# Relationship between Hardy Spaces Associated with Different Homogeneities and One-Parameter Hardy Spaces

Authors: Xinfeng Wu
Journal: Journal of Function Spaces and Applications
 (2013)

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## Abstract

We prove that the Hardy spaces associated with different homogeneitiesHcomp, 0<p<∞, are continuously embedded into the intersection of the isotropic Hardy spaces Hisop and the nonisotropic Hardy spaces Hnonp. As a consequence, we obtain that any operator bounded from either Hisop or Hnonp into Lp must be bounded from Hcomp to Lp.


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## Body

## 1. Introduction

Forx=(x′,xn)∈ℝn-1×ℝ and δ>0, we consider two kinds of homogeneities on ℝn:
(1)δ∘e(x′,xn)=(δx′,δxn),δ∘h(x′,xn)=(δx′,δ2xn).The first are the classical isotropic dilations occurring in the classical Calderón-Zygmund singular integrals, while the second are nonisotropic and related to the heat equations (also the Heisenberg groups). Lete(ξ) and h(ξ) be functions on ℝn homogeneous of degree 0 in the isotropic sense and in the nonisotropic sense, respectively, and both smooth away from the origin. Then, it is well known that the Fourier multipliers T1 defined by T1(f)^(ξ)=e(ξ)f^(ξ) and T2 given by T2(f)^(ξ)=h(ξ)f^(ξ) are both bounded on Lp for 1<p<∞, of weak-type (1,1), and bounded on the classical isotropic Hardy spaces Hisop and, nonisotropic Hardy spaces Hnonp, respectively. Riviere in [1] asked the question is the composition T1∘T2 still of weak-type (1,1)? Phong and Stein in [2] answered this question and gave a necessary and sufficient condition for which T1∘T2 is of weak-type (1,1). The operators Phong and Stein studied are in fact a composition of operators with different kinds of homogeneities which arise naturally in the ∂--Neumann problem. Recently, Han et al. [3] developed a theory of the Hardy spaces Hcomp, 0<p<∞, associated with the different homogeneities and proved that the composition of the two Calderón-Zygmund convolution operators with different homogeneities is bounded on Hcomp(ℝn). Weighted function spaces associated with different homogeneities and boundedness of composition of operators on them were recently investigated in [4–6].The Hardy spacesHcomp introduced in [3] have surprising multiparameter structures which reflect the mixed homogeneities arising from the two operators under consideration. A natural question arises: Is there any relationship between Hcomp and the two classical one-parameter Hardy spaces Hisop and Hnonp? The main purpose of this paper is to answer this question. We shall prove that Hcomp are continuously embedded into the intersection of Hisop and Hnonp. As an application, we show that any operator boundedness from either Hisop or Hnonp into Lp must be bounded from Hcomp into Lp. Our methods are to use the partially discrete Calderón-type formula and the Littlewood-Paley theory in this context, which are appropriately developed.Before stating the results more precisely, we first recall some notions and notations. Forx=(x′,xn)∈ℝn-1×ℝ, we denote |x|e=(|x′|2+|xn|2)1/2 and |x|h=(|x′|2+|xn|)1/2. Let ψ(1)∈𝒮(ℝn) satisfy
(2)suppψ(1)^(ξ)⊆{ξ:12<|ξ|e≤2},∑j∈ℤ‍ψ(1)^(2j∘eξ)=1∀ξ∈ℝn∖{0}.
The isotropic discrete square function gisod(f) is defined by
(3)gisod(f)(x):={∑j∈ℤ‍∑Q∈𝒬isoj‍|ψj(1)*f(xQ)|2χQ(x)}1/2,
where ψj(1)(x)=2-jnψ(1)(2-j∘ex), 𝒬isoj denotes the set of all dyadic cubes with sidelength 2j, and xQ denotes the left-lower corner of Q. The isotropic Hardy spaces Hisop(ℝn), 0<p<∞, are defined by
(4)Hisop(ℝn):={f∈𝒮′𝒫:∥gisod(f)∥Lp(ℝn)<∞}.
Similarly, let ψ(2)∈𝒮(ℝn) satisfy
(5)suppψ(2)^(ξ)⊆{ξ:12<|ξ|h≤2},∑k∈ℤ‍ψ(2)^(2k∘hξ)=1∀ξ∈ℝn∖{0}.
The nonisotropic discrete square function gnond(f) is defined by
(6)gnond(f)(x):={∑k∈ℤ‍∑Q∈𝒬nonk‍|ψk(2)*f(xQ)|2χQ(x)}1/2,
where ψk(2)(x)=2-k(n+1)ψ(2)(2-k∘hx), 𝒬nonk is the set of dyadic nonisotropic “cubes” Q=Q1×Q2⊂ℝn-1×ℝ with sidelength ℓ(Q1)=2k and |Q2|=22k, and xQ is the left-lower corner of Q. The nonisotropic Hardy spaces Hnonp, 0<p<∞, are defined by
(7)Hnonp(ℝn):={f∈𝒮′𝒫:∥gnond(f)∥Lp(ℝn)<∞}.Forj,k∈ℤ, let ψj,k=ψj(1)*ψk(2). The discrete square function gcomd associated with different homogeneities is given by
(8)gcomd(f)(x):={∑j,k∈ℤ‍∑R∈ℛj,k‍|ψj,k*f(xR)|2χR(x)}1/2,
where ℛj,k denote the set of dyadic rectangles R=Q1×Q2 in ℝn-1×ℝ with sidelength ℓ(Q1)=2j∨k and |Q2|=2j∨2k and xR is the left-lower corner of R. The Hardy spaces Hcomp, 0<p<∞, associated with different homogeneities are defined by
(9)Hcomp(ℝn):={f∈𝒮′𝒫:∥gcomd(f)∥Lp(ℝn)<∞}.The main result of this paper is as follows.Theorem 1.
Let0<p<∞. One has
(10)Hcomp(ℝn)↪Hisop(ℝn)∩Hnonp(ℝn).
More precisely, there is a constant C depending on p and n such that, for all f∈Hcomp(ℝn),
(11)∥f∥Hisop(ℝn)+∥f∥Hnonp(ℝn)≤C∥f∥Hcomp(ℝn).In [3], it was proved that the composition of the two Calderón-Zygmund convolution operators with different homogeneities is bounded on Hcomp (see [3, Theorem 1.9]). This result can be improved by the following.Corollary 2.
Let0<p<∞. One has that(1)
any operatorT bounded from either Hisop(ℝn) or Hnonp(ℝn) into Lp(ℝn) must be bounded from Hcomp(ℝn) to Lp(ℝn);(2)
ifT1 is bounded from Hisop(ℝn) to Lp(ℝn) and T2 is bounded from Hnonp(ℝn) to Lp(ℝn), then T1∘T2 is bounded from Hcomp(ℝn) to Lp(ℝn).


## 2. New Square Function Characterizations ofHcomp

Letϕ(1)∈𝒮(ℝn) satisfy suppϕ(1)⊆B(0,1),
(12)∫ℝn‍ϕ(1)(x)xαdx=0∀|α|≤Mp,∑j∈ℤ‍ϕ(1)^(2j∘eξ)=1∀ξ∈ℝn∖{0},
where Mp is a sufficiently large constant depending on p and n. Similarly, let ϕ(2)∈𝒮(ℝn) satisfy supp ϕ(2)⊆B(0,1),
(13)∫ℝn‍ϕ(2)(x)xβdx=0∀|β|≤Mp,∑k∈ℤ‍ψ(2)^(2k∘hξ)=1∀ξ∈ℝn∖{0}.
Let ϕj(1)(x)=2-jnϕ(2-j∘ex), ϕk(2)(x)=2-k(n+1)ϕ(2-k∘hx), and ϕj,k=ϕj(1)*ϕk(2). Then, Hcomp can be characterized via the continuous square function gcom defined by
(14)gcom(f)(x)={∑j,k∈ℤ‍|ϕj,k*f(x)|2}1/2.
The following Calderón-type identities are well known (cf. [7–9]).Lemma 3.
Forf∈L2(ℝn), there is a sufficiently large integer N such that
(15)f=∑j∈ℤ‍∑Q∈𝒬isoj-N‍ϕj(1)(·-ϑQ)ϕj(1)*f(ϑQ)=∑k∈ℤ‍∑Q∈𝒬nonk-N‍ϕk(2)(·-ϑQ)ϕk(2)*f(ϑQ),
where the two series converge in L2(ℝn) and ϑQ denotes any fixed point of Q.The Calderón-type identities in Lemma3 lead to the following square functions:
(16)gcom(1)(f)(x):={∑j,k∈ℤ‍∑Q∈𝒬isoj-N‍|ϕj,k*f(xQ)|2χQ(x)}1/2,gcom(2)(f)(x):={∑j,k∈ℤ‍∑Q∈𝒬nonk-N‍|ϕj,k*f(xQ)|2χQ(x)}1/2,
where N is the sufficiently large integer in Lemma 3.The purpose of this section is to prove the following.Theorem 4.
Forp∈(0,∞), one has that
(17)∥f∥Hcomp~∥gcom(1)(f)∥Lp~∥gcom(2)(f)∥Lp.Proof.
We only prove that(18)∥f∥Hcomp~∥gcom(1)(f)∥Lp,
as ∥f∥Hcomp~∥gcom(2)(f)∥Lp can be obtained in the same manner. To prove (18), it suffices to show that
(19)∥{∑j,k‍∑Q∈𝒬isoj-N‍supu∈Q|ϕj,k*f(u)|2χQ}1/2∥Lp≲∥{∑j′,k‍∑Q∈𝒬isoj′-N‍infu∈Q|ϕj′,k*f(u)|2χQ}1/2∥Lp.
To verify (19), we need the following lemma.Lemma 5 (see [10]).
Let
Q
∈
𝒬
iso
j
-
N
and
M
>
0
. Then, for any
u
,
ϑ
Q
∈
Q
,
ϑ
Q
′
∈
Q
′
, and
δ
∈
(
n
/
(
n
+
M
)
,
1
]
, one has that
(20)
∑
Q
′
∈
𝒬
iso
j
′
-
N
‍
2
M
(
j
∨
j
′
)
(
2
j
∨
j
′
+
|
u
-
ϑ
Q
′
|
)
n
+
M
|
g
j
′
(
ϑ
Q
′
)
|
≤
C
N
2
n
|
j
-
j
′
|
(
(
1
/
δ
)
-
1
)
×
{
ℳ
[
(
∑
Q
′
∈
𝒬
iso
j
′
-
N
‍
|
g
j
′
(
ϑ
Q
′
)
|
2
χ
Q
′
)
δ
/
2
]
(
ϑ
Q
)
}
1
/
δ
,
where
ℳ
denotes the classical isotropic Hardy-Littlewood maximal operator and
a
∨
b
=
max
(
a
,
b
)
.We first assume thatf∈L2∩Hcomp(ℝn). Set fk=ϕk(2)*f. Applying the discrete Calderón-type identity in (15), the classical almost orthogonality estimate, and Lemma 5, we deduce that for any u,ϑQ∈Q, ϑQ′∈Q′, and δ∈(n/(n+M),min(p,1)),
(21)|ϕj,k*f(u)|=|∑j′∈ℤ‍∑Q′∈𝒬isoj′-N‍ϕj(1)*ϕj′(1)(u-ϑQ′)ϕj′(1)*fk(ϑQ′)|≲∑j′∈ℤ‍2-|j-j′|10Mdddddd×∑Q′∈𝒬isoj′-N‍2M(j∨j′)(2j∨j′+|u-ϑQ′|)n+M|ϕj′(1)*fk(ϑQ′)|≲∑j′∈ℤ‍2-|j-j′|M×{×χQ′∑Q′∈𝒬isoj′-N‍)δ/2ℳ[(∑Q′∈𝒬isoj′-N‍|ϕj′(1)*fk(ϑQ′)|2ddddddddddddddd×χQ′∑Q′∈𝒬isoj′-N‍)δ/2](ϑQ)}1/δ.
Since u and ϑQ′ are arbitrary points in Q and Q′, respectively,
(22)supu∈Q|ϕj,k*f(u)|≲∑j′∈ℤ‍2-|j-j′|M×{×χQ′∑Q′∈𝒬isoj′-N‍)δ/2ℳ[(∑Q′∈𝒬isoj′-N‍infv∈Q′|ϕj′(1)*fk(v)|2dddddddddddddd×χQ′∑Q′∈𝒬isoj′-N‍)δ/2](ϑQ)}1/δ,
which, by the Cauchy-Schwarz inequality, implies that
(23)supu∈Q|ϕj,k*f(u)|2≲∑j′∈ℤ‍2-|j-j′|M×{×χQ′∑Q′∈𝒬isoj′-N‍)δ/2ℳ[(∑Q′∈𝒬isoj′-N‍infv∈Q′|ϕj′(1)*fk(v)|2ddddddddddddddd×χQ′∑Q′∈𝒬isoj′-N‍)δ/2](ϑQ)}2/δ,
where we have used ∑j′∈ℤ‍2-|j-j′|M≤C. Multiplying χQ from both sides and summing over j,k∈ℤ,Q∈𝒬isoj-N yield that, for any ϑQ∈Q,
(24)∑j,k∈ℤ‍∑Q∈𝒬isoj-N‍supu∈Q|ϕj,k*f(u)|2χQ(x)≲∑j,k∈ℤ‍∑j′∈ℤ‍2-|j-j′|Mddddddddd×{ℳ[(∑Q′∈𝒬isoj′-N‍infv∈Q′|ϕj′(1)*fk(v)|2dddddddddddddddddddddd×χQ′∑Q′∈𝒬isoj′-N‍infv∈Q′)δ/2](ϑQ)}2/δ≲∑j′∈ℤ‍∑k∈ℤ‍×{ℳ[(∑Q′∈𝒬isoj′-N‍infv∈Q′|ϕj′(1)*fk(v)|2ddddddddddddddddddddddd×χQ′∑Q′∈𝒬isoj′-N‍)δ/2](ϑQ)}.2/δ
It follows that
(25){∑j,k‍∑Q∈𝒬isoj-N‍supu∈Q|ϕj,k*f(u)|2χQ(x)}1/2≲{∑j′∈ℤ‍∑k∈ℤ‍×{×χQ′∑Q′∈𝒬isoj′-N‍)δ/2](x)}2/δℳ[(∑Q′∈𝒬isoj′-N‍infv∈Q′|ϕj′(1)*fk(v)|2dddddddddddddddddd×χQ′∑Q′∈𝒬isoj′-N‍)δ/2](x)}2/δ}1/2.Finally, taking theLp norm on both sides and applying the Fefferman-Stein vector-valued inequality yield (19). Since L2∩Hcomp is dense in Hcomp (see [3]), a limiting argument concludes the proof of Theorem 4.


## 3. Proofs of Theorem1 and Corollary 2

We first give the following.Proof of Theorem1.
For1<p<∞, Theorem 1 is trivial since Hcomp=Hisop=Hnonp=Lp. We now assume that 0<p≤1. We only prove ∥f∥Hnonp≲∥f∥Hcomp as the inequality ∥f∥Hisop≲∥f∥Hcomp can be proved similarly. Since L2∩Hcomp is dense in Hcomp, we may assume that f∈L2∩Hcomp. For i∈ℤ, set
(26)Ωi={x∈ℝn:gcom(1)(f)(x)>2i}.
Denote that
(27)ℬi={12|Q|(j,Q):Q∈𝒬isoj-N,|Q∩Ωi|>12|Q|,|Q∩Ωi+1|≤12|Q|},
where N is the sufficiently large integer in Lemma 3.
Applying the discrete Calderón-type reproducing formula in Lemma3,
(28)fk(x)=∑i∈ℤ‍∑(j,Q)∈ℬi‍|Q|ϕj(1)*fk(xQ)ϕj(1)(x-xQ),k∈ℤ,
where fk:=ϕk(2)*f and the series converges in the L2(ℝn) norm. We claim that
(29)∥{∑k∈ℤ‍|∑(j,Q)∈ℬi‍|Q|ϕj(1)*fk(xQ)ϕj(1)(·-xQ)|2}1/2∥pp≤C2ip|Ωi|.
Assume the claim for the moment. Then, by the continuous square function characterization of Hnonp,
(30)∥f∥Hnonpp~∫ℝn{∑k∈ℤ|∑i∈ℤ‍∑(j,Q)∈ℬi‍|Q|ϕj(1)*fk(xQ)ϕj(1)‍‍dddddddddddddddd×|(x-xQ)|2∑k∈ℤ|∑i∈ℤ‍∑(j,Q)∈ℬi‍|Q|ϕj(1)*fk(xQ)ϕj(1)‍}p/2dx≤∫ℝn{∑i∈ℤ[∑k∈ℤ‍|∑(j,Q)∈ℬi‍|Q|ϕj(1)*fk(xQ)ϕj(1)‍‍dddddddddddddddddddd×(x-xQ)∑k∈Z‍|2]1/2}pdx≤∑i∈ℤ‍∫‍{∑k∈ℤ|∑(j,Q)∈ℬi‍|Q|ϕj(1)*fk(xQ)ϕj(1)‍ddddddddddddddddd×(x-xQ)∑(j,Q)∈ℬi‍|2}p/2dx≲∑i∈ℤ‍2ip|Ωi|≲∥gcom(1)(f)∥pp~∥f∥Hcompp,
where in the second inequality, we have used Minkowski's inequality, for ℓ2 and in the third inequality used the inequality (∑i‍|ai|)p≤∑i‍|ai|p for 0<p≤1.
Thus, to finish the proof of Theorem1, it suffices to verify claim (29). Since ϕ(1) is supported in unit ball of ℝn, for (j,Q)∈ℬi, {ϕj(1)(·-xQ)} are supported in
(31)Ω~i={x:ℳ(χΩi)(x)>12}.
Thus, by Hölder's inequality and |Ω~i|≤C|Ωi| (which follows from L2 boundedness of ℳ),
(32)∥{∑k‍|∑(j,Q)∈ℬi‍|Q|ϕj(1)*fk(xQ)ϕj(1)(·-xQ)|2}1/2∥pp=∫Ω~i‍{∑k‍|∑(j,Q)∈ℬi‍|Q|ϕj(1)*fk(xQ)ϕj(1)ddddddddddddd×(x-xQ)∑(j,Q)∈ℬi‍|2}p/2dx≤|Ω~i|1-p/2∫Ω~i‍∑k‍|∑(j,Q)∈ℬi‍|Q|ϕj(1)*fk(xQ)ϕj(1)dddddddddddddddddddd×(x-xQ)∑(j,Q)∈ℬi‍|2dx≲|Ωi|1-p/2∑k‍∥∑(j,Q)∈ℬi‍|Q|ϕj(1)*fk(xQ)ϕj(1)dddddddddddddddddddd×(·-xQ)∑(j,Q)∈ℬi‍∥22.
We now estimate the last L2 norm by duality argument. For ζ with ∥ζ∥2≤1, applying the Cauchy-Schwarz inequality yields
(33)|〈∑(j,Q)∈ℬi‍|Q|ϕj(1)*fk(xQ)ϕj(1)(·-xQ),ζ〉|=∑(j,Q)∈ℬi‍|Q|ϕj(1)*fk(xQ)∫‍ϕj(1)(x-xQ)ζk(x)dx≤(∑(j,Q)∈ℬi‍|Q||ϕj(1)*fk(xQ)|2)1/2×(∑(j,Q)∈ℬi‍|Q||ϕj(1)~*ζ(xQ)|2)1/2,
where ϕj(1)~(x)=ϕj(1)(-x)¯. Since
(34)∑(j,Q)∈ℬi‍|Q||ϕj(1)~*ζ(xQ)|2=∫ℝn‍{∑(j,Q)∈ℬi‍|ϕj(1)*ζ(xQ)|2χQ(x)}(1/2)·2dx≤∫ℝn‍|gisod(ζ)(x)|2dx≲∥ζ∥L22≤1,
we now have that
(35)∑k‍∥∑(j,Q)∈ℬi‍|Q|ϕj(1)*fk(xQ)ϕj(1)(·-xQ)∥L22≲∑k‍∑(j,Q)∈ℬi‍|Q||ϕj(1)*fk(xQ)|2,
which is, in turn, bounded by
(36)2∑k‍∑(j,Q)∈ℬi‍|ϕj(1)*fk(xQ)|2|Q∩(Ω~i-Ωi+1)|=2∫Ω~i∖Ωi+1‍{∑k‍∑(j,Q)∈ℬi‍|ϕj,k*f(xQ)|2χQ(x)}(1/2)·2dx≤2∫Ω~i∖Ωi+1‍[gcom(1)(f)(x)]2dx≲22i|Ωi|,
where in the first inequality, we have used |Q|≤2|Q∩(Ω~i∖Ωi+1)| when (j,Q)∈ℬi. Substituting this estimate back to (32) verifies the claim (29), and hence, Theorem 1 follows.Finally, we give the following.Proof of Corollary2.
(1) follows directly from Theorem1, while (2) follows from (1).



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