From dyadic $\Lambda_{\alpha}$ to $\Lambda_{\alpha}$
FROM DYADIC Λα TO Λα
WAEL ABU-SHAMMALA AND ALBERTO TORCHINSKY
Abstract. In this paper we show how to compute the Λα norm , α ≥ 0,
using the dyadic grid. This result is a consequence of the description of
the Hardy spaces Hp(RN ) in terms of dyadic and special atoms.
Recently, several novel methods for computing the BMO norm of a function
f in two dimensions were discussed in [9]. Given its importance, it is also of
interest to explore the possibility of computing the norm of a BMO function,
or more generally a function in the Lipschitz class Λα, using the dyadic grid
in RN . It turns out that the BMO question is closely related to that of
approximating functions in the Hardy space H1(RN ) by the Haar system.
The approximation in H1(RN ) by affine systems was proved in [2], but this
result does not apply to the Haar system. Now, if HA(R) denotes the closure
of the Haar system in H1(R), it is not hard to see that the distance d(f,HA)
of f ∈ H1(R) to HA is ∼
f(x) dx
∣, see [1]. Thus, neither dyadic atoms
suffice to describe the Hardy spaces, nor the evaluation of the norm in BMO
can be reduced to a straightforward computation using the dyadic intervals.
In this paper we address both of these issues. First, we give a characterization
of the Hardy spaces Hp(RN ) in terms of dyadic and special atoms, and then,
by a duality argument, we show how to compute the norm in Λα(R
N ), α ≥ 0,
using the dyadic grid.
We begin by introducing some notations. Let J denote a family of cubes
Q in RN , and Pd the collection of polynomials in R
N of degree less than or
equal to d. Given α ≥ 0, Q ∈ J , and a locally integrable function g, let pQ(g)
denote the unique polynomial in P[α] such that [g − pQ(g)]χQ has vanishing
moments up to order [α].
For a locally square-integrable function g, we consider the maximal function
α,J g(x) given by
α,J g(x) = sup
x∈Q,Q∈J
|Q|α/N
|g(y)− pQ(g)(y)|
1991 Mathematics Subject Classification. 42B30,42B35.
http://arxiv.org/abs/0704.0005v1
2 WAEL ABU-SHAMMALA AND ALBERTO TORCHINSKY
The Lipschitz space Λα,J consists of those functions g such that M
α,J g is
in L∞, ‖g‖Λα,J = ‖M
α,J g‖∞; when the family in question contains all cubes
in RN , we simply omit the subscript J . Of course, Λ0 = BMO.
Two other families, of dyadic nature, are of interest to us. Intervals in R of
the form In,k = [ (k−1)2
n, k2n], where k and n are arbitrary integers, positive,
negative or 0, are said to be dyadic. In RN , cubes which are the product of
dyadic intervals of the same length, i.e., of the form Qn,k = In,k1 ×· · ·×In,kN ,
are called dyadic, and the collection of all such cubes is denoted D.
There is also the family D0. Let I
n,k = [(k− 1)2
n, (k+ 1)2n], where k and
n are arbitrary integers. Clearly I ′n,k is dyadic if k is odd, but not if k is even.
Now, the collection {I ′n,k : n, k integers} contains all dyadic intervals as well
as the shifts [(k − 1)2n + 2n−1, k 2n + 2n−1] of the dyadic intervals by their
half length. In RN , put D0 = {Q
n,k : Q
n,k = I
× · · · × I ′n,kN }; Q
n,k is
called a special cube. Note that D0 contains D properly.
Finally, given I ′n,k, let I
n,k = [(k − 1)2
n, k2n], and I
n,k = [k2
n, (k + 1)2n].
The 2N subcubes of Q′n,k = I
× · · · × I ′n,kN of the form I
× · · · × I
Sj = L or R, 1 ≤ j ≤ N , are called the dyadic subcubes of Q
Let Q0 denote the special cube [−1, 1]
N . Given α ≥ 0, we construct a
family Sα of piecewise polynomial splines in L
2(Q0) that will be useful in
characterizing Λα. Let A be the subspace of L
2(Q0) consisting of all functions
with vanishing moments up to order [α] which coincide with a polynomial
in P[α] on each of the 2
N dyadic subcubes of Q0. A is a finite dimensional
subspace of L2(Q0), and, therefore, by the Graham-Schmidt orthogonalization
process, say, A has an orthonormal basis in L2(Q0) consisting of functions
p1, . . . , pM with vanishing moments up to order [α], which coincide with a
polynomial in P[α] on each dyadic subinterval of Q0. Together with each p
we also consider all dyadic dilations and integer translations given by
pLn,k,α(x) = 2
n(N+α)pL(2nx1 + k1, . . . , 2
nxN + kN ) , 1 ≤ L ≤ M ,
and let
Sα = {p
n,k,α : n, k integers, 1 ≤ L ≤ M} .
Our first result shows how the dyadic grid can be used to compute the
norm in Λα.
Theorem A. Let g be a locally square-integrable function and α ≥ 0. Then,
g ∈ Λα if, and only if, g ∈ Λα,D and Aα(g) = supp∈Sα
∣〈g, p〉
∣ < ∞. Moreover,
‖g‖Λα ∼ ‖g‖Λα,D +Aα(g) .
Furthermore, it is also true, and the proof is given in Proposition 2.1 be-
low, that ‖g‖Λα ∼ ‖g‖Λα,D0 . However, in this simpler formulation, the tree
structure of the cubes in D has been lost.
FROM DYADIC Λα TO Λα 3
The proof of Theorem A relies on a close investigation of the predual of
Λα, namely, the Hardy space H
p(RN ) with 0 < p = (α + N)/N ≤ 1. In the
process we characterize Hp in terms of simpler subspaces: H
, or dyadic Hp,
and H
, the space generated by the special atoms in Sα. Specifically, we
Theorem B. Let 0 < p ≤ 1, and α = N(1/p− 1). We then have
Hp = H
where the sum is understood in the sense of quasinormed Banach spaces.
The paper is organized as follows. In Section 1 we show that individual
Hp atoms can be written as a superposition of dyadic and special atoms;
this fact may be thought of as an extension of the one-dimensional result of
Fridli concerning L∞ 1- atoms, see [5] and [1]. Then, we prove Theorem B.
In Section 2 we discuss how to pass from Λα,D, and Λα,D0 , to the Lipschitz
space Λα.
1. Characterization of the Hardy spaces Hp
We adopt the atomic definition of the Hardy spaces Hp, 0 < p ≤ 1, see
[6] and [10]. Recall that a compactly supported function a with [N(1/p− 1)]
vanishing moments is an L2 p -atom with defining cube Q if supp(a) ⊆ Q, and
|Q|1/p
| a(x) |2dx
≤ 1 .
The Hardy space Hp(RN ) = Hp consists of those distributions f that can be
written as f =
λjaj , where the aj ’s are H
p atoms,
|λj |
p < ∞, and the
convergence is in the sense of distributions as well as in Hp. Furthermore,
‖f‖Hp ∼ inf
|λj |
where the infimum is taken over all possible atomic decompositions of f . This
last expression has traditionally been called the atomic Hp norm of f .
Collections of atoms with special properties can be used to gain a better
understanding of the Hardy spaces. Formally, let A be a non-empty subset
of L2 p -atoms in the unit ball of Hp. The atomic space H
spanned by A
consists of those ϕ in Hp of the form
λjaj , aj ∈ A ,
|λj |
p < ∞ .
It is readily seen that, endowed with the atomic norm
‖ϕ‖Hp
= inf
|λj |
: ϕ =
λj aj , aj ∈ A
becomes a complete quasinormed space. Clearly, H
⊆ Hp, and, for
f ∈ H
, ‖f‖Hp ≤ ‖f‖Hp
4 WAEL ABU-SHAMMALA AND ALBERTO TORCHINSKY
Two families are of particular interest to us. When A is the collection
of all L2 p -atoms whose defining cube is dyadic, the resulting space is H
or dyadic Hp. Now, although ‖f‖Hp ≤ ‖f‖Hp
, the two quasinorms are not
equivalent on H
. Indeed, for p = 1 and N = 1, the functions
fn(x) = 2
n[χ[1−2−n,1](x) − χ[1,1+2−n](x)] ,
satisfy ‖fn‖H1 = 1, but ‖fn‖H1
∼ |n| tends to infinity with n.
Next, when Sα is the family of piecewise polynomial splines constructed
above with α = N(1/p − 1), in analogy with the one-dimensional results in
[4] and [1], H
is referred to as the space generated by special atoms.
We are now ready to describe Hp atoms as a superposition of dyadic and
special atoms.
Lemma 1.1. Let a be an L2 p -atom with defining cube Q, 0 < p ≤ 1,
and α = N(1/p − 1). Then a can be written as a linear combination of 2N
dyadic atoms ai, each supported in one of the dyadic subcubes of the smallest
special cube Qn,k containing Q, and a special atom b in Sα. More precisely,
a(x) =
i=1 di ai(x) +
L=1 cL p
−n,−k,α(x), with |di| , |cL| ≤ c.
Proof. Suppose first that the defining cube of a is Q0, and let Q1, . . . , Q2N
denote the dyadic subcubes of Q0. Furthermore, let {e
i , . . . , e
i } denote an
orthonormal basis of the subspace Ai of L
2(Qi) consisting of polynomials in
P[α], 1 ≤ i ≤ 2
N . Put
αi(x) = a(x)χQi (x)−
〈aχQi , e
j(x) , 1 ≤ i ≤ 2
and observe that 〈αi, e
j〉 = 0 for 1 ≤ j ≤ M . Therefore, αi has [α] vanishing
moments, is supported in Qi, and
‖αi‖2 ≤ ‖aχQi‖2 +
‖aχQi‖2 ≤ (M + 1) ‖aχQi‖2 .
ai(x) =
2N(1/2−1/p)
M + 1
αi(x) , 1 ≤ i ≤ N ,
is an L2 p - dyadic atom. Finally, put
b(x) = a(x) −
M + 1
2N(1/2−1/p)
ai(x) .
FROM DYADIC Λα TO Λα 5
Clearly b has [α] vanishing moments, is supported in Q0, coincides with a
polynomial in P[α] on each dyadic subcube of Q0, and
‖b‖22 ≤
|〈aχQi , e
2 ≤ M ‖a‖22 .
So, b ∈ A, and, consequently, b(x) =
L=1 cL p
L(x), where
|cL| = |〈b, p
L〉| ≤ c , 1 ≤ L ≤ M .
In the general case, let Q be the defining cube of a, side-length Q = ℓ, and
let n and k = (k1, . . . , kN ) be chosen so that 2
n−1 ≤ ℓ < 2n, and
Q ⊂ [(k1 − 1)2
n, (k1 + 1)2
n]× · · · × [(kN − 1)2
n, (kN + 1)2
Then, (1/2)N ≤ |Q|/2nN < 1.
Now, given x ∈ Q0, let a
′ be the translation and dilation of a given by
a′(x) = 2nN/pa(2nx1 − k1, . . . , 2
nxN − kN ) .
Clearly, [α] moments of a′ vanish, and
‖a′‖2 = 2
nN/p 2−nN/2‖a‖2 ≤ c |Q|
1/p|Q|−1/2‖a‖2 ≤ c .
Thus, a′ is a multiple of an atom with defining cube Q0. By the first part of
the proof,
a′(x) =
i(x) +
L(x) , x ∈ Q0 .
The support of each a′i is contained in one of the dyadic subcubes of Q0, and,
consequently, there is a k such that
ai(x) = 2
−nN/pa′i(2
−nx1 − k1, . . . , 2
−nxN − kN )
ai is an L
2p -atom supported in one of the dyadic subcubes of Q. Similarly
for the pL’s. Thus,
a(x) =
di ai(x) +
−n,−k,N(1/p−1)(x) ,
and we have finished. �
Theorem B follows readily from Lemma 1.1. Clearly, H
→֒ Hp.
Conversely, let f =
j λj aj be in H
p. By Lemma 1.1 each aj can be written
as a sum of dyadic and special atoms, and, by distributing the sum, we can
write f = fd + fs, with fd in H
, fs in H
, and
‖fd‖Hp
, ‖fs‖Hp
|λj |
Taking the infimum over the decompositions of f we get ‖f‖Hp
c ‖f‖Hp , and H
p →֒ H
. This completes the proof.
6 WAEL ABU-SHAMMALA AND ALBERTO TORCHINSKY
The meaning of this decomposition is the following. Cubes in D are con-
tained in one of the 2N non-overlapping quadrants of RN . To allow for the
information carried by a dyadic cube to be transmitted to an adjacent dyadic
cube, they must be connected. The pLn,k,α’s channel information across ad-
jacent dyadic cubes which would otherwise remain disconnected. The reader
will have no difficulty in proving the quantitative version of this observation:
Let T be a linear mapping defined on Hp, 0 < p ≤ 1, that assumes values in
a quasinormed Banach space X . Then, T is continuous if, and only if, the
restrictions of T to H
and H
are continuous.
2. Characterizations of Λα
Theorem A describes how to pass from Λα,D to Λα, and we prove it next.
Since (Hp)∗ = Λα and (H
)∗ = Λα,D, from Theorem B it follows readily that
Λα = Λα,D ∩ (H
)∗, so it only remains to show that (H
)∗ is characterized
by the condition Aα(g) < ∞.
First note that if g is a locally square-integrable function with Aα(g) < ∞
and f =
j,L cj,L p
nj ,kj ,α
, since 0 < p ≤ 1,
|〈g, f〉| ≤
|cj,L| |〈g, p
nj ,kj ,α
≤ Aα(g)
|cj,L|
and, consequently, taking the infimum over all atomic decompositions of f in
, we get g ∈ (H
)∗ and ‖g‖(Hp
)∗ ≤ Aα(g).
To prove the converse we proceed as in [3]. Let Qn = [−2
n, 2n]N . We begin
by observing that functions f in L2(Qn) that have vanishing moments up to
order [α] and coincide with polynomials of degree [α] on the dyadic subcubes
of Qn belong to H
‖f‖Hp
≤ |Qn|
1/p−1/2‖f‖2 .
Given ℓ ∈ (H
)∗, for a fixed n let us consider the restriction of ℓ to the space
of L2 functions f with [α] vanishing moments that are supported in Qn. Since
|ℓ(f)| ≤ ‖ℓ‖ ‖f‖Hp
≤ ‖ℓ‖ |Qn|
1/p−1/2‖f‖2 ,
this restriction is continuous with respect to the norm in L2 and, consequently,
it can be extended to a continuous linear functional in L2 and represented as
ℓ(f) =
f(x) gn(x) dx ,
FROM DYADIC Λα TO Λα 7
where gn ∈ L
2(Qn) and satisfies ‖gn‖2 ≤ ‖ℓ‖ |Qn|
1/p−1/2. Clearly, gn is
uniquely determined in Qn up to a polynomial pn in P[α]. Therefore,
gn(x) − pn(x) = gm(x)− pm(x) , a.e. x ∈ Qmin(n,m) .
Consequently, if
g(x) = gn(x)− pn(x) , x ∈ Qn ,
g(x) is well defined a.e. and, if f ∈ L2 has [α] vanishing moments and is
supported in Qn, we have
ℓ(f) =
f(x) gn(x) dx
f(x) [gn(x)− pn(x)] dx
f(x) g(x) dx .
Moreover, since each 2nN/ppL(2n ·+k) is an L2 p-atom, 1 ≤ L ≤ M , it readily
follows that
Aα(g) = sup
1≤L≤M
n,k∈Z
|〈g, 2−n/ppL(2n ·+k)〉|
≤ ‖ℓ‖ sup
‖pL‖Hp ≤ ‖ℓ‖ ,
and, consequently, Aα(g) ≤ ‖ℓ‖ , and (H
)∗ is the desired space. �
The reader will have no difficulty in showing that this result implies the
following: Let T be a bounded linear operator from a quasinormed space X
into Λα,D. Then, T is bounded from X into Λα if, and only if, Aα(Tx) ≤
c ‖x‖X for every x ∈ X .
The process of averaging the translates of dyadic BMO functions leads to
BMO, and is an important tool in obtaining results in BMO once they are
known to be true in its dyadic counterpart, BMOd, see [7]. It is also known
that BMO can be obtained as the intersection of BMOd and one of its shifted
counterparts, see [8]. These results motivate our next proposition, which
essentially says that g ∈ Λα if, and only if, g ∈ Λα,D and g is in the Lipschitz
class obtained from the shifted dyadic grid. Note that the shifts involved in
this class are in all directions parallel to the coordinate axis and depend on
the side-length of the cube.
Proposition 2.1. Λα = Λα,D0 , and ‖g‖Λα ∼ ‖g‖Λα,D0 .
Proof. It is obvious that ‖g‖Λα,D0 ≤ ‖g‖Λα . To show the other inequality we
invoke Theorem A. Since D ⊂ D0, it suffices to estimate Aα(g), or, equiva-
lently, |〈g, p〉| for p ∈ Sα, α = N(1/p − 1). So, pick p = p
n,k,α in Sα. The
defining cube Q of pLn,k,α is in D0, and, since p
n,k,α has [α] vanishing moments,
8 WAEL ABU-SHAMMALA AND ALBERTO TORCHINSKY
〈pLn,k,α, pQ(g)〉 = 0. Therefore,
|〈g, pLn,k,α〉| = |〈g − pQ(g), p
n,k,α〉|
≤ ‖pLn,k,α‖2 ‖g − pQ(g)‖L2(Q)
≤ |Q|α/N |Q|1/2‖pLn,k,α‖2 ‖g‖Λα,D0 .
Now, a simple change of variables gives |Q|α/N |Q|1/2‖pLn,k,α‖2 ≤ 1, and, con-
sequently, also Aα(g) ≤ ‖g‖Λα,D0 . �
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Department of Mathematics, Indiana University, Bloomington IN 47405
E-mail address: wabusham@indiana.edu
Department of Mathematics, Indiana University, Bloomington IN 47405
E-mail address: torchins@indiana.edu
	1. Characterization of the Hardy spaces Hp
	2. Characterizations of 
	References
