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On the weak* convergence in HF1(Rn)
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Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa
On the weak* convergence in HF1 (Rn ) ✩ Chin-Cheng Lin Department of Mathematics, National Central University, Chung-Li, 320, Taiwan, Republic of China
a r t i c l e
i n f o
Article history: Received 8 August 2016 Available online xxxx Submitted by C. Gutierrez Keywords: Ap weights BMO Hardy spaces Monge–Ampère equation VMO
a b s t r a c t Let VMO F denote the closure of Cc ∩ Lip with respect to the seminorm · BMOF , where F is a family of sections and the space BMO F associated to the family F was introduced by Caffarelli and Gutiérrez (1996) [4]. Sections play an important role in the investigation of Monge–Ampère equation and the linearized Monge–Ampère equation. 1 defined in Ding and Lin We show that the dual of VMO F is the Hardy space HF (2005) [8]. As an application, we prove that μ-almost everywhere convergence of a 1 sequence of functions bounded in HF to a function in L1 (dμ) implies the weak* convergence. © 2017 Elsevier Inc. All rights reserved.
1. Introduction In 1996, Caffarelli and Gutiérrez [4] studied real analysis related to the Monge–Ampère equation and introduced axiomatically a family of bounded convex sets F = {S(x, t) ⊂ Rn : x ∈ Rn and t > 0}, which they called sections, and a Borel measure μ satisfying the doubling property with respect to the parameter t, μ(S(x, 2t)) ≤ Cμ(S(x, t)). The sections S(x, t) satisfy some properties modeled on the solutions of the real Monge–Ampère equation det D2 φ = μ for a convex function φ in Rn and the generated measure μ. Caffarelli and Gutiérrez gave a Besicovitch type covering lemma for the family F and set up a variant of the Calderón–Zygmund decomposition by applying this covering lemma together with the doubling condition of μ. Such a decomposition plays an important role in the study of the linearized Monge–Ampère equation (see [5]). As applications of the decomposition, Caffarelli and Gutiérrez [4] defined the Hardy–Littlewood maximal operator MF and BMO F (Rn ) space associated to the family F, and obtained the weak type (1, 1) boundedness of MF and the John–Nirenberg inequality for BMO F (Rn ). Let us recall the definition of sections. For each x ∈ Rn , we have a one-parameter family {S(x, t)}t>0 of open and bounded convex subsets of Rn containing x, called sections, which is monotone increasing in t, i.e., S(x, t) ⊂ S(x, t ) for t ≤ t , and satisfies the following three conditions: ✩
Research supported by Ministry of Science and Technology of Taiwan under Grant #MOST 103-2115-M-008-003-MY3. E-mail address: [email protected].
http://dx.doi.org/10.1016/j.jmaa.2017.06.005 0022-247X/© 2017 Elsevier Inc. All rights reserved.
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(A) There exist positive constants K1 , K2 , K3 and 1 , 2 such that given two sections S(x0 , t0 ), S(x, t) with t ≤ t0 satisfying S(x0 , t0 ) ∩ S(x, t) = ∅, and given T an affine transformation that “normalizes” S(x0 , t0 ); that is, B(0, 1/n) ⊂ T (S(x0 , t0 )) ⊂ B(0, 1), there exists z ∈ B(0, K3 ) depending on S(x0 , t0 ) and S(x, t), which satisfies B z, K2 (t/t0 )2 ⊂ T (S(x, t)) ⊂ B z, K1 (t/t0 )1 and T (x) ∈ B z, (1/2)K2 (t/t0 )2 . Here and below B(x, t) denotes the Euclidean ball centered at x with radius t. (B) There exists a constant δ > 0 such that given a section S(x, t) and y ∈ / S(x, t), if T is an affine transformation that normalizes S(x, t), then B(T (y), δ ) ∩ T (S(x, (1 − )t)) = ∅ (C)
t>0
S(x, t) = {x} and
t>0
for any 0 < < 1.
S(x, t) = Rn .
In addition, we also assume that a Borel measure μ is given, which is finite on compact sets, μ(Rn ) = ∞, and satisfies the following doubling property with respect to the parameter t; that is, there exists a constant A such that μ(S(x, 2t)) ≤ Aμ(S(x, t))
for any x ∈ Rn and t > 0.
(1.1)
Sections play an important role in the investigation of Monge–Ampère equation and the linearized Monge– Ampère equation (see [2–5]). An example of a family of sections was given in [4]. Caffarelli and Gutiérrez [4] defined the space BMO F (Rn ) associated with the family F := {S(x, t) ⊂ Rn : x ∈ Rn and t > 0} and the Borel measure μ satisfying the doubling condition (1.1). Let f be a real-valued function defined on Rn . We say that f ∈ BMO F (Rn ) if f BMOF := sup
S∈F
1 μ(S)
|f (x) − mS (f )|dμ(x) < ∞, S
where mS (f ) denotes the mean of f over the section S defined by mS (f ) =
1 μ(S)
f (x)dμ(x). S
In 2005, Ding and Lin [8] extended the study to the context of Hardy spaces HF1 (Rn ) associated to the family F and a doubling measure μ. They showed the duality of HF1 and BMO F . In this paper, we will investigate the predual of HF1 . We first recall the definition of HF1 . Let 1 < q ≤ ∞. A function a ∈ Lq (dμ) is called a (1, q)-atom if there exists a section S(x0 , t0 ) ∈ F such that
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(i) supp(a) ⊂ S(x0 , t0 ); a(x)dμ(x) = 0; (ii) Rn
(iii) aLq (dμ) ≤ μ(S(x0 , t0 ))1/q−1 . The atomic Hardy space HF1,q (Rn ) is defined by HF1,q (Rn )
=
λj aj (x) : each aj is a (1, q)-atom and
j
|λj | < ∞
j
with norm f H 1,q = inf j |λj |, where the infimum is taken over all decompositions of f = j λj aj above. F It is clear that a ∈ L1 (dμ) with aL1 (dμ) ≤ 1 for each (1, q)-atom a, and HF1 (Rn ) is a subspace of L1 (dμ). It was shown in [8, Theorem 1.1] that all atomic Hardy spaces HF1,q (Rn ), 1 < q ≤ ∞, are equivalent. Hence, one defines the Hardy space HF1 to be HF1 (Rn ) := HF1,∞ (Rn )
and
f HF1 := f H 1,∞ . F
Hereafter, an “atom” will mean a (1, ∞)-atom. As usual, we use Cc (Rn ) to denote all continuous functions with compact support in Rn and use Lip(Rn ) to denote the collection of functions satisfying that there is a constant C such that |f (x) − f (y)| ≤ Cρ(x, y)
for all x, y ∈ Rn ,
where ρ is the quasi-metric described in (2.6) below. Let VMO F (Rn ) denote the closure of Cc (Rn ) ∩Lip(Rn ) with respect to the seminorm · BMOF . Our first result is about the predual of HF1 . Theorem 1.1. HF1 (Rn ) is the dual of VMO F (Rn ). As an application, we prove that μ-almost everywhere convergence of a sequence of functions bounded in HF1 implies weak* convergence, which is an HF1 version of the Jones–Journé theorem [10]. The Jones–Journé theorem is useful in the application of Hardy spaces to compensated compactness (see [7]). Theorem 1.2. Suppose that a sequence of functions {fk } ⊂ HF1 (Rn ) satisfies fk HF1 ≤ 1 for all k and fk (x) → f (x) for μ-almost every x ∈ Rn . Then f ∈ HF1 (Rn ), f HF1 ≤ 1, and
fk (x)φ(x)dμ(x) −→ Rn
f (x)φ(x)dμ(x)
for all φ ∈ VMO F (Rn ).
(1.2)
Rn
2. Proof of Theorem 1.1 From properties (A) and (B) of sections, Aimar, Forzani, and Toledano [1] obtained the following engulfing property: There exists a constant θ > 1, depending only on δ, K1 , and 1 , such that for each y ∈ S(x, t) ∈ F we have S(y, t) ⊂ S(x, θt)
and
S(x, t) ⊂ S(y, θt).
(2.1)
They also showed that there exists a quasi-metric d(x, y) on Rn with respect to F defined by d(x, y) = inf{t : x ∈ S(y, t) and y ∈ S(x, t)}.
(2.2)
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The triangular constant of the quasi-metric d is just the θ appeared in (2.1): d(x, y) ≤ θ d(x, z) + d(z, y)
for any x, y, z ∈ Rn .
(2.3)
for any x ∈ Rn and t > 0,
(2.4)
Also, t S x, ⊂ Bd (x, t) ⊂ S(x, t) 2θ
where Bd (x, t) is the d-ball centered at x with radius t defined by Bd (x, t) := {y ∈ Rn : d(x, y) < t}. By (1.1) and (2.4), if we choose k0 ∈ N satisfying 2k0 −2 ≥ θ, then μ(Bd (x, 2t)) ≤ Ak0 μ(Bd (x, t))
for any x ∈ Rn and t > 0.
(2.5)
Hence, (Rn , d, μ) is a space of homogeneous type introduced by Coifman and Weiss [6]. Macías and Segovia [11, Theorems 2 and 3] have shown that one can replace d by another quasi-metric ρ such that there exist constants C > 0 and ε ∈ (0, 1) satisfying ⎧ ⎪ ⎨ ρ(z, w) ≈ inf{μ(Bd ) : Bd are d-balls containing z and w}; μ(Bρ (z, r)) ≈ r, ∀ z ∈ Rn , r > 0, where Bρ (z, r) := {w ∈ Rn : ρ(z, w) < r}; ⎪ ⎩ |ρ(z, w) − ρ(z , w)| ≤ C(ρ(z, z ))ε [ρ(z, w) + ρ(z , w)]1−ε , ∀ z, z , w ∈ Rn .
(2.6)
In order to show the first theorem, we need some lemmas. We may use a diagonalization argument to obtain the following lemma, which also can be seen in [6, Lemma (4.3)].
∞ Lemma 2.1. Suppose that λkj ≥ 0, j, k = 1, 2, · · · , satisfy j=1 λkj ≤ 1 for each k = 1, 2, · · · . Then there exists an increasing sequence of natural numbers, k1 < k2 < · · · < kn < · · · , such that limm→∞ λkj m = λj
∞ for each j and j=1 λj ≤ 1. The following lemma is of a geometric nature. Lemma 2.2. Suppose that m is an integer and θ > 1 is given in (2.1). Then Rn is the union of sections 2m {S(xm ) : j = 1, 2, · · · } such that each point of Rn belongs to at most N of these sections (N is j ,θ independent of m) and each f ∈ HF1 has the representation f=
∞ ∞
m λm j aj ,
j=1 m=−∞ m 2m+2 where am ) and j is a (1, ∞)-atom supported in S(xj , θ
j∈N, m∈Z
|λm j | ≤ C f H 1 . F
Proof. Given a section S(x, t), where x ∈ Rn and t > 0, let S(y, t) satisfy that S(x, t) ∩ S(y, t) = ∅. Choose z ∈ S(x, t) ∩ S(y, t). By (2.1), we have S(y, t) ⊂ S(z, θt) and S(z, t) ⊂ S(x, θt), and hence S(y, t) ⊂ S(z, θt) ⊂ S(x, θ2 t). Therefore, y∈Rn
{S(y, t) | S(y, t) ∩ S(x, t) = ∅} ⊂ S(x, θ2 t).
(2.7)
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Plug x = x11 and t = 1 into (2.7) to give
S(y, 1) | S(y, 1) ∩ S(x11 , 1) = ∅ ⊂ S(x11 , θ2 ).
y∈Rn
Similarly, we pick x12 ∈ y ∈ Rn | S(y, 1) ∩ S(x11 , 1) = ∅ such that
S(y, 1) | S(y, 1) ∩ S(x12 , 1) = ∅ ⊂ S(x12 , θ2 );
y∈Rn
we pick x13 ∈ y ∈ Rn | S(y, 1) ∩ [S(x11 , 1) ∪ S(x12 , 1)] = ∅ such that
S(y, 1) | S(y, 1) ∩ S(x13 , 1) = ∅ ⊂ S(x13 , θ2 ).
y∈Rn
Continuing the same method for choosing points, we obtain a sequence of sections {S(x1j , θ2 )}j∈N such that 1 2 n 1 j∈N S(xj , θ ) = R . Since these sections {S(xj , 1)}j∈N are disjoint, it follows from (2.4) that the d-balls 1 1 {Bd (xj , 1)}j∈N are disjoint, and hence {Bd (xj , 2θ3 )}j∈N are finitely overlapping. By (2.4) again, the sections {S(x1j , θ2 )}j∈N are finitely overlapping. Using the same argument as above (by choosing t = θ2 ), we can also obtain a finitely overlapping sequence of sections {S(x2j , θ4 )}j∈N such that j∈N S(x2j , θ4 ) = Rn . By induction, for each m ∈ Z, there exists a sequence {xm j }j∈N , such that 2m ∞ ) j=1 is a finitely overlapping sequence of sections; (i) S(xm j ,θ ∞ 2m ) = Rn . (ii) j=1 S(xm j ,θ
For f = k∈N λk ak ∈ HF1 , where |λk | < ∞ and ak are (1, ∞)-atom supporting on S(xk , tk ), we may assume supp(ak ) ⊂ S(xk , θ2m ) for some m = m(k) due to θ2m → ∞ as m → ∞. We choose the smallest integer m such that the choice of m is unique for each ak . Note that, the mapping k → m of indices form N to Z is neither injective nor surjective. That is, for each m, it may have more than one corresponding ak ’s and may have no corresponding ak ’s. For each m, which has corresponding ak ’s, the supporting section S(xk , θ2m ) of each ak satisfies one of the followings: 2m 1 S(xk , θ 2m ) ∩ S(xm ) = ∅; 1 ,θ 2m m 2m 2m 2 S(xk , θ ) ∩ S(x1 , θ ) = ∅ and S(xk , θ2m ) ∩ S(xm ) = ∅; 2 ,θ .. . i−1 2m 2m i S(xk , θ 2m ) ∩ [ j=1 S(xm )] = ∅ and S(xk , θ2m ) ∩ S(xm ) = ∅; j ,θ i ,θ .. .
i above, we set For the general case
i {k: case }
m λk ak = λm i ai ,
m 2m+2 where λm ) i = μ S(xi , θ
i {k: case }
|λk | . μ(S(xk , θ2m ))
m m 2m+2 Then am ). We set λm i is a (1, ∞)-atom with supp(ai ) ⊂ S(xi , θ i = 0, i ∈ N, if there is no corresponding 2m m 2m 2m+2 ak ’s for this m. For S(xk , θ ) ∩ S(xi , θ ) = ∅, we have S(xk , θ2m+2 ) ∩ S(xm ) = ∅ and hence i ,θ m 2m+2 2m+4 S(xi , θ ) ⊂ S(xk , θ ). By (1.1),
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|λm i |≤
|λk |
i {k: case }
μ(S(xk , θ2m+4 )) ≤ A1+4 log2 θ μ(S(xk , θ2m ))
|λk |.
i {k: case }
Therefore,
i∈N, m∈Z
i {k: case }
1+4 log2 θ |λm i |≤A
i∈N, m∈Z
|λk | = A1+4 log2 θ
|λk | ≤ C f H 1 , F
k∈N
and the proof is completed. 2 Lemma 2.3. Suppose fk H 1 ≤ 1 for all k ∈ N. Then there exist a subsequence {fkl }l∈N and f ∈ HF1 (Rn ) F such that
lim
l→∞ Rn
fkl gdμ =
for all g ∈ Cc (Rn ).
f gdμ
(2.8)
Rn
Proof. For each k ∈ N, we apply Lemma 2.2 to express fk =
∞ ∞
m λm i (k)ai (k),
i=1 m=−∞ m 2m+2 where am ) for all k ∈ N and i (k) are (1, ∞)-atoms supported on S(xi , θ
i∈N, m∈Z
1 |λm i (k)| ≤ Cfk HF
≤ C. By Lemma 2.1, there exists an strictly increasing sequence {k }∈N of natural numbers, such that
m m m liml→∞ λm i (kl ) = λi for each (i, m) ∈ N × Z and i,m |λi | ≤ C. The definition of ai (k) gives m 2m+2 −1 am ) i (k)L∞ (dμ) ≤ μ S(xi , θ
for all k ∈ N.
∗ Since L1μ is separable, Banach–Alaoglu theorem shows that there exists a subsequence {am i (kl )}l∈N weak −1 m m m 2m+2 convergent to a function ai satisfying ai L∞ (dμ) ≤ μ S(xi , θ ) . By using the diagonalization argument, there exists a subsequence of {kl } (for simplicity we still use the same indices {kl } to denote this subsequence of {kl }) such that {am (kl )}l∈N converges to am , as l → ∞, for all (i, m). It is easy to check i
i m m m m that ai are (1, ∞)-atoms. Let f = i,m λi ai . Since i,m |λi | ≤ C, we have f ∈ HF1 (Rn ). To show (2.8), we write
fkl gdμ =
m λm i (kl )ai (kl )gdμ
Rn i,m
Rn
=
m
⎛ =⎝
λm i (kl )
i
m<−M
am i (kl )gdμ
Rn
+
−M ≤m≤M
+
m>M
⎞ ⎠
i
λm i (kl )
am i (kl )gdμ
Rn
:= I + II + III , where M is a large number that will be determined later. For term I, given ε > 0, the continuity of g yields m 2m+2 |g(x) − g(xm ) whenever M is large enough. Hence, the vanishing conditions of i )| < ε for x ∈ S(xi , θ m ai (k) show that
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m m m |I| = λi (kl ) ai (kl )(x) {g(x) − g(xi )} dμ(x) m<−M i 2m+2 ) S(xm i ,θ m ≤ |λm |am i (kl )| i (kl )||g(x) − g(xi )|dμ(x) < Cε. i
m<−M
(2.9)
2m+2 ) S(xm i ,θ
2m+2 For each m with −M ≤ m ≤ M , the compact support of g interests a finite number of {S(xm )}i∈N i ,θ m 2m since {S(xi , θ )}i∈N are finitely overlapping. The sum in II involves finitely many terms only, independently of k. Thus,
−M ≤m≤M
i
II =
=
Rn −M ≤m≤M
λm i (kl )
am i (kl )gdμ
Rn
m λm i (kl )ai (kl )gdμ −→
i
(2.10) f gdμ
Rn
m as l → ∞ and M → ∞. To estimate III , we note that i,m |λm i (k)|ai (k)L1 (dμ) gL∞ (dμ) ≤ CgL∞ (dμ) . m 2m+2 −1 We use the estimate am ) which is uniformly small for large m. Given i (k)L∞ (dμ) ≤ μ S(xi , θ ε > 0, |III | ≤
m>M
m |λm i (k)|ai (k)L1 (dμ) gL∞ (dμ) < ε
for M large enough.
(2.11)
i
Combining (2.9)–(2.11), we finish the proof. 2 To prove the duality between VMO F and HF1 , we need a functional analysis result that can be found in [9, p. 439, Exercise 41]. Proposition 2.4. Let X be a locally convex linear topological space and Y be a linear subspace of X ∗ . Then Y is X-dense in X ∗ if and only if Y is a total set of functionals on X. Proof of Theorem 1.1. By definition, VMO F is a subspace of BMO F . Since BMO F is the dual space of HF1 , the space HF1 is a subspace of VMO ∗F . Conversely, we first note that, if f, g = 0 for all f ∈ HF1 , then g is the zero element of BMO F and hence g must be the zero of VMO F . Thus, HF1 is a total set of functionals on VMO F . Proposition 2.4 shows that HF1 is dense in VMO ∗F . For each x∗ ∈ VMO ∗F , there exists a sequence {fk } in HF1 such that fk , g → x∗ , g for all g ∈ C. It follows from Banach–Steinhaus theorem that {fk HF1 }k∈N is bounded. By Lemma 2.3, there exists f ∈ HF1 and a subsequence {fkl }l∈N such that x∗ , g = lim fkl , g = lim l→∞
l→∞ Rn
f gdμ = f, g
=
fkl gdμ for all x∗ ∈ Cc (Rn ).
Rn
Thus, the linear functional x∗ ∈ VMO ∗F is represented by f ∈ HF1 . Therefore, the proof is concluded. 2
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3. Proof of Theorem 1.2 Caffarelli and Gutiérrez [4] introduced the (centered) Hardy–Littlewood maximal operator MF with respect to F and the measure μ by 1 MF f (x) = sup |f (y)|dμ(y), t>0 μ(S(x, t)) S(x,t)
and proved that MF is of weak type (1, 1) with respect to the measure μ. Since MF is obviously bounded on L∞ (dμ), we may obtain the Lp (dμ), 1 < p < ∞, boundedness of MF by applying the Marcinkiewicz interpolation theorem. To show Theorem 1.2, we need to define the noncentered Hardy–Littlewood maximal F with respect to F and the measure μ by operator M 1 F f (x) = sup |f (y)|dμ(y), M x∈S∈F μ(S) S
where the supremum is taken over all sections S ∈ F containing x. Given S = S(y, t) ∈ F that contains x, the engulfing property yields 1 1 μ(S(x, θt)) |f (y)|dμ(y) ≤ |f (y)|dμ(y) μ(S) μ(S) μ(S(x, θt)) S
S(x,θt)
≤
μ(S(x, θt)) MF f (x) μ(S)
≤
μ(S(y, θ2 t)) MF f (x). μ(S(y, t))
By the doubling property (1.1), F f (x) ≤ A1+2 log2 θ MF f (x), M
(3.1)
F is of weak type (1, 1) and is of type (p, p), 1 < p ≤ ∞, with respect to the measure μ. and hence M We also need a weight class associated with the family F. A nonnegative locally integrable function ω is said to belong to Ap,F , 1 < p < ∞, if sup S∈F
1 μ(S)
ω(x)dμ(x)
1 μ(S)
S
p−1
ω(x)− p−1 dμ(x) 1
< ∞,
S
and ω is said to belong to A1,F if sup S∈F
1 μ(S)
ω(x)dμ(x)
ess sup ω −1 (x) < ∞. x∈S
S
Similar to the classical case, Ap,F classes have the following properties that will be used to prove Theorem 1.2. Lemma 3.1. Let ω ∈ Ap,F , 1 ≤ p < ∞. There exists a constant C > 0 such that, for any subset E of S ∈ F,
μ(E) μ(S)
p
≤C
ω(x)dμ(x) E
−1 ω(x)dμ(x)
S
.
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Proof. For p = 1, the definition of A1,F yields −1 −1 μ(E) = ω(x)ω(x) dμ(x) ≤ ess sup ω (x) ω(x)dμ(x) x∈S
E
≤ Cμ(S)
E
−1 ω(x)dμ(x)
ω(x)dμ(x).
S
E
For 1 < p < ∞, Hölder’s inequality and the definition of Ap,F give μ(E) = ω(x)1/p ω(x)−1/p dμ(x) E
1/p
≤
ω(x)dμ(x) E
≤C
1/p ω(x)dμ(x)
(p−1)/p
ω(x)−1/(p−1) dμ(x)
S
μ(S)
E
−1/p ω(x)dμ(x)
.
S
Hence the lemma follows. 2 F is of ω-weak type (1, 1) with respect to μ; that is, there exists a constant Theorem 3.2. If ω ∈ A1,F , then M C > 0 such that, for any λ > 0 and f ∈ L1ω (dμ), ω(x)dμ(x) ≤ F f (x)>λ} {x∈Rn :M
C λ
|f (x)|ω(x)dμ(x). Rn
To show Theorem 3.2, we need a covering lemma. Lemma 3.3. For 0 < t0 < ∞ and E ⊂ Rn , let FE = {S(x, t) : x ∈ E, 0 < t ≤ t0 } be a family in F. Then there exists a disjoint countable subfamily {S(xi , ti )} ⊂ FE satisfying the property: for any S(x, t) ∈ FE , there is an S(xi , ti ) such that S(x, t) ⊂ S(xi , 2θ2 ti ), where θ is the constant given in (2.1). Proof. We construct the subfamily inductively. First we take a set S(x1,1 , t1,1 ) ∈ FE , where 2−1 t0 < t1,1 ≤ t0 . Then we chose S(x1,2 , t1,2 ) ∈ FE such that 2−1 t0 < t1,2 ≤ t0 and S(x1,2 , t1,2 ) ∩ S(x1,1 , t1,1 ) = ∅. In general, if S(x1,j , t1,j ), j ≥ 1, have been chosen, we select S(x1,j+1 , t1,j+1 ) such that j (1-i) S(x1,j+1 , t1,j+1 ) i=1 S(x1,i , t1,i ) = ∅; (1-ii) 2−1 t0 < t1,j+1 ≤ t0 . Thus, we obtain a “maximal” sequence {S(x1,j , t1,j )}j∈N with respect to properties (1-i) and (1-ii), where the “maximal” is under the sense that for any S(x, t) ∈ FE with 2−1 t0 < t ≤ t0 , there is a j such that S(x, t) ∩ S(x1,j , t1,j ) = ∅. For k ∈ N, if the countable sequence {S(xk,j , tk,j )}j∈N of disjoint sets in FE has been chosen following the above way, then we may choose {S(xk+1,j , tk+1,j )}j∈N such that (k-i) S(xk+1,j , tk+1,j ) ∈ FE and 2−k−1 t0 < tk+1,j ≤ 2−k t0 for all j ∈ N; (k-ii) S(xk+1,j , tk+1,j ) S(xh,i , th,i ) = ∅ for all h ≤ k + 1 and i ∈ N (if h = k + 1, then i = j); (k-iii) the sequence {S(xk+1,j , tk+1,j )}j∈N is maximal with respect to properties (k-i) and (k-ii).
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For S(x, t) ∈ FE with 2−k0 t0 < t ≤ 2−k0 +1 t0 for some k0 ∈ N, there exists S(xk0 ,j0 , tk0 ,j0 ) such that S(x, t) ∩ S(xk0 ,j0 , tk0 ,j0 ) = ∅ and t < 2tk0 ,j0 . Suppose that y ∈ S(x, t) ∩ S(xk0 ,j0 , tk0 ,j0 ). By (2.1), S(x, t) ⊂ S(y, θt) ⊂ S(y, 2θtk0 ,j0 ). On the other hand, y ∈ S(xk0 ,j0 , tk0 ,j0 ) ⊂ S(xk0 ,j0 , 2θtk0 ,j0 ). Applying (2.1) again, we have S(x, t) ⊂ S(y, 2θtk0 ,j0 ) ⊂ S(xk0 ,j0 , 2θ2 tk0 ,j0 ). The disjoint countable subfamily {S(xk,j , tk,j )}k,j∈N is what we need. 2 We return to show Theorem 3.2. Proof of Theorem 3.2. By (3.1), it suffices to show that MF is of ω-weak type (1, 1) with respect to μ. For fixed t0 > 0, we define another maximal operator by
1 μ(S(x, t))
MF,t0 f (x) = sup
0
|f (y)|dμ(y). S(x,t)
Then {x ∈ R : MF,t0 f (x) > λ} ⊂ n
S(x, t) : t ≤ t0 and λμ(S(x, t)) <
|f (y)|dμ(y) . S(x,t)
For ω ∈ A1,F and f ∈ L1ω (dμ), we note that λμ(S(x, t)) < λμ(S(x, t)) ≤
S(x,t)
|f (y)|dμ(y) implies
ess sup ω −1 (y)
|f (y)|ω(y)dμ(y)
y∈S(x,t) S(x,t)
≤ Cμ(S(x, t))
−1 |f (y)|ω(y)dμ(y),
ω(y)dμ(y)
S(x,t)
S(x,r)
which is
ω(y)dμ(y) ≤ C
λ S(x,t)
|f (y)|ω(y)dμ(y).
(3.2)
S(x,t)
By Lemma 3.3, there exists a disjoint family of sections {S(xi , ti )} satisfying (3.2) such that {x ∈ Rn : MF,t0 f (x) > λ} ⊂
S(xi , 2θ2 ti ).
i
Using Lemma 3.1 and the doubling property (1.1), we get
−1
ω(y)dμ(y) S(xi ,2θ 2 ti )
so that have
S(xi ,2θ 2 ti )
ω(x)dμ(x) ≤ Cθ,ω
ω(y)dμ(y)
≤C
μ(S(xi , 2θ2 ti ) ≤ Cθ,ω μ(S(xi , ti ))
S(xi ,ti )
S(xi ,ti )
ω(y)dμ(y). Since {S(xi , ti )} are disjoint and satisfy (3.2), we
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ω(y)dμ(y) ≤ {x∈Rn :M
F ,t0 f (x)>λ}
i
ω(y)dμ(y)
S(xi
≤ Cθ,ω
,2θ 2 t
≤
i)
i
≤
ω(y)dμ(y)
S(xi ,ti )
Cθ,ω λ i Cθ,ω λ
11
|f (y)|ω(y)dμ(y)
S(xi ,ti )
|f (y)|ω(y)dμ(y). Rn
Since the constant Cθ,ω is independent of t0 , we take the limit t0 → ∞ and the proof is done. 2 Lemma 3.4. The weight ω ∈ A1,F if and only if there is a constant C > 0 such that F ω(x) ≤ Cω(x) M
μ-almost everywhere x ∈ Rn .
F ω(x) ≤ Cω(x) μ-almost everywhere. It is clear Proof. Suppose that there is a constant C > 0 such that M that 1 μ(S)
ω(y)dμ(y) ≤ Cω(x)
μ-almost everywhere x ∈ S, for any S ∈ F.
S
Hence ω ∈ A1,F . Conversely, Theorem 3.2 shows that there exists C > 0 such that, for any λ > 0 and f ∈ L1ω (dμ), F f (x)>λ} {x∈Rn :M
C ω(x)dμ(x) ≤ λ
|f (x)|ω(x)dμ(x). Rn
Suppose x ∈ S1 ⊂ S2 , where S1 , S2 ∈ F. Let f = χS1 and z ∈ S2 . Then F f (z) ≥ M
1 μ(S2 )
f (y)dμ(y) =
μ(S1 ) . μ(S2 )
S2
F f (x) ≥ μ(S1 )/μ(S2 )}. Hence, The above inequality shows that S2 ⊂ {x : M
ω(x)dμ(x) ≤ S2
ω(x)dμ(x) F f (x)≥μ(S1 )/μ(S2 )} {x:M
μ(S2 ) ≤C μ(S1 )
ω(x)dμ(x). S1
By Lebesgue’s differentiation theorem, we conclude the lemma. 2 F f δ ∈ A1,F for F f (x) < ∞ μ-almost everywhere. Then M Lemma 3.5. Let f ∈ L1loc (Rn ) such that M 0 ≤ δ < 1.
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Proof. By Lemma 3.4, it suffice to show that there exists a constant C such that, for any S ∈ F and μ-almost every x ∈ S, 1 μ(S)
F f M
δ
F f (x) δ . dμ ≤ C M
S
Fix S = S(x0 , t0 ) and decompose f as f = f1 + f2 , where f1 = f χθ2 S and f2 = f χ(θ2 S)c with θ2 S = F f1 (y) + M F f2 (y) and F f (y) ≤ M S(x0 , θ2 t0 ). Then M
δ
F f (y) M
F f1 (y) δ + M F f2 (y) δ ≤ M
for 0 ≤ δ < 1.
F is weak (1, 1) with respect to the measure μ, Kolmogorov’s inequality shows Since M 1 μ(S)
F f1 (y) δ dμ(y) ≤ M
1 C μ(S)1−δ f1 δL1 (dμ) = C μ(S) μ(S)
|f |dμ
δ
F f (x) δ . ≤C M
θ2 S
S
F f2 , given y ∈ S, for any S(y0 , R) ∈ F that contains y, we have S ⊂ S(y0 , θ2 max{t0 , R}). To estimate M If R < t0 , we have S(y0 , t0 ) ∩ S(x0 , t0 ) = ∅ and hence S(y0 , t0 ) ⊂ S(x0 , θ2 t0 ). Since S(y0 , R) ⊂ S(y0 , t0 ) ⊂ S(x0 , θ2 t0 ), S(y0 ,R) |f2 |dμ = 0 when R < t0 . It is clear that S ⊂ S(y0 , θ2 R) when R ≥ t0 . Thus,
1 μ(S(y0 , R))
C |f2 |dμ ≤ μ(S(y0 , θ2 R))
F f (x), |f2 |dμ ≤ C M
S(y0 ,θ 2 R)
S(y0 ,R)
F f (x) for any y ∈ S. Therefore, F f2 (y) ≤ C M so that M 1 μ(S)
F f2 (y) δ dμ(y) ≤ C M F f (x) δ M
S
and the proof is completed. 2 Lemma 3.6. If ω ∈ A2,F , then log ω ∈ BMO F . Proof. Assume ω ∈ A2,F and write f := log ω, which is equivalent to exp(f ) ∈ A2,F . Given S ∈ F, the definition of A2,F gives 1 1 exp(f )dμ exp(−f )dμ ≤ C; μ(S) μ(S) S
S
equivalently, 1 1 exp f − mS (f ) dμ exp mS (f ) − f dμ ≤ C, μ(S) μ(S) S
(3.3)
S
1 where mS (f ) = μ(S) f dμ. By Jensen’s inequality, each factor is at least 1 and at most C. Therefore, S inequality (3.3) implies 1 μ(S)
exp(|f − mS (f )|)dμ ≤ 2C, S
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and so 1 μ(S)
|f − mS (f )|dμ ≤ 2C. S
Hence, f ∈ BMO F .
2
We are ready to show the second main result. Proof of Theorem 1.2. Since HF1 is a subspace of L1 (dμ), it follows from Fatou’s lemma that f ∈ L1 (dμ). To show (1.2), it suffices to consider for all φ ∈ Cc ∩ Lip. We can assume that φL1 (dμ) ≤ 1, φL∞ (dμ) ≤ 1, |φ(x) − φ(y)| ≤ ρ(x, y) for x, y ∈ Rn and that supp(φ) is compact. Fix δ ∈ (0, 1) and pick η > 0 such that η exp(δ −1 ) ≤ δ and E |f |dμ ≤ δ whenever μ(E) ≤ Cη exp(δ −1 ). Now choose k large enough so that
μ(Ek ) := μ
x ∈ supp(φ) : |fk (x) − f (x)| > η
≤ η.
Define F χ )(x) . τ (x) = max 0, 1 + δ log(M Ek It is clear that 0 ≤ τ (x) ≤ 1 and τ = 1 μ-almost everywhere on Ek . Also, τ BMOF ≤ F χ )1/2 BMO ≤ Cδ due to Lemmas 3.5 and 3.6. By the weak (1, 1) estimate for M F with 2δ log(M Ek F respect to μ, μ supp(τ ) ≤ Cμ(Ek ) exp(δ −1 ) ≤ Cη exp(δ −1 ). Consequently, |f |dμ ≤ δ. supp(τ )
We now write (f − fk )φdμ ≤ (f − fk )φ(1 − τ )dμ + (f − fk )φτ dμ Rn
Rn
Rn
|f |dμ + fk φτ dμ
supp(τ )
Rn
≤ ηφL1 (dμ) +
≤ δ + δ + fk φτ dμ. Rn
The proof of (1.2) will therefore be established provided we verify φτ BMOF ≤ Cδ. Note that |φ(x) − mS (φ)| ≤ sup ρ(x, y) for x ∈ S = S(x0 , t0 ). Thus, y∈S
(3.4)
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1 μ(S)
2 |φτ − mS (φτ )|dμ ≤ μ(S)
S
|φτ − mS (φ)mS (τ )|dμ S
2 ≤ μ(S)
2|mS (φ)| |φτ − mS (φ)τ |dμ + μ(S)
S
2 ≤ sup ρ(x, y) μ(S) x,y∈S
|τ − mS (τ )|dμ S
|τ |dμ + 2φL∞ (dμ) τ BMOF . S
By (2.4)–(2.6), sup ρ(x, y)
sup ρ(x, y) x,y∈S
μ(S)
≤C
x,y∈S
μ(Bd (x0 , 2θt0 ))
≤ C.
Hence, 1 μ(S)
|φτ − mS (φτ )|dμ ≤ C
S
|τ |dμ + Cδ ≤ Cμ supp(τ ) + Cδ
S
≤ Cη exp(δ −1 ) + Cδ ≤ Cδ, and (3.4) follows. By weak∗ compactness of the ball in HF1 , there exists a g ∈ HF1 with gHF1 ≤ 1 and a subsequence {fkl }l∈N such that {fkl }l∈N weak∗ converges to g. By (1.2), we have f φ = gφ for all φ ∈ Cc ∩ Lip, and hence f = g ∈ HF1 . 2 Acknowledgment The author is grateful to the referee for valuable suggestions. References [1] H. Aimar, L. Forzani, R. Toledano, Balls and quasi-metrics: a space of homogeneous type modeling the real analysis related to the Monge–Ampère equation, J. Fourier Anal. Appl. 4 (1998) 377–381. [2] L.A. Caffarelli, Some regularity properties of solutions of Monge–Ampère equation, Comm. Pure Appl. Math. XLIV (1991) 965–969. [3] L.A. Caffarelli, Boundary regularity of maps with convex potentials, Comm. Pure Appl. Math. XLV (1992) 1141–1151. [4] L.A. Caffarelli, C.E. Gutiérrez, Real analysis related to the Monge–Ampère equation, Trans. Amer. Math. Soc. 348 (1996) 1075–1092. [5] L.A. Caffarelli, C.E. Gutiérrez, Properties of the solutions of the linearized Monge–Ampère equation, Amer. J. Math. 119 (1997) 423–465. [6] R.R. Coifman, G. Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977) 569–645. [7] R. Coifman, P.-L. Lions, Y. Meyer, S. Semmes, Compensated compactness and Hardy spaces, J. Math. Pures Appl. 72 (1993) 247–286. [8] Y. Ding, C.-C. Lin, Hardy spaces associated to the sections, Tohoku Math. J. 57 (2005) 147–170. [9] N. Dunford, J. Schwartz, Linear Operators. I, Interscience, New York and London, 1964. [10] P.W. Jones, J.-L. Journé, On weak convergence in H 1 (Rd ), Proc. Amer. Math. Soc. 120 (1994) 137–138. [11] R.A. Macías, C. Segovia, Lipschitz functions on spaces of homogeneous type, Adv. Math. 33 (1979) 257–270.
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J. Math. Anal. Appl. ••• (••••) •••–•••
Contents lists available at ScienceDirect
Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa
On the weak* convergence in HF1 (Rn ) ✩ Chin-Cheng Lin Department of Mathematics, National Central University, Chung-Li, 320, Taiwan, Republic of China
a r t i c l e
i n f o
Article history: Received 8 August 2016 Available online xxxx Submitted by C. Gutierrez Keywords: Ap weights BMO Hardy spaces Monge–Ampère equation VMO
a b s t r a c t Let VMO F denote the closure of Cc ∩ Lip with respect to the seminorm · BMOF , where F is a family of sections and the space BMO F associated to the family F was introduced by Caffarelli and Gutiérrez (1996) [4]. Sections play an important role in the investigation of Monge–Ampère equation and the linearized Monge–Ampère equation. 1 defined in Ding and Lin We show that the dual of VMO F is the Hardy space HF (2005) [8]. As an application, we prove that μ-almost everywhere convergence of a 1 sequence of functions bounded in HF to a function in L1 (dμ) implies the weak* convergence. © 2017 Elsevier Inc. All rights reserved.
1. Introduction In 1996, Caffarelli and Gutiérrez [4] studied real analysis related to the Monge–Ampère equation and introduced axiomatically a family of bounded convex sets F = {S(x, t) ⊂ Rn : x ∈ Rn and t > 0}, which they called sections, and a Borel measure μ satisfying the doubling property with respect to the parameter t, μ(S(x, 2t)) ≤ Cμ(S(x, t)). The sections S(x, t) satisfy some properties modeled on the solutions of the real Monge–Ampère equation det D2 φ = μ for a convex function φ in Rn and the generated measure μ. Caffarelli and Gutiérrez gave a Besicovitch type covering lemma for the family F and set up a variant of the Calderón–Zygmund decomposition by applying this covering lemma together with the doubling condition of μ. Such a decomposition plays an important role in the study of the linearized Monge–Ampère equation (see [5]). As applications of the decomposition, Caffarelli and Gutiérrez [4] defined the Hardy–Littlewood maximal operator MF and BMO F (Rn ) space associated to the family F, and obtained the weak type (1, 1) boundedness of MF and the John–Nirenberg inequality for BMO F (Rn ). Let us recall the definition of sections. For each x ∈ Rn , we have a one-parameter family {S(x, t)}t>0 of open and bounded convex subsets of Rn containing x, called sections, which is monotone increasing in t, i.e., S(x, t) ⊂ S(x, t ) for t ≤ t , and satisfies the following three conditions: ✩
Research supported by Ministry of Science and Technology of Taiwan under Grant #MOST 103-2115-M-008-003-MY3. E-mail address: [email protected].
http://dx.doi.org/10.1016/j.jmaa.2017.06.005 0022-247X/© 2017 Elsevier Inc. All rights reserved.
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(A) There exist positive constants K1 , K2 , K3 and 1 , 2 such that given two sections S(x0 , t0 ), S(x, t) with t ≤ t0 satisfying S(x0 , t0 ) ∩ S(x, t) = ∅, and given T an affine transformation that “normalizes” S(x0 , t0 ); that is, B(0, 1/n) ⊂ T (S(x0 , t0 )) ⊂ B(0, 1), there exists z ∈ B(0, K3 ) depending on S(x0 , t0 ) and S(x, t), which satisfies B z, K2 (t/t0 )2 ⊂ T (S(x, t)) ⊂ B z, K1 (t/t0 )1 and T (x) ∈ B z, (1/2)K2 (t/t0 )2 . Here and below B(x, t) denotes the Euclidean ball centered at x with radius t. (B) There exists a constant δ > 0 such that given a section S(x, t) and y ∈ / S(x, t), if T is an affine transformation that normalizes S(x, t), then B(T (y), δ ) ∩ T (S(x, (1 − )t)) = ∅ (C)
t>0
S(x, t) = {x} and
t>0
for any 0 < < 1.
S(x, t) = Rn .
In addition, we also assume that a Borel measure μ is given, which is finite on compact sets, μ(Rn ) = ∞, and satisfies the following doubling property with respect to the parameter t; that is, there exists a constant A such that μ(S(x, 2t)) ≤ Aμ(S(x, t))
for any x ∈ Rn and t > 0.
(1.1)
Sections play an important role in the investigation of Monge–Ampère equation and the linearized Monge– Ampère equation (see [2–5]). An example of a family of sections was given in [4]. Caffarelli and Gutiérrez [4] defined the space BMO F (Rn ) associated with the family F := {S(x, t) ⊂ Rn : x ∈ Rn and t > 0} and the Borel measure μ satisfying the doubling condition (1.1). Let f be a real-valued function defined on Rn . We say that f ∈ BMO F (Rn ) if f BMOF := sup
S∈F
1 μ(S)
|f (x) − mS (f )|dμ(x) < ∞, S
where mS (f ) denotes the mean of f over the section S defined by mS (f ) =
1 μ(S)
f (x)dμ(x). S
In 2005, Ding and Lin [8] extended the study to the context of Hardy spaces HF1 (Rn ) associated to the family F and a doubling measure μ. They showed the duality of HF1 and BMO F . In this paper, we will investigate the predual of HF1 . We first recall the definition of HF1 . Let 1 < q ≤ ∞. A function a ∈ Lq (dμ) is called a (1, q)-atom if there exists a section S(x0 , t0 ) ∈ F such that
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(i) supp(a) ⊂ S(x0 , t0 ); a(x)dμ(x) = 0; (ii) Rn
(iii) aLq (dμ) ≤ μ(S(x0 , t0 ))1/q−1 . The atomic Hardy space HF1,q (Rn ) is defined by HF1,q (Rn )
=
λj aj (x) : each aj is a (1, q)-atom and
j
|λj | < ∞
j
with norm f H 1,q = inf j |λj |, where the infimum is taken over all decompositions of f = j λj aj above. F It is clear that a ∈ L1 (dμ) with aL1 (dμ) ≤ 1 for each (1, q)-atom a, and HF1 (Rn ) is a subspace of L1 (dμ). It was shown in [8, Theorem 1.1] that all atomic Hardy spaces HF1,q (Rn ), 1 < q ≤ ∞, are equivalent. Hence, one defines the Hardy space HF1 to be HF1 (Rn ) := HF1,∞ (Rn )
and
f HF1 := f H 1,∞ . F
Hereafter, an “atom” will mean a (1, ∞)-atom. As usual, we use Cc (Rn ) to denote all continuous functions with compact support in Rn and use Lip(Rn ) to denote the collection of functions satisfying that there is a constant C such that |f (x) − f (y)| ≤ Cρ(x, y)
for all x, y ∈ Rn ,
where ρ is the quasi-metric described in (2.6) below. Let VMO F (Rn ) denote the closure of Cc (Rn ) ∩Lip(Rn ) with respect to the seminorm · BMOF . Our first result is about the predual of HF1 . Theorem 1.1. HF1 (Rn ) is the dual of VMO F (Rn ). As an application, we prove that μ-almost everywhere convergence of a sequence of functions bounded in HF1 implies weak* convergence, which is an HF1 version of the Jones–Journé theorem [10]. The Jones–Journé theorem is useful in the application of Hardy spaces to compensated compactness (see [7]). Theorem 1.2. Suppose that a sequence of functions {fk } ⊂ HF1 (Rn ) satisfies fk HF1 ≤ 1 for all k and fk (x) → f (x) for μ-almost every x ∈ Rn . Then f ∈ HF1 (Rn ), f HF1 ≤ 1, and
fk (x)φ(x)dμ(x) −→ Rn
f (x)φ(x)dμ(x)
for all φ ∈ VMO F (Rn ).
(1.2)
Rn
2. Proof of Theorem 1.1 From properties (A) and (B) of sections, Aimar, Forzani, and Toledano [1] obtained the following engulfing property: There exists a constant θ > 1, depending only on δ, K1 , and 1 , such that for each y ∈ S(x, t) ∈ F we have S(y, t) ⊂ S(x, θt)
and
S(x, t) ⊂ S(y, θt).
(2.1)
They also showed that there exists a quasi-metric d(x, y) on Rn with respect to F defined by d(x, y) = inf{t : x ∈ S(y, t) and y ∈ S(x, t)}.
(2.2)
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The triangular constant of the quasi-metric d is just the θ appeared in (2.1): d(x, y) ≤ θ d(x, z) + d(z, y)
for any x, y, z ∈ Rn .
(2.3)
for any x ∈ Rn and t > 0,
(2.4)
Also, t S x, ⊂ Bd (x, t) ⊂ S(x, t) 2θ
where Bd (x, t) is the d-ball centered at x with radius t defined by Bd (x, t) := {y ∈ Rn : d(x, y) < t}. By (1.1) and (2.4), if we choose k0 ∈ N satisfying 2k0 −2 ≥ θ, then μ(Bd (x, 2t)) ≤ Ak0 μ(Bd (x, t))
for any x ∈ Rn and t > 0.
(2.5)
Hence, (Rn , d, μ) is a space of homogeneous type introduced by Coifman and Weiss [6]. Macías and Segovia [11, Theorems 2 and 3] have shown that one can replace d by another quasi-metric ρ such that there exist constants C > 0 and ε ∈ (0, 1) satisfying ⎧ ⎪ ⎨ ρ(z, w) ≈ inf{μ(Bd ) : Bd are d-balls containing z and w}; μ(Bρ (z, r)) ≈ r, ∀ z ∈ Rn , r > 0, where Bρ (z, r) := {w ∈ Rn : ρ(z, w) < r}; ⎪ ⎩ |ρ(z, w) − ρ(z , w)| ≤ C(ρ(z, z ))ε [ρ(z, w) + ρ(z , w)]1−ε , ∀ z, z , w ∈ Rn .
(2.6)
In order to show the first theorem, we need some lemmas. We may use a diagonalization argument to obtain the following lemma, which also can be seen in [6, Lemma (4.3)].
∞ Lemma 2.1. Suppose that λkj ≥ 0, j, k = 1, 2, · · · , satisfy j=1 λkj ≤ 1 for each k = 1, 2, · · · . Then there exists an increasing sequence of natural numbers, k1 < k2 < · · · < kn < · · · , such that limm→∞ λkj m = λj
∞ for each j and j=1 λj ≤ 1. The following lemma is of a geometric nature. Lemma 2.2. Suppose that m is an integer and θ > 1 is given in (2.1). Then Rn is the union of sections 2m {S(xm ) : j = 1, 2, · · · } such that each point of Rn belongs to at most N of these sections (N is j ,θ independent of m) and each f ∈ HF1 has the representation f=
∞ ∞
m λm j aj ,
j=1 m=−∞ m 2m+2 where am ) and j is a (1, ∞)-atom supported in S(xj , θ
j∈N, m∈Z
|λm j | ≤ C f H 1 . F
Proof. Given a section S(x, t), where x ∈ Rn and t > 0, let S(y, t) satisfy that S(x, t) ∩ S(y, t) = ∅. Choose z ∈ S(x, t) ∩ S(y, t). By (2.1), we have S(y, t) ⊂ S(z, θt) and S(z, t) ⊂ S(x, θt), and hence S(y, t) ⊂ S(z, θt) ⊂ S(x, θ2 t). Therefore, y∈Rn
{S(y, t) | S(y, t) ∩ S(x, t) = ∅} ⊂ S(x, θ2 t).
(2.7)
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Plug x = x11 and t = 1 into (2.7) to give
S(y, 1) | S(y, 1) ∩ S(x11 , 1) = ∅ ⊂ S(x11 , θ2 ).
y∈Rn
Similarly, we pick x12 ∈ y ∈ Rn | S(y, 1) ∩ S(x11 , 1) = ∅ such that
S(y, 1) | S(y, 1) ∩ S(x12 , 1) = ∅ ⊂ S(x12 , θ2 );
y∈Rn
we pick x13 ∈ y ∈ Rn | S(y, 1) ∩ [S(x11 , 1) ∪ S(x12 , 1)] = ∅ such that
S(y, 1) | S(y, 1) ∩ S(x13 , 1) = ∅ ⊂ S(x13 , θ2 ).
y∈Rn
Continuing the same method for choosing points, we obtain a sequence of sections {S(x1j , θ2 )}j∈N such that 1 2 n 1 j∈N S(xj , θ ) = R . Since these sections {S(xj , 1)}j∈N are disjoint, it follows from (2.4) that the d-balls 1 1 {Bd (xj , 1)}j∈N are disjoint, and hence {Bd (xj , 2θ3 )}j∈N are finitely overlapping. By (2.4) again, the sections {S(x1j , θ2 )}j∈N are finitely overlapping. Using the same argument as above (by choosing t = θ2 ), we can also obtain a finitely overlapping sequence of sections {S(x2j , θ4 )}j∈N such that j∈N S(x2j , θ4 ) = Rn . By induction, for each m ∈ Z, there exists a sequence {xm j }j∈N , such that 2m ∞ ) j=1 is a finitely overlapping sequence of sections; (i) S(xm j ,θ ∞ 2m ) = Rn . (ii) j=1 S(xm j ,θ
For f = k∈N λk ak ∈ HF1 , where |λk | < ∞ and ak are (1, ∞)-atom supporting on S(xk , tk ), we may assume supp(ak ) ⊂ S(xk , θ2m ) for some m = m(k) due to θ2m → ∞ as m → ∞. We choose the smallest integer m such that the choice of m is unique for each ak . Note that, the mapping k → m of indices form N to Z is neither injective nor surjective. That is, for each m, it may have more than one corresponding ak ’s and may have no corresponding ak ’s. For each m, which has corresponding ak ’s, the supporting section S(xk , θ2m ) of each ak satisfies one of the followings: 2m 1 S(xk , θ 2m ) ∩ S(xm ) = ∅; 1 ,θ 2m m 2m 2m 2 S(xk , θ ) ∩ S(x1 , θ ) = ∅ and S(xk , θ2m ) ∩ S(xm ) = ∅; 2 ,θ .. . i−1 2m 2m i S(xk , θ 2m ) ∩ [ j=1 S(xm )] = ∅ and S(xk , θ2m ) ∩ S(xm ) = ∅; j ,θ i ,θ .. .
i above, we set For the general case
i {k: case }
m λk ak = λm i ai ,
m 2m+2 where λm ) i = μ S(xi , θ
i {k: case }
|λk | . μ(S(xk , θ2m ))
m m 2m+2 Then am ). We set λm i is a (1, ∞)-atom with supp(ai ) ⊂ S(xi , θ i = 0, i ∈ N, if there is no corresponding 2m m 2m 2m+2 ak ’s for this m. For S(xk , θ ) ∩ S(xi , θ ) = ∅, we have S(xk , θ2m+2 ) ∩ S(xm ) = ∅ and hence i ,θ m 2m+2 2m+4 S(xi , θ ) ⊂ S(xk , θ ). By (1.1),
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|λm i |≤
|λk |
i {k: case }
μ(S(xk , θ2m+4 )) ≤ A1+4 log2 θ μ(S(xk , θ2m ))
|λk |.
i {k: case }
Therefore,
i∈N, m∈Z
i {k: case }
1+4 log2 θ |λm i |≤A
i∈N, m∈Z
|λk | = A1+4 log2 θ
|λk | ≤ C f H 1 , F
k∈N
and the proof is completed. 2 Lemma 2.3. Suppose fk H 1 ≤ 1 for all k ∈ N. Then there exist a subsequence {fkl }l∈N and f ∈ HF1 (Rn ) F such that
lim
l→∞ Rn
fkl gdμ =
for all g ∈ Cc (Rn ).
f gdμ
(2.8)
Rn
Proof. For each k ∈ N, we apply Lemma 2.2 to express fk =
∞ ∞
m λm i (k)ai (k),
i=1 m=−∞ m 2m+2 where am ) for all k ∈ N and i (k) are (1, ∞)-atoms supported on S(xi , θ
i∈N, m∈Z
1 |λm i (k)| ≤ Cfk HF
≤ C. By Lemma 2.1, there exists an strictly increasing sequence {k }∈N of natural numbers, such that
m m m liml→∞ λm i (kl ) = λi for each (i, m) ∈ N × Z and i,m |λi | ≤ C. The definition of ai (k) gives m 2m+2 −1 am ) i (k)L∞ (dμ) ≤ μ S(xi , θ
for all k ∈ N.
∗ Since L1μ is separable, Banach–Alaoglu theorem shows that there exists a subsequence {am i (kl )}l∈N weak −1 m m m 2m+2 convergent to a function ai satisfying ai L∞ (dμ) ≤ μ S(xi , θ ) . By using the diagonalization argument, there exists a subsequence of {kl } (for simplicity we still use the same indices {kl } to denote this subsequence of {kl }) such that {am (kl )}l∈N converges to am , as l → ∞, for all (i, m). It is easy to check i
i m m m m that ai are (1, ∞)-atoms. Let f = i,m λi ai . Since i,m |λi | ≤ C, we have f ∈ HF1 (Rn ). To show (2.8), we write
fkl gdμ =
m λm i (kl )ai (kl )gdμ
Rn i,m
Rn
=
m
⎛ =⎝
λm i (kl )
i
m<−M
am i (kl )gdμ
Rn
+
−M ≤m≤M
+
m>M
⎞ ⎠
i
λm i (kl )
am i (kl )gdμ
Rn
:= I + II + III , where M is a large number that will be determined later. For term I, given ε > 0, the continuity of g yields m 2m+2 |g(x) − g(xm ) whenever M is large enough. Hence, the vanishing conditions of i )| < ε for x ∈ S(xi , θ m ai (k) show that
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m m m |I| = λi (kl ) ai (kl )(x) {g(x) − g(xi )} dμ(x) m<−M i 2m+2 ) S(xm i ,θ m ≤ |λm |am i (kl )| i (kl )||g(x) − g(xi )|dμ(x) < Cε. i
m<−M
(2.9)
2m+2 ) S(xm i ,θ
2m+2 For each m with −M ≤ m ≤ M , the compact support of g interests a finite number of {S(xm )}i∈N i ,θ m 2m since {S(xi , θ )}i∈N are finitely overlapping. The sum in II involves finitely many terms only, independently of k. Thus,
−M ≤m≤M
i
II =
=
Rn −M ≤m≤M
λm i (kl )
am i (kl )gdμ
Rn
m λm i (kl )ai (kl )gdμ −→
i
(2.10) f gdμ
Rn
m as l → ∞ and M → ∞. To estimate III , we note that i,m |λm i (k)|ai (k)L1 (dμ) gL∞ (dμ) ≤ CgL∞ (dμ) . m 2m+2 −1 We use the estimate am ) which is uniformly small for large m. Given i (k)L∞ (dμ) ≤ μ S(xi , θ ε > 0, |III | ≤
m>M
m |λm i (k)|ai (k)L1 (dμ) gL∞ (dμ) < ε
for M large enough.
(2.11)
i
Combining (2.9)–(2.11), we finish the proof. 2 To prove the duality between VMO F and HF1 , we need a functional analysis result that can be found in [9, p. 439, Exercise 41]. Proposition 2.4. Let X be a locally convex linear topological space and Y be a linear subspace of X ∗ . Then Y is X-dense in X ∗ if and only if Y is a total set of functionals on X. Proof of Theorem 1.1. By definition, VMO F is a subspace of BMO F . Since BMO F is the dual space of HF1 , the space HF1 is a subspace of VMO ∗F . Conversely, we first note that, if f, g = 0 for all f ∈ HF1 , then g is the zero element of BMO F and hence g must be the zero of VMO F . Thus, HF1 is a total set of functionals on VMO F . Proposition 2.4 shows that HF1 is dense in VMO ∗F . For each x∗ ∈ VMO ∗F , there exists a sequence {fk } in HF1 such that fk , g → x∗ , g for all g ∈ C. It follows from Banach–Steinhaus theorem that {fk HF1 }k∈N is bounded. By Lemma 2.3, there exists f ∈ HF1 and a subsequence {fkl }l∈N such that x∗ , g = lim fkl , g = lim l→∞
l→∞ Rn
f gdμ = f, g
=
fkl gdμ for all x∗ ∈ Cc (Rn ).
Rn
Thus, the linear functional x∗ ∈ VMO ∗F is represented by f ∈ HF1 . Therefore, the proof is concluded. 2
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3. Proof of Theorem 1.2 Caffarelli and Gutiérrez [4] introduced the (centered) Hardy–Littlewood maximal operator MF with respect to F and the measure μ by 1 MF f (x) = sup |f (y)|dμ(y), t>0 μ(S(x, t)) S(x,t)
and proved that MF is of weak type (1, 1) with respect to the measure μ. Since MF is obviously bounded on L∞ (dμ), we may obtain the Lp (dμ), 1 < p < ∞, boundedness of MF by applying the Marcinkiewicz interpolation theorem. To show Theorem 1.2, we need to define the noncentered Hardy–Littlewood maximal F with respect to F and the measure μ by operator M 1 F f (x) = sup |f (y)|dμ(y), M x∈S∈F μ(S) S
where the supremum is taken over all sections S ∈ F containing x. Given S = S(y, t) ∈ F that contains x, the engulfing property yields 1 1 μ(S(x, θt)) |f (y)|dμ(y) ≤ |f (y)|dμ(y) μ(S) μ(S) μ(S(x, θt)) S
S(x,θt)
≤
μ(S(x, θt)) MF f (x) μ(S)
≤
μ(S(y, θ2 t)) MF f (x). μ(S(y, t))
By the doubling property (1.1), F f (x) ≤ A1+2 log2 θ MF f (x), M
(3.1)
F is of weak type (1, 1) and is of type (p, p), 1 < p ≤ ∞, with respect to the measure μ. and hence M We also need a weight class associated with the family F. A nonnegative locally integrable function ω is said to belong to Ap,F , 1 < p < ∞, if sup S∈F
1 μ(S)
ω(x)dμ(x)
1 μ(S)
S
p−1
ω(x)− p−1 dμ(x) 1
< ∞,
S
and ω is said to belong to A1,F if sup S∈F
1 μ(S)
ω(x)dμ(x)
ess sup ω −1 (x) < ∞. x∈S
S
Similar to the classical case, Ap,F classes have the following properties that will be used to prove Theorem 1.2. Lemma 3.1. Let ω ∈ Ap,F , 1 ≤ p < ∞. There exists a constant C > 0 such that, for any subset E of S ∈ F,
μ(E) μ(S)
p
≤C
ω(x)dμ(x) E
−1 ω(x)dμ(x)
S
.
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Proof. For p = 1, the definition of A1,F yields −1 −1 μ(E) = ω(x)ω(x) dμ(x) ≤ ess sup ω (x) ω(x)dμ(x) x∈S
E
≤ Cμ(S)
E
−1 ω(x)dμ(x)
ω(x)dμ(x).
S
E
For 1 < p < ∞, Hölder’s inequality and the definition of Ap,F give μ(E) = ω(x)1/p ω(x)−1/p dμ(x) E
1/p
≤
ω(x)dμ(x) E
≤C
1/p ω(x)dμ(x)
(p−1)/p
ω(x)−1/(p−1) dμ(x)
S
μ(S)
E
−1/p ω(x)dμ(x)
.
S
Hence the lemma follows. 2 F is of ω-weak type (1, 1) with respect to μ; that is, there exists a constant Theorem 3.2. If ω ∈ A1,F , then M C > 0 such that, for any λ > 0 and f ∈ L1ω (dμ), ω(x)dμ(x) ≤ F f (x)>λ} {x∈Rn :M
C λ
|f (x)|ω(x)dμ(x). Rn
To show Theorem 3.2, we need a covering lemma. Lemma 3.3. For 0 < t0 < ∞ and E ⊂ Rn , let FE = {S(x, t) : x ∈ E, 0 < t ≤ t0 } be a family in F. Then there exists a disjoint countable subfamily {S(xi , ti )} ⊂ FE satisfying the property: for any S(x, t) ∈ FE , there is an S(xi , ti ) such that S(x, t) ⊂ S(xi , 2θ2 ti ), where θ is the constant given in (2.1). Proof. We construct the subfamily inductively. First we take a set S(x1,1 , t1,1 ) ∈ FE , where 2−1 t0 < t1,1 ≤ t0 . Then we chose S(x1,2 , t1,2 ) ∈ FE such that 2−1 t0 < t1,2 ≤ t0 and S(x1,2 , t1,2 ) ∩ S(x1,1 , t1,1 ) = ∅. In general, if S(x1,j , t1,j ), j ≥ 1, have been chosen, we select S(x1,j+1 , t1,j+1 ) such that j (1-i) S(x1,j+1 , t1,j+1 ) i=1 S(x1,i , t1,i ) = ∅; (1-ii) 2−1 t0 < t1,j+1 ≤ t0 . Thus, we obtain a “maximal” sequence {S(x1,j , t1,j )}j∈N with respect to properties (1-i) and (1-ii), where the “maximal” is under the sense that for any S(x, t) ∈ FE with 2−1 t0 < t ≤ t0 , there is a j such that S(x, t) ∩ S(x1,j , t1,j ) = ∅. For k ∈ N, if the countable sequence {S(xk,j , tk,j )}j∈N of disjoint sets in FE has been chosen following the above way, then we may choose {S(xk+1,j , tk+1,j )}j∈N such that (k-i) S(xk+1,j , tk+1,j ) ∈ FE and 2−k−1 t0 < tk+1,j ≤ 2−k t0 for all j ∈ N; (k-ii) S(xk+1,j , tk+1,j ) S(xh,i , th,i ) = ∅ for all h ≤ k + 1 and i ∈ N (if h = k + 1, then i = j); (k-iii) the sequence {S(xk+1,j , tk+1,j )}j∈N is maximal with respect to properties (k-i) and (k-ii).
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For S(x, t) ∈ FE with 2−k0 t0 < t ≤ 2−k0 +1 t0 for some k0 ∈ N, there exists S(xk0 ,j0 , tk0 ,j0 ) such that S(x, t) ∩ S(xk0 ,j0 , tk0 ,j0 ) = ∅ and t < 2tk0 ,j0 . Suppose that y ∈ S(x, t) ∩ S(xk0 ,j0 , tk0 ,j0 ). By (2.1), S(x, t) ⊂ S(y, θt) ⊂ S(y, 2θtk0 ,j0 ). On the other hand, y ∈ S(xk0 ,j0 , tk0 ,j0 ) ⊂ S(xk0 ,j0 , 2θtk0 ,j0 ). Applying (2.1) again, we have S(x, t) ⊂ S(y, 2θtk0 ,j0 ) ⊂ S(xk0 ,j0 , 2θ2 tk0 ,j0 ). The disjoint countable subfamily {S(xk,j , tk,j )}k,j∈N is what we need. 2 We return to show Theorem 3.2. Proof of Theorem 3.2. By (3.1), it suffices to show that MF is of ω-weak type (1, 1) with respect to μ. For fixed t0 > 0, we define another maximal operator by
1 μ(S(x, t))
MF,t0 f (x) = sup
0
|f (y)|dμ(y). S(x,t)
Then {x ∈ R : MF,t0 f (x) > λ} ⊂ n
S(x, t) : t ≤ t0 and λμ(S(x, t)) <
|f (y)|dμ(y) . S(x,t)
For ω ∈ A1,F and f ∈ L1ω (dμ), we note that λμ(S(x, t)) < λμ(S(x, t)) ≤
S(x,t)
|f (y)|dμ(y) implies
ess sup ω −1 (y)
|f (y)|ω(y)dμ(y)
y∈S(x,t) S(x,t)
≤ Cμ(S(x, t))
−1 |f (y)|ω(y)dμ(y),
ω(y)dμ(y)
S(x,t)
S(x,r)
which is
ω(y)dμ(y) ≤ C
λ S(x,t)
|f (y)|ω(y)dμ(y).
(3.2)
S(x,t)
By Lemma 3.3, there exists a disjoint family of sections {S(xi , ti )} satisfying (3.2) such that {x ∈ Rn : MF,t0 f (x) > λ} ⊂
S(xi , 2θ2 ti ).
i
Using Lemma 3.1 and the doubling property (1.1), we get
−1
ω(y)dμ(y) S(xi ,2θ 2 ti )
so that have
S(xi ,2θ 2 ti )
ω(x)dμ(x) ≤ Cθ,ω
ω(y)dμ(y)
≤C
μ(S(xi , 2θ2 ti ) ≤ Cθ,ω μ(S(xi , ti ))
S(xi ,ti )
S(xi ,ti )
ω(y)dμ(y). Since {S(xi , ti )} are disjoint and satisfy (3.2), we
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ω(y)dμ(y) ≤ {x∈Rn :M
F ,t0 f (x)>λ}
i
ω(y)dμ(y)
S(xi
≤ Cθ,ω
,2θ 2 t
≤
i)
i
≤
ω(y)dμ(y)
S(xi ,ti )
Cθ,ω λ i Cθ,ω λ
11
|f (y)|ω(y)dμ(y)
S(xi ,ti )
|f (y)|ω(y)dμ(y). Rn
Since the constant Cθ,ω is independent of t0 , we take the limit t0 → ∞ and the proof is done. 2 Lemma 3.4. The weight ω ∈ A1,F if and only if there is a constant C > 0 such that F ω(x) ≤ Cω(x) M
μ-almost everywhere x ∈ Rn .
F ω(x) ≤ Cω(x) μ-almost everywhere. It is clear Proof. Suppose that there is a constant C > 0 such that M that 1 μ(S)
ω(y)dμ(y) ≤ Cω(x)
μ-almost everywhere x ∈ S, for any S ∈ F.
S
Hence ω ∈ A1,F . Conversely, Theorem 3.2 shows that there exists C > 0 such that, for any λ > 0 and f ∈ L1ω (dμ), F f (x)>λ} {x∈Rn :M
C ω(x)dμ(x) ≤ λ
|f (x)|ω(x)dμ(x). Rn
Suppose x ∈ S1 ⊂ S2 , where S1 , S2 ∈ F. Let f = χS1 and z ∈ S2 . Then F f (z) ≥ M
1 μ(S2 )
f (y)dμ(y) =
μ(S1 ) . μ(S2 )
S2
F f (x) ≥ μ(S1 )/μ(S2 )}. Hence, The above inequality shows that S2 ⊂ {x : M
ω(x)dμ(x) ≤ S2
ω(x)dμ(x) F f (x)≥μ(S1 )/μ(S2 )} {x:M
μ(S2 ) ≤C μ(S1 )
ω(x)dμ(x). S1
By Lebesgue’s differentiation theorem, we conclude the lemma. 2 F f δ ∈ A1,F for F f (x) < ∞ μ-almost everywhere. Then M Lemma 3.5. Let f ∈ L1loc (Rn ) such that M 0 ≤ δ < 1.
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Proof. By Lemma 3.4, it suffice to show that there exists a constant C such that, for any S ∈ F and μ-almost every x ∈ S, 1 μ(S)
F f M
δ
F f (x) δ . dμ ≤ C M
S
Fix S = S(x0 , t0 ) and decompose f as f = f1 + f2 , where f1 = f χθ2 S and f2 = f χ(θ2 S)c with θ2 S = F f1 (y) + M F f2 (y) and F f (y) ≤ M S(x0 , θ2 t0 ). Then M
δ
F f (y) M
F f1 (y) δ + M F f2 (y) δ ≤ M
for 0 ≤ δ < 1.
F is weak (1, 1) with respect to the measure μ, Kolmogorov’s inequality shows Since M 1 μ(S)
F f1 (y) δ dμ(y) ≤ M
1 C μ(S)1−δ f1 δL1 (dμ) = C μ(S) μ(S)
|f |dμ
δ
F f (x) δ . ≤C M
θ2 S
S
F f2 , given y ∈ S, for any S(y0 , R) ∈ F that contains y, we have S ⊂ S(y0 , θ2 max{t0 , R}). To estimate M If R < t0 , we have S(y0 , t0 ) ∩ S(x0 , t0 ) = ∅ and hence S(y0 , t0 ) ⊂ S(x0 , θ2 t0 ). Since S(y0 , R) ⊂ S(y0 , t0 ) ⊂ S(x0 , θ2 t0 ), S(y0 ,R) |f2 |dμ = 0 when R < t0 . It is clear that S ⊂ S(y0 , θ2 R) when R ≥ t0 . Thus,
1 μ(S(y0 , R))
C |f2 |dμ ≤ μ(S(y0 , θ2 R))
F f (x), |f2 |dμ ≤ C M
S(y0 ,θ 2 R)
S(y0 ,R)
F f (x) for any y ∈ S. Therefore, F f2 (y) ≤ C M so that M 1 μ(S)
F f2 (y) δ dμ(y) ≤ C M F f (x) δ M
S
and the proof is completed. 2 Lemma 3.6. If ω ∈ A2,F , then log ω ∈ BMO F . Proof. Assume ω ∈ A2,F and write f := log ω, which is equivalent to exp(f ) ∈ A2,F . Given S ∈ F, the definition of A2,F gives 1 1 exp(f )dμ exp(−f )dμ ≤ C; μ(S) μ(S) S
S
equivalently, 1 1 exp f − mS (f ) dμ exp mS (f ) − f dμ ≤ C, μ(S) μ(S) S
(3.3)
S
1 where mS (f ) = μ(S) f dμ. By Jensen’s inequality, each factor is at least 1 and at most C. Therefore, S inequality (3.3) implies 1 μ(S)
exp(|f − mS (f )|)dμ ≤ 2C, S
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and so 1 μ(S)
|f − mS (f )|dμ ≤ 2C. S
Hence, f ∈ BMO F .
2
We are ready to show the second main result. Proof of Theorem 1.2. Since HF1 is a subspace of L1 (dμ), it follows from Fatou’s lemma that f ∈ L1 (dμ). To show (1.2), it suffices to consider for all φ ∈ Cc ∩ Lip. We can assume that φL1 (dμ) ≤ 1, φL∞ (dμ) ≤ 1, |φ(x) − φ(y)| ≤ ρ(x, y) for x, y ∈ Rn and that supp(φ) is compact. Fix δ ∈ (0, 1) and pick η > 0 such that η exp(δ −1 ) ≤ δ and E |f |dμ ≤ δ whenever μ(E) ≤ Cη exp(δ −1 ). Now choose k large enough so that
μ(Ek ) := μ
x ∈ supp(φ) : |fk (x) − f (x)| > η
≤ η.
Define F χ )(x) . τ (x) = max 0, 1 + δ log(M Ek It is clear that 0 ≤ τ (x) ≤ 1 and τ = 1 μ-almost everywhere on Ek . Also, τ BMOF ≤ F χ )1/2 BMO ≤ Cδ due to Lemmas 3.5 and 3.6. By the weak (1, 1) estimate for M F with 2δ log(M Ek F respect to μ, μ supp(τ ) ≤ Cμ(Ek ) exp(δ −1 ) ≤ Cη exp(δ −1 ). Consequently, |f |dμ ≤ δ. supp(τ )
We now write (f − fk )φdμ ≤ (f − fk )φ(1 − τ )dμ + (f − fk )φτ dμ Rn
Rn
Rn
|f |dμ + fk φτ dμ
supp(τ )
Rn
≤ ηφL1 (dμ) +
≤ δ + δ + fk φτ dμ. Rn
The proof of (1.2) will therefore be established provided we verify φτ BMOF ≤ Cδ. Note that |φ(x) − mS (φ)| ≤ sup ρ(x, y) for x ∈ S = S(x0 , t0 ). Thus, y∈S
(3.4)
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14
1 μ(S)
2 |φτ − mS (φτ )|dμ ≤ μ(S)
S
|φτ − mS (φ)mS (τ )|dμ S
2 ≤ μ(S)
2|mS (φ)| |φτ − mS (φ)τ |dμ + μ(S)
S
2 ≤ sup ρ(x, y) μ(S) x,y∈S
|τ − mS (τ )|dμ S
|τ |dμ + 2φL∞ (dμ) τ BMOF . S
By (2.4)–(2.6), sup ρ(x, y)
sup ρ(x, y) x,y∈S
μ(S)
≤C
x,y∈S
μ(Bd (x0 , 2θt0 ))
≤ C.
Hence, 1 μ(S)
|φτ − mS (φτ )|dμ ≤ C
S
|τ |dμ + Cδ ≤ Cμ supp(τ ) + Cδ
S
≤ Cη exp(δ −1 ) + Cδ ≤ Cδ, and (3.4) follows. By weak∗ compactness of the ball in HF1 , there exists a g ∈ HF1 with gHF1 ≤ 1 and a subsequence {fkl }l∈N such that {fkl }l∈N weak∗ converges to g. By (1.2), we have f φ = gφ for all φ ∈ Cc ∩ Lip, and hence f = g ∈ HF1 . 2 Acknowledgment The author is grateful to the referee for valuable suggestions. References [1] H. Aimar, L. Forzani, R. Toledano, Balls and quasi-metrics: a space of homogeneous type modeling the real analysis related to the Monge–Ampère equation, J. Fourier Anal. Appl. 4 (1998) 377–381. [2] L.A. Caffarelli, Some regularity properties of solutions of Monge–Ampère equation, Comm. Pure Appl. Math. XLIV (1991) 965–969. [3] L.A. Caffarelli, Boundary regularity of maps with convex potentials, Comm. Pure Appl. Math. XLV (1992) 1141–1151. [4] L.A. Caffarelli, C.E. Gutiérrez, Real analysis related to the Monge–Ampère equation, Trans. Amer. Math. Soc. 348 (1996) 1075–1092. [5] L.A. Caffarelli, C.E. Gutiérrez, Properties of the solutions of the linearized Monge–Ampère equation, Amer. J. Math. 119 (1997) 423–465. [6] R.R. Coifman, G. Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977) 569–645. [7] R. Coifman, P.-L. Lions, Y. Meyer, S. Semmes, Compensated compactness and Hardy spaces, J. Math. Pures Appl. 72 (1993) 247–286. [8] Y. Ding, C.-C. Lin, Hardy spaces associated to the sections, Tohoku Math. J. 57 (2005) 147–170. [9] N. Dunford, J. Schwartz, Linear Operators. I, Interscience, New York and London, 1964. [10] P.W. Jones, J.-L. Journé, On weak convergence in H 1 (Rd ), Proc. Amer. Math. Soc. 120 (1994) 137–138. [11] R.A. Macías, C. Segovia, Lipschitz functions on spaces of homogeneous type, Adv. Math. 33 (1979) 257–270.