2 right-parenthesis EndLabel script upper C left-parenthesis f right-parenthesis left-parenthesis x right-parenthesis equals sup Underscript x element-of upper B Endscripts left-parenthesis StartFraction 1 Over StartAbsoluteValue upper B EndAbsoluteValue EndFraction integral Underscript ModifyingAbove upper B With caret Endscripts StartAbsoluteValue f left-parenthesis y comma t right-parenthesis EndAbsoluteValue squared StartFraction d y d t Over t EndFraction right-parenthesis Superscript 1 slash 2 Baseline period EndLayout

As in Reference8, the “tent space” upper T 2 Superscript p is defined as the space of functions f such that script upper A left-parenthesis f right-parenthesis element-of upper L Superscript p Baseline left-parenthesis double-struck upper R Superscript n Baseline right-parenthesis comma when p less-than normal infinity period The resulting equivalence classes are then equipped with the norm StartAbsoluteValue EndAbsoluteValue StartAbsoluteValue f EndAbsoluteValue StartAbsoluteValue EndAbsoluteValue Subscript upper T 2 Sub Superscript p Baseline equals double-vertical-bar script upper A left-parenthesis f right-parenthesis double-vertical-bar Subscript p . When p equals normal infinity comma the space upper T 2 Superscript normal infinity is the class of functions f for which script upper C left-parenthesis f right-parenthesis element-of upper L Superscript normal infinity Baseline left-parenthesis double-struck upper R Superscript n Baseline right-parenthesis and the norm double-vertical-bar vertical-bar f double-vertical-bar vertical-bar Subscript upper T 2 Sub Superscript normal infinity Subscript Baseline equals double-vertical-bar script upper C left-parenthesis f right-parenthesis double-vertical-bar Subscript normal infinity Baseline . Thus, f element-of upper H Subscript upper L Superscript 1 if and only if upper Q Subscript t Sub Superscript m Baseline f element-of upper T 2 Superscript 1 , i.e., script upper A left-parenthesis upper Q Subscript t Sub Superscript m Subscript Baseline f right-parenthesis equals script upper S Subscript upper L Baseline left-parenthesis f right-parenthesis element-of upper L Superscript 1 Baseline period

Next, a function a left-parenthesis t comma x right-parenthesis is called a upper T 2 Superscript 1 -atom if

left-parenthesis normal i right-parenthesis the function a left-parenthesis t comma x right-parenthesis is supported in ModifyingAbove upper B With caret (for some ball upper B subset-of double-struck upper R Superscript n Baseline right-parenthesis ;

left-parenthesis normal i normal i right-parenthesis integral Underscript ModifyingAbove upper B With caret Endscripts StartAbsoluteValue a left-parenthesis t comma x right-parenthesis EndAbsoluteValue squared StartFraction d x d t Over t EndFraction less-than-or-equal-to StartAbsoluteValue upper B EndAbsoluteValue Superscript negative 1 Baseline period

The following proposition on duality and atomic decomposition for functions in upper T 2 Superscript 1 was proved in Reference8.

Proposition 4.1

(a) The following inequality holds, whenever f element-of upper T 2 Superscript 1 and g element-of upper T 2 Superscript normal infinity Baseline colon

StartLayout 1st Row with Label left-parenthesis 4.3 right-parenthesis EndLabel integral Underscript double-struck upper R Subscript plus Superscript n plus 1 Baseline Endscripts StartAbsoluteValue f left-parenthesis x comma t right-parenthesis g left-parenthesis x comma t right-parenthesis EndAbsoluteValue StartFraction d x d t Over t EndFraction less-than-or-equal-to c integral Underscript double-struck upper R Superscript n Baseline Endscripts script upper A left-parenthesis f right-parenthesis left-parenthesis x right-parenthesis script upper C left-parenthesis g right-parenthesis left-parenthesis x right-parenthesis d x period EndLayout

(b) The pairing

mathematical left-angle f comma g mathematical right-angle right-arrow integral Underscript double-struck upper R Subscript plus Superscript n plus 1 Baseline Endscripts f left-parenthesis x comma t right-parenthesis g left-parenthesis x comma t right-parenthesis StartFraction d x d t Over t EndFraction

realizes upper T 2 Superscript normal infinity as equivalent to the Banach space dual of upper T 2 Superscript 1 Baseline period

(c) Every element f element-of upper T 2 Superscript 1 can be written as f equals sigma-summation lamda Subscript j Baseline a Subscript j Baseline comma where the a Subscript j are upper T 2 Superscript 1 atoms, lamda Subscript j Baseline element-of double-struck upper C , and sigma-summation StartAbsoluteValue lamda Subscript j Baseline EndAbsoluteValue less-than-or-equal-to c StartAbsoluteValue EndAbsoluteValue StartAbsoluteValue f EndAbsoluteValue StartAbsoluteValue EndAbsoluteValue Subscript upper T 2 Sub Superscript 1 Subscript Baseline period

Proof.

For the proof of Proposition 4.1, we refer to Theorem 1 of Reference8. See also Theorem 1 of Reference11 for a proof of (a).

Proposition 4.1 gives a quick proof of the atomic decomposition for the classical Hardy space upper H Superscript 1 . Let upper L equals StartRoot white up pointing triangle EndRoot . For any f element-of upper H Superscript 1 , we denote by upper P Subscript t Baseline f left-parenthesis x right-parenthesis the Poisson integral upper P Subscript t Baseline f equals e Superscript minus t StartRoot white up pointing triangle EndRoot Baseline f and set upper F equals upper Q Subscript t Baseline f left-parenthesis x right-parenthesis equals minus t StartFraction d Over d t EndFraction upper P Subscript t Baseline f element-of upper T 2 Superscript 1 . The atomic decomposition of upper F in upper T 2 Superscript 1 leads to the atomic decomposition of f in upper H Superscript 1 by using the following identity on upper H Superscript 1 :

StartLayout 1st Row with Label left-parenthesis 4.4 right-parenthesis EndLabel f left-parenthesis x right-parenthesis equals pi Subscript phi Baseline left-parenthesis upper F right-parenthesis left-parenthesis x right-parenthesis equals integral Subscript 0 Superscript normal infinity Baseline upper F left-parenthesis x comma t right-parenthesis asterisk phi Subscript t Baseline StartFraction d t Over t EndFraction comma EndLayout

where phi Subscript t Baseline equals t Superscript negative n Baseline phi left-parenthesis dot slash t right-parenthesis for all t greater-than 0 comma the function phi is radial and in upper C 0 Superscript normal infinity with integral phi left-parenthesis x right-parenthesis d x equals 0 , and minus 2 pi integral Subscript 0 Superscript normal infinity Baseline ModifyingAbove phi With caret left-parenthesis xi t right-parenthesis StartAbsoluteValue xi EndAbsoluteValue e Superscript minus 2 pi StartAbsoluteValue xi EndAbsoluteValue t Baseline d t equals 1 for all xi not-equals 0 . Note that instead of the condition phi element-of upper C 0 Superscript normal infinity , we may assume that StartAbsoluteValue phi left-parenthesis x right-parenthesis EndAbsoluteValue plus StartAbsoluteValue nabla phi left-parenthesis x right-parenthesis EndAbsoluteValue less-than-or-equal-to upper M left-parenthesis 1 plus StartAbsoluteValue x EndAbsoluteValue right-parenthesis Superscript negative n minus 1 for some upper M greater-than 0 . Then, the operator pi Subscript phi maps upper T 2 Superscript 1 atoms to appropriate “molecules”. See Lemma 7 of Reference7.

We now give a short discussion of the Hardy space upper H Subscript upper L Superscript 1 . For more details, see Reference4. First, we need a variant of formula (Equation4.4), which is inspired from the upper H Subscript normal infinity -calculus for upper L . We start from the identity:

StartFraction 1 Over 4 m EndFraction equals integral Subscript 0 Superscript normal infinity Baseline left-parenthesis t Superscript m Baseline z e Superscript minus t Super Superscript m Superscript z Baseline right-parenthesis left-parenthesis t Superscript m Baseline z e Superscript minus t Super Superscript m Superscript z Baseline right-parenthesis StartFraction d t Over t EndFraction comma

which is valid for all z not-equals 0 in a sector upper S Subscript mu Superscript 0 with mu element-of left-parenthesis omega comma pi right-parenthesis . As a consequence, one has

StartLayout 1st Row with Label left-parenthesis 4.5 right-parenthesis EndLabel normal upper I normal d equals 4 m integral Subscript 0 Superscript normal infinity Baseline upper Q Subscript t Sub Superscript m Subscript Baseline upper Q Subscript t Sub Superscript m Subscript Baseline StartFraction d t Over t EndFraction comma EndLayout

where the integral converges strongly in upper L squared . See Reference26. For any f element-of upper H Subscript upper L Superscript 1 , we let upper F left-parenthesis x comma t right-parenthesis equals left-parenthesis upper Q Subscript t Sub Superscript m Subscript Baseline f right-parenthesis left-parenthesis x right-parenthesis . We then have the following identity for all f element-of upper H Subscript upper L Superscript 1 intersection upper L squared :

StartLayout 1st Row with Label left-parenthesis 4.6 right-parenthesis EndLabel f left-parenthesis x right-parenthesis equals pi Subscript upper L Baseline left-parenthesis upper F right-parenthesis left-parenthesis x right-parenthesis equals 4 m integral Subscript 0 Superscript normal infinity Baseline upper Q Subscript t Sub Superscript m Subscript Baseline left-parenthesis upper Q Subscript t Sub Superscript m Subscript Baseline f right-parenthesis left-parenthesis x right-parenthesis StartFraction d t Over t EndFraction period EndLayout

Recall that in Reference4, a function alpha left-parenthesis x right-parenthesis is called an upper L -molecule if

StartLayout 1st Row with Label left-parenthesis 4.7 right-parenthesis EndLabel alpha left-parenthesis x right-parenthesis equals integral Subscript 0 Superscript normal infinity Baseline upper Q Subscript t Sub Superscript m Subscript Baseline left-parenthesis a left-parenthesis t comma dot right-parenthesis right-parenthesis left-parenthesis x right-parenthesis StartFraction d t Over t EndFraction comma EndLayout

where a left-parenthesis t comma x right-parenthesis is a upper T 2 Superscript 1 -atom supported in the tent ModifyingAbove upper B With caret of some ball upper B subset-of double-struck upper R Superscript n , and a left-parenthesis t comma x right-parenthesis satisfies the condition integral Underscript ModifyingAbove upper B With caret Endscripts StartAbsoluteValue a left-parenthesis t comma x right-parenthesis EndAbsoluteValue squared d x d t slash t less-than-or-equal-to StartAbsoluteValue upper B EndAbsoluteValue Superscript negative 1 Baseline period By using the identity (Equation4.6) in place of (Equation4.4), an upper L -molecule decomposition of f in the space upper H Subscript upper L Superscript 1 is obtained in Theorem 7 of Reference4 as follows.

Proposition 4.2

Let f element-of upper H Subscript upper L Superscript 1 intersection upper L squared . There exist upper L -molecules alpha Subscript k Baseline left-parenthesis x right-parenthesis and numbers lamda Subscript k for k equals 0 comma 1 comma 2 comma midline-horizontal-ellipsis such that

StartLayout 1st Row with Label left-parenthesis 4.8 right-parenthesis EndLabel f left-parenthesis x right-parenthesis equals sigma-summation Underscript k Endscripts lamda Subscript k Baseline alpha Subscript k Baseline left-parenthesis x right-parenthesis period EndLayout

The sequence lamda Subscript k satisfies sigma-summation Underscript k Endscripts StartAbsoluteValue lamda Subscript k Baseline EndAbsoluteValue less-than-or-equal-to c double-vertical-bar f double-vertical-bar Subscript upper H Sub Subscript upper L Sub Superscript 1 . Conversely, the decomposition Equation4.8 satisfies

double-vertical-bar f double-vertical-bar Subscript upper H Sub Subscript upper L Sub Superscript 1 Subscript Baseline less-than-or-equal-to c sigma-summation Underscript k Endscripts StartAbsoluteValue lamda Subscript k Baseline EndAbsoluteValue period

Proof.

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.

4.2. Properties for upper H Subscript upper L Superscript 1 and BMO Subscript upper L spaces

Let upper T Subscript 2 comma c Superscript p be the set of all f element-of upper T 2 Superscript p with compact support in double-struck upper R Subscript plus Superscript n plus 1 Baseline period Consider the operator pi Subscript upper L of (Equation4.6) initially defined on upper T Subscript 2 comma c Superscript p by

StartLayout 1st Row with Label left-parenthesis 4.9 right-parenthesis EndLabel pi Subscript upper L Baseline left-parenthesis f right-parenthesis left-parenthesis x right-parenthesis equals 4 m integral Subscript 0 Superscript normal infinity Baseline upper Q Subscript t Sub Superscript m Subscript Baseline left-parenthesis f left-parenthesis dot comma t right-parenthesis right-parenthesis left-parenthesis x right-parenthesis StartFraction d t Over t EndFraction period EndLayout

Note that for any compact set upper K in double-struck upper R Subscript plus Superscript n plus 1 Baseline comma

integral Underscript upper K Endscripts StartAbsoluteValue f left-parenthesis x comma t right-parenthesis EndAbsoluteValue squared d x d t less-than-or-equal-to c left-parenthesis upper K comma p right-parenthesis double-vertical-bar script upper A left-parenthesis f right-parenthesis double-vertical-bar Subscript p Superscript 2 Baseline period

This and the estimate (Equation3.2) imply that the integral (Equation4.9) is well-defined, and pi Subscript upper L Baseline left-parenthesis f right-parenthesis element-of upper L squared for f element-of upper T Subscript 2 comma c Superscript p Baseline period

Lemma 4.3

The operator pi Subscript upper L , initially defined on upper T Subscript 2 comma c Superscript p , extends to a bounded linear operator from

(a) upper T 2 Superscript p to upper L Superscript p , if 1 less-than p less-than normal infinity semicolon

(b) upper T 2 Superscript 1 to upper H Subscript upper L Superscript 1 Baseline semicolon

(c) upper T 2 Superscript normal infinity to normal upper B normal upper M normal upper O Subscript upper L .

Proof.

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 upper H Subscript upper L Superscript 1 and normal upper B normal upper M normal upper O Subscript upper L Sub Superscript asterisk spaces.

We now verify (a). By using (5.1) of Reference8, we have

integral Underscript double-struck upper R Subscript plus Superscript n plus 1 Baseline Endscripts StartAbsoluteValue f left-parenthesis x comma t right-parenthesis h left-parenthesis x comma t right-parenthesis EndAbsoluteValue StartFraction d x d t Over t EndFraction less-than-or-equal-to integral Underscript double-struck upper R Superscript n Baseline Endscripts script upper A left-parenthesis f right-parenthesis left-parenthesis x right-parenthesis script upper A left-parenthesis h right-parenthesis left-parenthesis x right-parenthesis d x period

This, together with (Equation4.9) and the estimate (Equation3.8), yield

StartLayout 1st Row 1st Column StartAbsoluteValue integral Underscript double-struck upper R Superscript n Baseline Endscripts pi Subscript upper L Baseline left-parenthesis f right-parenthesis left-parenthesis x right-parenthesis g left-parenthesis x right-parenthesis d x EndAbsoluteValue 2nd Column less-than-or-equal-to 3rd Column c StartAbsoluteValue integral Underscript double-struck upper R Subscript plus Superscript n plus 1 Baseline Endscripts f left-parenthesis x comma t right-parenthesis upper Q Subscript t Sub Superscript m Subscript Superscript asterisk Baseline g left-parenthesis x right-parenthesis StartFraction d x d t Over t EndFraction EndAbsoluteValue 2nd Row 1st Column Blank 2nd Column less-than-or-equal-to 3rd Column c StartAbsoluteValue integral Underscript double-struck upper R Superscript n Baseline Endscripts script upper A left-parenthesis f right-parenthesis left-parenthesis x right-parenthesis script upper A left-parenthesis upper Q Subscript t Sub Superscript m Subscript Superscript asterisk Baseline g right-parenthesis left-parenthesis x right-parenthesis d x EndAbsoluteValue 3rd Row 1st Column Blank 2nd Column less-than-or-equal-to 3rd Column c double-vertical-bar script upper A left-parenthesis f right-parenthesis double-vertical-bar Subscript p Baseline double-vertical-bar script upper A left-parenthesis upper Q Subscript t Sub Superscript m Subscript Superscript asterisk Baseline g right-parenthesis double-vertical-bar Subscript p prime 4th Row 1st Column Blank 2nd Column less-than-or-equal-to 3rd Column c StartAbsoluteValue EndAbsoluteValue StartAbsoluteValue f EndAbsoluteValue StartAbsoluteValue EndAbsoluteValue Subscript upper T 2 Sub Superscript p Baseline double-vertical-bar script upper S Subscript upper L Sub Superscript asterisk Subscript Baseline g double-vertical-bar Subscript p prime 5th Row 1st Column Blank 2nd Column less-than-or-equal-to 3rd Column c StartAbsoluteValue EndAbsoluteValue StartAbsoluteValue f EndAbsoluteValue StartAbsoluteValue EndAbsoluteValue Subscript upper T 2 Sub Superscript p Baseline double-vertical-bar g double-vertical-bar Subscript p prime EndLayout

for any g element-of upper L Superscript p Super Superscript prime Superscript Baseline comma StartFraction 1 Over p EndFraction plus StartFraction 1 Over p Superscript prime Baseline EndFraction equals 1 period Hence, we obtain double-vertical-bar pi Subscript upper L Baseline left-parenthesis f right-parenthesis double-vertical-bar Subscript p Baseline less-than-or-equal-to c StartAbsoluteValue EndAbsoluteValue StartAbsoluteValue f EndAbsoluteValue StartAbsoluteValue EndAbsoluteValue Subscript upper T 2 Sub Superscript p Subscript Baseline period

As a consequence of Lemma 4.3, we have the following corollary.

Corollary 4.4

The space upper H Subscript upper L Superscript 1 intersection upper L squared is dense in upper H Subscript upper L Superscript 1 .

Proof.

For any f element-of upper H Subscript upper L Superscript 1 , by the definition of upper H Subscript upper L Superscript 1 we have upper Q Subscript t Sub Superscript m Subscript Baseline f element-of upper T 2 Superscript 1 Baseline period Define upper O overTilde Subscript k Baseline equals StartSet left-parenthesis x comma t right-parenthesis element-of double-struck upper R Subscript plus Superscript n plus 1 Baseline colon StartAbsoluteValue x EndAbsoluteValue less-than-or-equal-to k comma k Superscript negative 1 Baseline less-than t less-than-or-equal-to k EndSet comma and let

f Subscript k Baseline left-parenthesis x right-parenthesis equals 4 m integral Subscript 0 Superscript normal infinity Baseline upper Q Subscript t Sub Superscript m Baseline left-parenthesis left-bracket upper Q Subscript t Sub Superscript m Subscript Baseline f right-bracket chi Subscript upper O overTilde Sub Subscript k Subscript Baseline right-parenthesis left-parenthesis x right-parenthesis StartFraction d t Over t EndFraction

for all k element-of double-struck upper N period This family of functions StartSet f Subscript k Baseline EndSet Subscript k element-of double-struck upper N satisfies

(i) f Subscript k Baseline element-of upper L squared intersection upper H Subscript upper L Superscript 1 Baseline semicolon

(ii) double-vertical-bar f minus f Subscript k Baseline double-vertical-bar Subscript upper H Sub Subscript upper L Sub Superscript 1 Baseline right-arrow 0 as k right-arrow normal infinity period

By (a) and (b) of Lemma 4.3, the estimate (i) is straightforward since for each k element-of double-struck upper N , left-bracket upper Q Subscript t Sub Superscript m Subscript Baseline f right-bracket chi Subscript upper O overTilde Sub Subscript k element-of upper T 2 Superscript 1 intersection upper T 2 squared . Moreover, by (b) of Lemma 4.3,

StartLayout 1st Row 1st Column double-vertical-bar f minus f Subscript k Baseline double-vertical-bar Subscript upper H Sub Subscript upper L Sub Superscript 1 2nd Column less-than-or-equal-to 3rd Column c StartAbsoluteValue EndAbsoluteValue StartAbsoluteValue upper Q Subscript t Sub Superscript m Subscript Baseline f left-parenthesis x right-parenthesis minus left-parenthesis upper Q Subscript t Sub Superscript m Subscript Baseline f right-parenthesis chi Subscript upper O overTilde Sub Subscript k Subscript Baseline left-parenthesis x right-parenthesis EndAbsoluteValue StartAbsoluteValue EndAbsoluteValue Subscript upper T 2 Sub Superscript 1 2nd Row 1st Column Blank 2nd Column less-than-or-equal-to 3rd Column c StartAbsoluteValue EndAbsoluteValue StartAbsoluteValue left-parenthesis upper Q Subscript t Sub Superscript m Subscript Baseline f right-parenthesis chi Subscript left-parenthesis upper O overTilde Sub Subscript k Subscript right-parenthesis Sub Superscript c Subscript Baseline left-parenthesis x right-parenthesis EndAbsoluteValue StartAbsoluteValue EndAbsoluteValue Subscript upper T 2 Sub Superscript 1 3rd Row 1st Column Blank 2nd Column right-arrow 3rd Column 0 EndLayout

as k right-arrow normal infinity period This proves property (ii) and completes the proof of Corollary 4.4.

Remark

From Corollary 4.4, it follows from a standard argument that for any f element-of upper H Subscript upper L Superscript 1 , f has an upper L -molecular decomposition (Equation4.8). See, for example, Chapter III of Reference31.

We next prove the following upper H Subscript upper L Superscript 1 -estimate for functions in the space upper H Subscript upper L Superscript 1 , which will be useful in proving our Theorems 3.1 and 3.2 in Section 5.

Lemma 4.5

For any upper L squared -function f supported on a ball upper B with radius r Subscript upper B , there exists a positive constant c such that

StartLayout 1st Row with Label left-parenthesis 4.10 right-parenthesis EndLabel double-vertical-bar left-parenthesis script upper I minus upper P Subscript r Sub Subscript upper B Sub Superscript m Subscript Baseline right-parenthesis f double-vertical-bar Subscript upper H Sub Subscript upper L Sub Superscript 1 Subscript Baseline less-than-or-equal-to c StartAbsoluteValue upper B EndAbsoluteValue Superscript 1 slash 2 Baseline double-vertical-bar f double-vertical-bar Subscript upper L squared Baseline period EndLayout

Proof.

Assume that upper B equals upper B left-parenthesis z 0 comma r Subscript upper B Baseline right-parenthesis is a ball of radius r Subscript upper B and centered at z 0 . One writes

StartLayout 1st Row 1st Column double-vertical-bar script upper S Subscript upper L Baseline left-parenthesis script upper I minus upper P Subscript r Sub Subscript upper B Sub Superscript m Subscript Baseline right-parenthesis f double-vertical-bar Subscript upper L Sub Superscript 1 2nd Column equals 3rd Column integral Underscript 4 upper B Endscripts script upper S Subscript upper L Baseline left-parenthesis script upper I minus upper P Subscript r Sub Subscript upper B Sub Superscript m Subscript Baseline right-parenthesis f left-parenthesis x right-parenthesis d x plus integral Underscript left-parenthesis 4 upper B right-parenthesis Superscript c Endscripts script upper S Subscript upper L Baseline left-parenthesis script upper I minus upper P Subscript r Sub Subscript upper B Sub Superscript m Subscript Baseline right-parenthesis f left-parenthesis x right-parenthesis d x 2nd Row 1st Column Blank 2nd Column equals 3rd Column normal upper I plus normal upper I normal upper I comma normal r normal e normal s normal p normal e normal c normal t normal i normal v normal e normal l normal y period EndLayout

Note that double-vertical-bar upper P Subscript t Baseline f double-vertical-bar Subscript upper L squared Baseline less-than-or-equal-to c double-vertical-bar f double-vertical-bar Subscript upper L squared for any t greater-than 0 . Using Hölder’s inequality and the fact that the area integral function script upper S Subscript upper L is bounded on upper L squared , one obtains

StartLayout 1st Row 1st Column integral Underscript 4 upper B Endscripts script upper S Subscript upper L Baseline left-parenthesis script upper I minus upper P Subscript r Sub Subscript upper B Sub Superscript m Subscript Baseline right-parenthesis f left-parenthesis x right-parenthesis d x 2nd Column less-than-or-equal-to 3rd Column c StartAbsoluteValue upper B EndAbsoluteValue Superscript one-half Baseline double-vertical-bar script upper S Subscript upper L Baseline left-parenthesis script upper I minus upper P Subscript r Sub Subscript upper B Sub Superscript m Subscript Baseline right-parenthesis f double-vertical-bar Subscript upper L squared 2nd Row 1st Column Blank 2nd Column less-than-or-equal-to 3rd Column c StartAbsoluteValue upper B EndAbsoluteValue Superscript one-half Baseline double-vertical-bar left-parenthesis script upper I minus upper P Subscript r Sub Subscript upper B Sub Superscript m Subscript Baseline right-parenthesis f double-vertical-bar Subscript upper L squared 3rd Row 1st Column Blank 2nd Column less-than-or-equal-to 3rd Column c StartAbsoluteValue upper B EndAbsoluteValue Superscript 1 slash 2 Baseline double-vertical-bar f double-vertical-bar Subscript upper L squared Baseline period EndLayout

We now estimate the term normal upper I normal upper I . First, we will show that there exists a constant c greater-than 0 such that for any x not-an-element-of 4 upper B ,

StartLayout 1st Row with Label left-parenthesis 4.11 right-parenthesis EndLabel left-parenthesis script upper S Subscript upper L Baseline left-parenthesis script upper I minus e Superscript minus r Super Subscript upper B Super Superscript m Superscript upper L Baseline right-parenthesis f right-parenthesis squared left-parenthesis x right-parenthesis less-than-or-equal-to c r Subscript upper B Superscript 2 epsilon Baseline double-vertical-bar f double-vertical-bar Subscript upper L Sub Superscript 1 Subscript Superscript 2 Baseline StartAbsoluteValue x minus z 0 EndAbsoluteValue Superscript minus 2 left-parenthesis n plus epsilon right-parenthesis Baseline period EndLayout

Let us verify (Equation4.11). Let

normal upper Psi Subscript t comma s Baseline left-parenthesis upper L right-parenthesis f left-parenthesis x right-parenthesis equals left-parenthesis t Superscript m Baseline plus s Superscript m Baseline right-parenthesis squared left-parenthesis StartFraction d squared upper P Subscript r Baseline Over d r squared EndFraction vertical-bar Subscript r equals t Sub Superscript m Subscript plus s Sub Superscript m Subscript Baseline f right-parenthesis left-parenthesis x right-parenthesis

and h left-parenthesis x right-parenthesis equals m x Superscript m Baseline left-parenthesis 1 plus x Superscript m Baseline right-parenthesis Superscript negative 2 Baseline period Since left-parenthesis script upper I minus upper P Subscript r Sub Subscript m Sub Superscript upper B Subscript Baseline right-parenthesis equals m integral Subscript 0 Superscript r Subscript upper B Baseline Baseline upper Q Subscript s Sub Superscript m Subscript Baseline StartFraction d s Over s EndFraction comma we obtain

StartLayout 1st Row upper Q Subscript t Sub Superscript m Subscript Baseline left-parenthesis script upper I minus upper P Subscript r Sub Subscript upper B Sub Superscript m Subscript Baseline right-parenthesis equals m integral Subscript 0 Superscript r Subscript upper B Baseline Baseline upper Q Subscript t Sub Superscript m Subscript Baseline upper Q Subscript s Sub Superscript m Subscript Baseline StartFraction d s Over s EndFraction equals integral Subscript 0 Superscript r Subscript upper B Baseline Baseline h left-parenthesis StartFraction s Over t EndFraction right-parenthesis normal upper Psi Subscript t comma s Baseline left-parenthesis upper L right-parenthesis StartFraction d s Over s EndFraction period EndLayout

It follows from the estimate (Equation3.1) that the kernel normal upper Psi Subscript t comma s Baseline left-parenthesis upper L right-parenthesis left-parenthesis y comma z right-parenthesis of the operator normal upper Psi Subscript t comma s Baseline left-parenthesis upper L right-parenthesis satisfies

StartAbsoluteValue normal upper Psi Subscript t comma s Baseline left-parenthesis upper L right-parenthesis left-parenthesis y comma z right-parenthesis EndAbsoluteValue less-than-or-equal-to c StartFraction left-parenthesis t plus s right-parenthesis Superscript epsilon Baseline Over left-parenthesis t plus s plus StartAbsoluteValue y minus z EndAbsoluteValue right-parenthesis Superscript n plus epsilon Baseline EndFraction comma

where epsilon is the positive constant in (Equation2.3). Therefore,

StartLayout 1st Row 1st Column Blank 2nd Column left-parenthesis script upper S Subscript upper L Baseline left-parenthesis script upper I minus upper P Subscript r Sub Subscript upper B Sub Superscript m Subscript Baseline right-parenthesis f right-parenthesis squared left-parenthesis x right-parenthesis 2nd Row 1st Column Blank 2nd Column less-than-or-equal-to integral Subscript 0 Superscript normal infinity Baseline integral Underscript StartAbsoluteValue y minus x EndAbsoluteValue less-than-or-equal-to t Endscripts left-bracket integral Subscript 0 Superscript r Subscript upper B Baseline Baseline h left-parenthesis StartFraction s Over t EndFraction right-parenthesis normal upper Psi Subscript t comma s Baseline left-parenthesis upper L right-parenthesis f left-parenthesis y right-parenthesis StartFraction d s Over s EndFraction right-bracket squared StartFraction d y d t Over t Superscript n plus 1 Baseline EndFraction 3rd Row 1st Column Blank 2nd Column less-than-or-equal-to c left-parenthesis integral Subscript 0 Superscript r Subscript upper B Baseline Baseline plus integral Subscript r Subscript upper B Baseline Superscript normal infinity Baseline right-parenthesis integral Underscript StartAbsoluteValue y minus x EndAbsoluteValue less-than-or-equal-to t Endscripts left-bracket integral Subscript 0 Superscript r Subscript upper B Baseline Baseline h left-parenthesis StartFraction s Over t EndFraction right-parenthesis integral Underscript upper B Endscripts StartFraction left-parenthesis t plus s right-parenthesis Superscript epsilon Baseline Over left-parenthesis t plus s plus StartAbsoluteValue y minus z EndAbsoluteValue right-parenthesis Superscript n plus epsilon Baseline EndFraction StartAbsoluteValue f left-parenthesis z right-parenthesis EndAbsoluteValue StartFraction d z d s Over s EndFraction right-bracket squared StartFraction d y d t Over t Superscript n plus 1 Baseline EndFraction 4th Row 1st Column Blank 2nd Column equals normal upper I normal upper I 1 plus normal upper I normal upper I 2 period EndLayout

We only consider the term normal upper I normal upper I Subscript 2 since the estimate of the term normal upper I normal upper I Subscript 1 is even simpler. For x not-an-element-of 4 upper B and t greater-than-or-equal-to r Subscript upper B , we set upper B equals upper B 1 union upper B 2 , where upper B 1 equals upper B intersection StartSet z colon StartAbsoluteValue y minus z EndAbsoluteValue less-than-or-equal-to StartFraction StartAbsoluteValue x minus z 0 EndAbsoluteValue Over 2 EndFraction EndSet . For any z element-of upper B 1 and StartAbsoluteValue y minus x EndAbsoluteValue less-than t , we have

StartAbsoluteValue x minus z 0 EndAbsoluteValue less-than-or-equal-to StartAbsoluteValue y minus x EndAbsoluteValue plus StartAbsoluteValue y minus z EndAbsoluteValue plus StartAbsoluteValue z minus z 0 EndAbsoluteValue less-than-or-equal-to t plus StartFraction StartAbsoluteValue x minus z 0 EndAbsoluteValue Over 2 EndFraction plus r Subscript upper B Baseline less-than-or-equal-to 2 t plus StartFraction StartAbsoluteValue x minus z 0 EndAbsoluteValue Over 2 EndFraction comma

which implies t greater-than-or-equal-to StartAbsoluteValue x minus z 0 EndAbsoluteValue slash 4 ; hence left-parenthesis t plus s plus StartAbsoluteValue y minus z EndAbsoluteValue right-parenthesis greater-than-or-equal-to StartAbsoluteValue x minus z 0 EndAbsoluteValue slash 4 . Obviously, for any z element-of upper B 2 and StartAbsoluteValue y minus x EndAbsoluteValue less-than t , we also have left-parenthesis t plus s plus StartAbsoluteValue y minus z EndAbsoluteValue right-parenthesis greater-than-or-equal-to StartAbsoluteValue x minus z 0 EndAbsoluteValue slash 2 . Note that

StartLayout 1st Row left-parenthesis t plus s right-parenthesis Superscript epsilon Baseline h left-parenthesis StartFraction s Over t EndFraction right-parenthesis less-than-or-equal-to c left-parenthesis t plus s right-parenthesis Superscript epsilon Baseline left-parenthesis t s right-parenthesis Superscript m Baseline left-parenthesis t Superscript m Baseline plus s Superscript m Baseline right-parenthesis Superscript negative 2 Baseline less-than-or-equal-to c t Superscript negative epsilon slash 2 Baseline s Superscript 3 epsilon slash 2 Baseline period EndLayout

It follows from elementary integration that

StartLayout 1st Row 1st Column normal upper I normal upper I 2 2nd Column less-than-or-equal-to 3rd Column c integral Subscript r Subscript upper B Baseline Superscript normal infinity Baseline integral Underscript StartAbsoluteValue y minus x EndAbsoluteValue less-than-or-equal-to t Endscripts left-bracket integral Subscript 0 Superscript r Subscript upper B Baseline Baseline left-parenthesis t plus s right-parenthesis Superscript epsilon Baseline h left-parenthesis StartFraction s Over t EndFraction right-parenthesis StartFraction d s Over s EndFraction right-bracket squared StartFraction d y d t Over t Superscript n plus 1 Baseline EndFraction double-vertical-bar f double-vertical-bar Subscript upper L Sub Superscript 1 Superscript 2 Baseline StartAbsoluteValue x minus z 0 EndAbsoluteValue Superscript minus 2 left-parenthesis n plus epsilon right-parenthesis 2nd Row 1st Column Blank 2nd Column less-than-or-equal-to 3rd Column c left-parenthesis integral Subscript r Subscript upper B Baseline Superscript normal infinity Baseline left-bracket integral Subscript 0 Superscript r Subscript upper B Baseline Baseline t Superscript negative epsilon slash 2 Baseline s Superscript 3 epsilon slash 2 Baseline StartFraction d s Over s EndFraction right-bracket squared StartFraction d t Over t EndFraction right-parenthesis double-vertical-bar f double-vertical-bar Subscript upper L Sub Superscript 1 Superscript 2 Baseline StartAbsoluteValue x minus z 0 EndAbsoluteValue Superscript minus 2 left-parenthesis n plus epsilon right-parenthesis 3rd Row 1st Column Blank 2nd Column less-than-or-equal-to 3rd Column c r Subscript upper B Superscript 2 epsilon Baseline double-vertical-bar f double-vertical-bar Subscript upper L Sub Superscript 1 Subscript Superscript 2 Baseline StartAbsoluteValue x minus z 0 EndAbsoluteValue Superscript minus 2 left-parenthesis n plus epsilon right-parenthesis Baseline period EndLayout

The estimate (Equation4.11) then follows readily. Therefore,

StartLayout 1st Row 1st Column integral Underscript left-parenthesis 4 upper B right-parenthesis Superscript c Endscripts script upper S Subscript upper L Baseline left-parenthesis script upper I minus upper P Subscript r Sub Subscript upper B Sub Superscript m Subscript Baseline right-parenthesis f left-parenthesis x right-parenthesis d x 2nd Column less-than-or-equal-to 3rd Column c r Subscript upper B Superscript epsilon Baseline double-vertical-bar f double-vertical-bar Subscript upper L Sub Superscript 1 Baseline integral Underscript left-parenthesis 4 upper B right-parenthesis Superscript c Endscripts StartAbsoluteValue x minus z 0 EndAbsoluteValue Superscript minus left-parenthesis n plus epsilon right-parenthesis Baseline d x 2nd Row 1st Column Blank 2nd Column less-than-or-equal-to 3rd Column c double-vertical-bar f double-vertical-bar Subscript upper L Sub Superscript 1 3rd Row 1st Column Blank 2nd Column less-than-or-equal-to 3rd Column c StartAbsoluteValue upper B EndAbsoluteValue Superscript 1 slash 2 Baseline double-vertical-bar f double-vertical-bar Subscript upper L squared Baseline period EndLayout

Combining the estimates of the terms I and II, we obtain that double-vertical-bar script upper S Subscript upper L Baseline left-parenthesis script upper I minus upper P Subscript r Sub Subscript upper B Sub Superscript m Subscript Baseline right-parenthesis f double-vertical-bar Subscript upper L Sub Superscript 1 Subscript Baseline less-than-or-equal-to c StartAbsoluteValue upper B EndAbsoluteValue Superscript 1 slash 2 Baseline double-vertical-bar f double-vertical-bar Subscript upper L squared Baseline period The proof of Lemma 4.5 is complete.

We now follow Theorem 2.14 of Reference16 to prove the implication (i) right double arrow (ii) of Theorem 3.2. For the implication (ii) right double arrow (i) of Theorem 3.2, we will present its proof in Section 5.3.

Lemma 4.6

If f element-of BMO Subscript upper L , then mu Subscript f Baseline left-parenthesis x comma t right-parenthesis equals StartAbsoluteValue upper Q Subscript t Sub Superscript m Subscript Baseline left-parenthesis script upper I minus upper P Subscript t Sub Superscript m Subscript Baseline right-parenthesis f left-parenthesis x right-parenthesis EndAbsoluteValue squared StartFraction d x d t Over t EndFraction is a Carleson measure with double-vertical-bar vertical-bar mu Subscript f Baseline double-vertical-bar vertical-bar Subscript c Baseline tilde double-vertical-bar f double-vertical-bar Subscript normal upper B normal upper M normal upper O Sub Subscript upper L Subscript Superscript 2 .

Proof.

We will prove that there exists a positive constant c greater-than 0 such that for any ball upper B equals upper B left-parenthesis x Subscript upper B Baseline comma r Subscript upper B Baseline right-parenthesis on double-struck upper R Superscript n ,

StartLayout 1st Row with Label left-parenthesis 4.12 right-parenthesis EndLabel integral integral Underscript ModifyingAbove upper B With caret Endscripts StartAbsoluteValue upper Q Subscript t Sub Superscript m Subscript Baseline left-parenthesis script upper I minus upper P Subscript t Sub Superscript m Subscript Baseline right-parenthesis f left-parenthesis x right-parenthesis EndAbsoluteValue squared StartFraction d x d t Over t EndFraction less-than-or-equal-to c StartAbsoluteValue upper B EndAbsoluteValue double-vertical-bar f double-vertical-bar Subscript normal upper B normal upper M normal upper O Sub Subscript upper L Subscript Superscript 2 Baseline period EndLayout

Note that

upper Q Subscript t Sub Superscript m Subscript Baseline left-parenthesis script upper I minus upper P Subscript t Sub Superscript m Subscript Baseline right-parenthesis equals upper Q Subscript t Sub Superscript m Subscript Baseline left-parenthesis script upper I minus upper P Subscript t Sub Superscript m Subscript Baseline right-parenthesis left-parenthesis script upper I minus upper P Subscript r Sub Subscript upper B Sub Superscript m Subscript Baseline right-parenthesis plus upper Q Subscript t Sub Superscript m Subscript Baseline left-parenthesis upper I minus upper P Subscript t Sub Superscript m Subscript Baseline right-parenthesis upper P Subscript r Sub Subscript upper B Sub Superscript m Subscript Baseline period

Hence, (Equation4.12) follows from the following estimates (Equation4.13) and (Equation4.14):

StartLayout 1st Row with Label left-parenthesis 4.13 right-parenthesis EndLabel integral integral Underscript ModifyingAbove upper B With caret Endscripts StartAbsoluteValue upper Q Subscript t Sub Superscript m Subscript Baseline left-parenthesis script upper I minus upper P Subscript t Sub Superscript m Subscript Baseline right-parenthesis left-parenthesis script upper I minus upper P Subscript r Sub Subscript upper B Sub Superscript m Subscript Baseline right-parenthesis f left-parenthesis x right-parenthesis EndAbsoluteValue squared StartFraction d x d t Over t EndFraction less-than-or-equal-to c StartAbsoluteValue upper B EndAbsoluteValue double-vertical-bar f double-vertical-bar Subscript normal upper B normal upper M normal upper O Sub Subscript upper L Superscript 2 EndLayout