The Hardy-Lorentz Spaces Hp,q(Rn)
Wael Abu-Shammala and Alberto Torchinsky
Abstract
In this paper we consider the Hardy-Lorentz spaces Hp,q(Rn), with
0 < p ≤ 1, 0 < q ≤ ∞. We discuss the atomic decomposition of
the elements in these spaces, their interpolation properties, and the
behavior of singular integrals and other operators acting on them.
The real variable theory of the Hardy spaces represents a fruitful setting for
the study of maximal functions and singular integral operators. In fact, it
is because of the failure of these operators to preserve L1 that the Hardy
space H1 assumes its prominent role in harmonic analysis. Now, for many
of these operators, the role of L1 can just as well be played by H1,∞, or
Weak H1. However, although these operators are amenable to H1 − L1 and
H1,∞ − L1,∞ estimates, interpolation between H1 and H1,∞ has not been
available. Similar considerations apply to Hp and Weak Hp for 0 < p < 1.
The purpose of this paper is to provide an interpolation result for the
Hardy-Lorentz spaces Hp,q, 0 < p ≤ 1, 0 < q ≤ ∞, including the case of
Weak Hp as and end point for real interpolation. The atomic decomposition
is the key ingredient in dealing with interpolation since in this context neither
truncations are available, nor reiteration applies.
The paper is organized as follows. The Lorentz spaces, including criteria
that assure membership in Lp,q, 0 < p < ∞, 0 < q ≤ ∞, are discussed in
Section 1. In Section 2 we show that distributions in Hp,q have an atomic
decomposition in terms ofHp atoms with coefficients in an appropriate mixed
norm space. An interesting application of this decomposition is to Hp,q−Lp,∞
estimates for Calderón-Zygmund singular integral operators, p < q ≤ ∞.
Also, by manipulating the different levels of the atomic decomposition, we
show that, for 0 < q1 < q < q2 ≤ ∞, H
p,q is an intermediate space between
Hp,q1 and Hp,q2. This result applies to Calderón-Zygmund singular integral
operators, including those with variable kernels, Marcinkiewicz integrals, and
other operators.
http://arxiv.org/abs/0704.0054v1
1 The Lorentz spaces
The Lorentz space Lp,q(Rn) = Lp,q, 0 < p <∞, 0 < q ≤ ∞, consists of those
measurable functions f with finite quasinorm ‖f‖p,q given by
‖f‖p,q =
[t1/pf ∗(t)]q
, 0 < q <∞ ,
‖f‖p,∞ = sup
[t1/pf ∗(t)] , q = ∞ .
The Lorentz quasinorm may also be given in terms of the distribution func-
tion m(f, λ) = |{x ∈ Rn : |f(x)| > λ}|, loosely speaking, the inverse of the
non-increasing rearrangement f ∗ of f . Indeed, we have
‖f‖p,q =
λq−1m(f, λ)q/p dλ
2km(f, 2k)1/p
when 0 < q <∞, and
‖f‖p,∞ = sup
2km(f, 2k)1/p , q = ∞ .
Note that, in particular, Lp,p = Lp, and Lp,∞ is weak Lp.
The following two results are useful in verifying that a function is in Lp,q.
Lemma 1.1. Let 0 < p <∞, and 0 < q ≤ ∞. Assume that the non-negative
sequence {µk} satisfies {2
kµk} ∈ ℓ
q. Further suppose that the non-negative
function ϕ verifies the following property: there exists 0 < ε < 1 such that,
given an arbitrary integer k0, we have ϕ ≤ ψk0 + ηk0, where ψk0 is essentially
bounded and satisfies ‖ψk0‖∞ ≤ c 2
k0, and
2k0εpm(ηk0 , 2
k0) ≤ c
[2kεµk]
Then, ϕ ∈ Lp,q, and ‖ϕ‖p,q ≤ c ‖{2
kµk}‖ℓq.
Proof. It clearly suffices to verify that ‖{2k |{ϕ > γ 2k}|1/p}‖ℓq <∞, where γ
is an arbitrary positive constant. Now, given k0, let ψk0 and ηk0 be as above,
and put γ = c+ 1, where c is the constant in the above inequalities; for this
choice of γ, {ϕ > γ 2k0} ⊆ {ηk0 > 2
When q = ∞, we have
2k0εm(ηk0 , 2
k0)1/p ≤ c
[2−k(1−ε) 2k µk]
≤ c 2−k0(1−ε) sup
[ 2k µk] .
Thus, 2k0 m(ηk0 , 2
k0)1/p ≤ supk≥k0[ 2
k µk] , and, consequently,
2k0 m(ϕ, γ 2k0)1/p ≤ c ‖{2kµk}‖ℓ∞ , all k0.
When 0 < q < ∞, let 1 − ε = 2δ and rewrite the right-hand side above
[2k(1−δ)µk]
When p < q, by Hölder’s inequality with exponent r = q/p and its conjugate
r′, this expression is dominated by
2k δpr
)1/r′(
2k(1−δ)µk
]rp )1/r
≤ c 2−k0 δp
2k(1−δ)µk
]q )p/q
and, when 0 < q ≤ p, r < 1, and we get a similar bound by simply observing
that it does not exceed
2−k0δp
[2k(1−δ)µk]
≤ 2−k0δp
2k(1−δ)µk
]q )p/q
Whence, continuing with the estimate, we have
2k0εpm(ηk0 , 2
k0) ≤ c 2−k0δp
2k(1−δ)µk
]q )p/q
which yields, since 1− ε = 2 δ,
2k0 m(ϕ, γ 2k0)1/p ≤ c 2k0 δ
2k(1−δ)µk
]q )1/q
Thus, raising to the q and summing, we get
2k0 m(ϕ, γ 2k0)1/p
2k0 δ q
2k(1−δ)µk
which, upon changing the order of summation in the right-hand side of the
above inequality, is bounded by
2k(1−δ)µk
k0=−∞
2k0 δ q
The reader will have no difficulty in verifying that, for Lemma 1.1 to hold,
it suffices that ψx0 satisfies
m(ψx0 , 2
k0)1/p ≤ c µk0 , all k0 .
This holds, for instance, when ‖ψx0‖
r ≤ c 2
, 0 < r < ∞. In fact, the
assumptions of Lemma 1.1 correspond to the limiting case of this inequality
as r → ∞.
Another useful condition is given by our next result, the proof is left to
the reader.
Lemma 1.2. Let 0 < p < ∞, and let the non-negative sequence {µk} be
such that {2kµk} ∈ ℓ
q, 0 < q ≤ ∞. Further, suppose that the non-negative
function ϕ satisfies the following property: there exists 0 < ε < 1 such that,
given an arbitrary integer k0, we have ϕ ≤ ψk0+ηk0, where ψk0 and ηk0 satisfy
2k0pm(ψk0 , 2
k0)ε ≤ c
2kµεk
, 0 < ε < min(1, q/p) ,
2k0ε|{ηk0 > 2
k0}| ≤ c
2kεµk
Then, ϕ ∈ Lp,q, and ‖ϕ‖p,q ≤ c ‖{2
kµk}‖ℓq.
We will also require some basic concepts from the theory of real interpo-
lation. Let A0, A1, be a compatible couple of quasinormed Banach spaces,
i.e., both A0 and A1 are continuously embedded in a larger topological vector
space. The Peetre K functional of f ∈ A0 + A1 at t > 0 is defined by
K(t, f ;A0, A1) = inf
f=f0+f1
‖f0‖0 + t ‖f1‖1 ,
where f = f0 + f1, f0 ∈ A0 and f1 ∈ A1.
In the particular case of the Lq spaces, the K functional can be computed
by Holmstedt’s formula, see [12]. Specifically, for 0 < q0 < q1 ≤ ∞, let α be
given by 1/α = 1/q0 − 1/q1. Then,
K(t, f ;Lq0, Lq1) ∼
f ∗(s)q0ds
)1/q0
f ∗(s)q1ds
)1/q1
The intermediate space (A0, A1)η, q, 0 < η < 1, 0 < q < ∞, consists of
those f ’s in A0 + A1 with
‖f‖(A0,A1)η, q =
t−ηK(t, f ;A0, A1)
]q dt
‖f‖(A0,A1)η,∞ = sup
t−ηK(t, f ;A0, A1)
<∞ , q = ∞ .
Finally, for the Lq and Lp,q spaces, we have the following result. Let
0 < q1 < q < q2 ≤ ∞, and suppose that 1/q = (1 − η)/q1 + η/q2. Then,
Lq = (Lq1 , Lq2)η,q, and, L
1,q = (L1,q1, L1,q2)η,q, see [4].
2 The Hardy-Lorentz spaces Hp,q
In this paper we adopt the atomic characterization of the Hardy spaces Hp,
0 < p ≤ 1. Recall that a compactly supported function a with [n(1/p− 1)]
vanishing moments is an Hp atom with defining interval I (of course, I is
a cube in Rn), if supp(a) ⊆ I, and |I|1/p |a(x)| ≤ 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,
p < ∞, and the convergence is
in the sense of distributions as well as in Hp. Furthermore,
‖f‖Hp ∼ inf
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 .
C. Fefferman, Rivière and Sagher identified the intermediate spaces be-
tween the Hardy space Hp0, 0 < p0 < 1, and L
∞, as
(Hp0, L∞)η,q = H
p,q, 1/p = (1− η)/p0 , 0 < q ≤ ∞ ,
where Hp,q consists of those distributions f whose radial maximal function
Mf(x) = supt>0 |(f ∗ ϕt)(x)| belongs to L
p,q. Here ϕ is a compactly sup-
ported, smooth function with nonvanishing integral, see [10]. R. Fefferman
and Soria studied in detail the space H1,∞, which they called Weak H1, see
[11].
Just as in the case of Hp, Hp,q can be characterized in a number of
different ways, including in terms of non-tangential maximal functions and
Lusin functions. In what follows we will calculate the quasinorm of f in Hp,q
by the means of the expression
2km(Mf, 2k)1/p
, 0 < p ≤ 1, 0 < q ≤ ∞ ,
where Mf is an appropriate maximal function of f .
Passing to the atomic decomposition of Hp,q, the proof is divided in two
parts. First, we construct an essentially optimal atomic decomposition; Par-
ilov has obtained independently this result for H1,q when 1 ≤ q, see [14].
Also, R. Fefferman and Soria gave the atomic decomposition of Weak H1,
see [11], and Alvarez the atomic decomposition of Weak Hp, 0 < p < 1, see
Theorem 2.1. Let f ∈ Hp,q, 0 < p ≤ 1, 0 < q ≤ ∞. Then f has an atomic
decomposition f =
j,k λj,kaj,k, where the aj,k’s are H
p atoms with defining
intervals Ij,k that have bounded overlap uniformly for each k, the sequence
{λj,k} satisfies
j |λj,k|
< ∞, and the convergence is in the
sense of distributions. Furthermore,
j |λj,k|
∼ ‖f‖Hp,q .
Proof. The idea of constructing an atomic decomposition using Calderón’s
reproducing formula is well understood, so we will only sketch it here, for
further details, see [5] and [18]. Let Nf(x) = sup{|(f ∗ ψt)(y)| : |x− y| < t}
denote the non-tangential maximal function of f with respect to a suitable
smooth function ψ with nonvanishing integral. One considers the open sets
Ok = {Nf > 2
k}, all integers k, and builds the atoms with defining interval
associated to the intervals, actually cubes, of the Whitney decomposition
of Ok, and hence satisfying all the required properties. More precisely, one
constructs a sequence of bounded functions fk with norm not exceeding c 2
for each k, and such that f −
|k|≤n fk → 0 as n→ ∞ in the sense of distri-
butions. These functions have the further property that fk(x) =
j αj,k(x) ,
where |αj,k(x)| ≤ c 2
k, c is a constant, each αj,k has vanishing moments up
to order [n(1/p − 1)] and is supported in Ij,k - roughly one of the Whitney
cubes -, where the Ij,k’s have bounded overlaps for each k, uniformly in k. It
only remains now to scale αj,k,
αj,k(x) = λj,k aj,k(x) ,
and balance the contribution of each term to the sum. Let λj,k = 2
k|Ij,k|
Then, aj,k(x) is essentially an H
p atom with defining interval Ij,k, and one
j |λj,k|
∼ 2k |Ok|
1/p. Thus,
|λj,k|
)1/p∥
2k |Ok|
∼ ‖f‖Hp,q , 0 < q ≤ ∞ . �
As an application of this atomic decomposition, the reader should have
no difficulty in showing directly the C. Fefferman, Rivière, Sagher character-
ization of Hp,q, see [10].
Another interesting application of this decomposition is to Hp,q − Lp,∞
estimates for Calderón-Zygmund singular integral operators T , p < q ≤ ∞.
This approach combines the concept of p-quasi local operator of Weisz, see
[17], with the idea of variable dilations of R. Fefferman and Soria, see [11].
Intuitively, since Hörmander’s condition implies that T maps H1 into L1, say,
for T to be defined in H1,s, 1 < s ≤ ∞, some strengthening of this condition
is required. This is accomplished by the variable dilations. Moreover, since
we will include p < 1 in our discussion, as p gets smaller, more regularity of
the kernel of T will be required. This justifies the following definition.
Given 0 < p ≤ 1, let N = [n(1/p − 1)], and, associated to the kernel
k(x, y) of a Calderón-Zygmund singular integral operator T , consider the
modulus of continuity ωp given by
ωp(δ) = sup
Rn\(2/δ)I
| k(x, y)−
|α|≤N
(y − yI)
αkα(x, yI)| dy
where 0 < δ ≤ 1, and the sup is taken over the collection of arbitrary intervals
I of Rn centered at yI . Here, for a multi-index α = (α1, . . . , αn),
kα(x, yI) =
Dαk(x, y)
ωp(δ) controls the behavior of T on atoms. More precisely, if a is an H
p atom
with defining interval I, and 0 < δ < 1, observe that
T (a)(x) =
[k(x, y)−
|α|≤N
(y − yI)
αkα(x, yI)] a(y) dy ,
and, consequently,
Rn\(2/δ)I
|T (a)(x)|p dx ≤ ωp(δ) .
We are now ready to prove the Hp,q − Lp,∞ estimate for a Calderón-
Zygmund singular integral operator T with kernel k(x, y).
Theorem 2.2. Let 0 < p ≤ 1, and p < q ≤ ∞. Assume that a Calderón-
Zygmund singular integral operator T is of weak-type (r, r) for some 1 < r <
∞, and that the modulus of continuity ωp of the kernel k satisfies a Dini
condition of order q/(q − p), namely,
Ap,q =
ωp(δ)
q/(q−p)dδ
](q−p)/q
Then T maps Hp,q continuously into Lp,∞, and ‖Tf‖p,∞ ≤ cA
p,q ‖f‖Hp,q .
Proof. We need to show that
2k0pm(Tf, 2k0) ≤ c ‖f‖
Hp,q , all k0 .
Let f =
j λj,kaj,k , be the atomic decomposition of f given in Theorem
2.1, and set f1 =
j λj,kaj,k, and f2 = f − f1. Further, let µk =
j |λj,k|
, and recall that ‖{µk}‖ℓq ∼ ‖f‖Hp,q .
Since ‖f1‖
r ≤ c 2
k0(r−p)‖f‖
Hp,∞, we have
2pk0m(Tf1, 2
k0) ≤ c ‖f‖
Hp,∞ .
Next, put I∗j,k = 2
1/n(3/2)p(k−k0)/nIj,k, and let
I∗j,k .
Since |I∗j,k| = 2(3/2)
p(k−k0)|Ij,k| ∼ 2
−k0p(3/4)p(k−k0)|λj,k|
p, we get
|Ω| ≤
|I∗j,k| ≤ c 2
(3/4)p(k−k0)
|λj,k|
≤ c 2−k0p
≤ c 2−k0p‖f‖
Hp,∞ .
Also, since 0 < p ≤ 1, it readily follows that
|T (f2)(x)|
|λj,k|
p|T (aj,k)(x)|
and, by Tonelli and the estimate for T (a), we have
|T (f2)(x)|
p dx ≤
|λj,k|
Rn\I∗
|T (aj,k)(x)|
)p(k−k0)/n
)pk/n)q/(q−p))(q−p)/q
ωp(δ)
q/(q−p)dδ
](q−p)/q
Hp,q .
This bound gives at once
2pk0 |{x /∈ Ω : |T (f2)(x)| > 2
k0}| ≤ cAp,q ‖f‖
Hp,q ,
which implies that
2pk0m(Tf2, 2
k0−1) ≤ 2pk0
|Ω|+ |{x /∈ Ω : |T (f2)(x)| > 2
k0−1}|
≤ c ‖f‖
Hp,∞ + cAp,q ‖f‖
Hp,q .
Finally,
2k0pm(Tf, 2k0) ≤ 2k0pm(Tf1, 2
k0−1) + 2k0pm(Tf2, 2
k0−1)
≤ c ‖f‖
Hp,∞ + cAp,q ‖f‖
Hp,q ,
and, since ‖f‖Hp,∞ ≤ c ‖f‖Hp,q for all q, we have finished. �
We pass now to the converse of Theorem 2.1. It is apparent that a
condition that relates the coefficients λj with the corresponding atoms aj
involved in an atomic decomposition of the form
j λjaj(x) is relevant here.
More precisely, if Ij denotes the supporting interval of aj , let
Ik = {j : 2
k ≤ |λj|/|Ij|
1/p < 2k+1} ,
and, for λ = {λj}, put
‖λ‖[p,q] =
]q/p)1/q
We then have,
Theorem 2.3. Let 0 < p ≤ 1, 0 < q ≤ ∞, and let f be a distribution given
by f =
j λj aj(x) , where the aj’s are H
p atoms, and the convergence is in
the sense of distributions. Further, assume that the family {Ij} consisting of
the supports of the aj’s has bounded overlap at each level Ik uniformly in k,
and ‖λ‖[p,q] <∞. Then, f ∈ H
p,q, and ‖f‖Hp,q ≤ c ‖λ‖[p,q].
Proof. Let Mf(x) = supt>0 |(f ∗ ψt)(x)| denote the radial maximal function
of f with respect to a suitable smooth function ψ with support contained in
{|x| ≤ 1} and nonvanishing integral. We will verify that Mf satisfies the
conditions of Lemma 1.1 and is thus in Lp,q.
Fix an integer k0 and let
g(x) =
λjaj(x) .
Since ‖Mg‖∞ ≤ ‖g‖∞ it suffices to estimate |g(x)|. Let C be the bounded
overlap constant for the family of the supports of the aj’s. Then, for j ∈ Ik,
|λj| |aj(x)| =
|Ij|1/p
|λj | |Ij|
1/p |aj(x)| ≤ 2
kχIj (x) ,
and, consequently,
|g(x)| ≤
χIj(x) ≤ C 2
Next, let
h(x) =
λjaj(x) .
Since aj has N = [n(1/p − 1)] vanishing moments, it is not hard to see
that, if Ij is the defining interval of aj and Ij is centered at xj , and γ =
(n+N+1)/n > 1/p, then, with c independent of j, ϕj(x) =Maj(x) satisfies
ϕj(x) ≤ c
|Ij |
γ−1/p
(|Ij|+ |x− xj |n)γ
Thus, if 1/γ < εp < 1,
Mh(x)εp ≤ c
j∈Ik,k≥k0
(|λj| |Ij|
γ−1/p)εp
(|Ij |+ |x− xj |n)γεp
which, upon integration, yields
Mh(x)εp dx ≤ c
j∈Ik,k≥k0
(|λj| |Ij|
γ−1/p)εp
(|Ij |+ |x− xj |n)γεp
The integrals in the right-hand side above are of order |Ij|
1−γεp and, conse-
quently, by Chebychev’s inequality,
2k0εp|{Mh > 2k0}| ≤ c
j∈Ik,k≥k0
εp |Ij|
1−ε ≤ c
|Ij| .
Thus, Lemma 1.1 applies with ϕ = Mf , ψk0 = Mg, ηk0 = Mh, and µk =
, and we get
2km(Mf, 2k)1/p
)1/p}∥
which, since
|Ij| ∼
, j ∈ Ik ,
is bounded by c ‖λ‖[p,q], 0 < q ≤ ∞. �
The next result is of interest because it applies to arbitrary decomposi-
tions in Hp,q. The proof relies on Lemma 1.2, and is left to the reader.
Theorem 2.4. Let 0 < p ≤ 1, 0 < q ≤ ∞, and let f be a distribution given
by f =
j λj aj(x) , where the aj’s are H
p atoms, and the convergence is
in the sense of distributions. Further, assume that ‖λ‖[η,q] < ∞ for some
0 < η < min(p, q). Then, f ∈ Hp,q, and ‖f‖Hp,q ≤ c ‖λ‖[η,q].
2.1 Interpolation between Hardy-Lorentz spaces
We are now ready to identify the intermediate spaces of a couple of Hardy-
Lorentz spaces with the same first index p ≤ 1.
Theorem 2.5. Let 0 < p ≤ 1. Given 0 < q1 < q < q2 ≤ ∞, define 0 < η < 1
by the relation 1/q = (1− η)/q1 + η/q2. Then, with equivalent quasinorms,
Hp,q = (Hp,q1, Hp,q2)η,q .
Proof. Since the non-tangential maximal function Nf of a distribution f in
Hp,q1 is in Lp,q1, and that of f in Hp,q2 is in Lp,q2, we have
K(t, Nf ;Lp,q1, Lp,q2) ≤ cK(t, f ;Hp,q1, Hp,q2) .
Thus,
‖Nf‖p,q ∼ ‖Nf‖(Lp,q1 ,Lp,q2)η,q ≤ c ‖f‖(Hp,q1 ,Hp,q2 )η,q ,
and (Hp,q1, Hp,q2)η,q →֒ H
To show the other embedding, with the notation in the proof of Theorem
2.1, write f =
j λj,kaj,k , and recall that for every integer k, the level
set Ik = {j : |λj,k|/|Ij,k|
1/p ∼ 2k} contains exclusively the sequence {λj,k}.
Let µ
|λj,k|
p. By construction,
k ∼ ‖f‖
Hp,q . Now, rearrange
{µk} into {µ
l }, and, for each l ≥ 1, let kl be such that µkl = µ
l . For
l0 ≥ 1, let Kl0 = {k1, . . . , kl0}, and put f1,l0 =
k∈Kl0
j λj,kaj,k and f2,l0 =
f − f1,l0 . Then, by Theorem 2.2, f1,l0 ∈ H
p,q1, f2,l0 ∈ H
p,q2, and, with the
usual interpretation for q2 = ∞,
‖f1,l0‖Hp,q1 ≤ c
)1/q1
, ‖f2,l0‖Hp,q2 ≤ c
)1/q2
So, for t > 0 and every positive integer l0, we have
K(t, f ;Hp,q1, Hp,q2) ≤ c
)1/q1
)1/q2
Now, by Homstedt’s formula, there is a choice of l0 such that the right-hand
side above ∼ K(t, {µk}; ℓ
q1, ℓq2), and, consequently,
K(t, f ;Hp,q1, Hp,q2) ≤ cK(t, {µk}; ℓ
q1, ℓq2) .
Thus,
‖f‖(Hp,q1 ,Hp,q2 )η,q ≤ c ‖{µk}‖(ℓq1 ,ℓq2)η,q
≤ c ‖{µk}‖ℓq ≤ c ‖f‖Hp,q ,
and Hp,q →֒ (Hp,q1, Hp,q2)η,q. �
The reader will have no difficulty in verifying that Theorem 2.5 gives that
if T is a continuous, sublinear map fromH1 into L1, and fromH1,∞ into L1,∞,
then ‖Tf‖1,q ≤ c ‖f‖H1,q for 1 < q < ∞. This observation has numerous
applications. For instance, consider the Calderón-Zygmund singular integral
operators with variable kernel defined by
TΩ(f)(x) = p.v.
Ω(x, x−y)
|x− y|n
f(y) dy .
Under appropriate growth and smoothness assumptions on Ω, TΩ maps H
continuously into L1, see [6], and H1,∞ continuously into L1,∞, see [8]. Thus,
if Ω satisfies the assumptions of both of these results, TΩ maps H
1,q contin-
uously into L1,q for 1 < q < ∞. A similar result follows by invoking the
characterization of H1,q given by C. Fefferman, Rivière and Sagher. How-
ever, in this case the Hp−Lp estimate requires additional smoothness of Ω, as
shown, for instance, in [6]. Similar considerations apply to the Marcinkiewicz
integral, see [9], and [7].
Finally, when p < 1, our results cover, for instance, the δ-CZ operators
satisfying T ∗(1) = 0 discussed by Alvarez and Milman, see [3]. These oper-
ators, as well as a more general related class introduced in [15], preserve Hp
and Hp,∞ for n/(n + δ) < p ≤ 1, and, consequently, by Theorem 2.5, they
also preserve Hp,q for p in that same range, and q > p.
References
[1] W. Abu-Shammala and A. Torchinsky, The atomic decomposition for
H1,q(Rn), Proceedings of the International Conference on Harmonic
Analysis and Ergodic Theory, (2005), to appear.
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DEPARTMENT OF MATHEMATICS, INDIANA UNIVERSITY,
BLOOMINGTON, IN 47405
E-mail: wabusham@indiana.edu, torchins@indiana.edu
	The Lorentz spaces
	The Hardy-Lorentz spaces Hp,q
	Interpolation between Hardy-Lorentz spaces

  In this paper we consider the Hardy-Lorentz spaces $H^{p,q}(R^n)$, with
$0<p\le 1$, $0<q\le \infty$. We discuss the atomic decomposition of the
elements in these spaces, their interpolation properties, and the behavior of
singular integrals and other operators acting on them.

The Hardy-Lorentz Spaces Hp,q(Rn)
Wael Abu-Shammala and Alberto Torchinsky
Abstract
In this paper we consider the Hardy-Lorentz spaces Hp,q(Rn), with
0 < p ≤ 1, 0 < q ≤ ∞. We discuss the atomic decomposition of
the elements in these spaces, their interpolation properties, and the
behavior of singular integrals and other operators acting on them.
The real variable theory of the Hardy spaces represents a fruitful setting for
the study of maximal functions and singular integral operators. In fact, it
is because of the failure of these operators to preserve L1 that the Hardy
space H1 assumes its prominent role in harmonic analysis. Now, for many
of these operators, the role of L1 can just as well be played by H1,∞, or
Weak H1. However, although these operators are amenable to H1 − L1 and
H1,∞ − L1,∞ estimates, interpolation between H1 and H1,∞ has not been
available. Similar considerations apply to Hp and Weak Hp for 0 < p < 1.
The purpose of this paper is to provide an interpolation result for the
Hardy-Lorentz spaces Hp,q, 0 < p ≤ 1, 0 < q ≤ ∞, including the case of
Weak Hp as and end point for real interpolation. The atomic decomposition
is the key ingredient in dealing with interpolation since in this context neither
truncations are available, nor reiteration applies.
The paper is organized as follows. The Lorentz spaces, including criteria
that assure membership in Lp,q, 0 < p < ∞, 0 < q ≤ ∞, are discussed in
Section 1. In Section 2 we show that distributions in Hp,q have an atomic
decomposition in terms ofHp atoms with coefficients in an appropriate mixed
norm space. An interesting application of this decomposition is to Hp,q−Lp,∞
estimates for Calderón-Zygmund singular integral operators, p < q ≤ ∞.
Also, by manipulating the different levels of the atomic decomposition, we
show that, for 0 < q1 < q < q2 ≤ ∞, H
p,q is an intermediate space between
Hp,q1 and Hp,q2. This result applies to Calderón-Zygmund singular integral
operators, including those with variable kernels, Marcinkiewicz integrals, and
other operators.
http://arxiv.org/abs/0704.0054v1
1 The Lorentz spaces
The Lorentz space Lp,q(Rn) = Lp,q, 0 < p <∞, 0 < q ≤ ∞, consists of those
measurable functions f with finite quasinorm ‖f‖p,q given by
‖f‖p,q =
[t1/pf ∗(t)]q
, 0 < q <∞ ,
‖f‖p,∞ = sup
[t1/pf ∗(t)] , q = ∞ .
The Lorentz quasinorm may also be given in terms of the distribution func-
tion m(f, λ) = |{x ∈ Rn : |f(x)| > λ}|, loosely speaking, the inverse of the
non-increasing rearrangement f ∗ of f . Indeed, we have
‖f‖p,q =
λq−1m(f, λ)q/p dλ
2km(f, 2k)1/p
when 0 < q <∞, and
‖f‖p,∞ = sup
2km(f, 2k)1/p , q = ∞ .
Note that, in particular, Lp,p = Lp, and Lp,∞ is weak Lp.
The following two results are useful in verifying that a function is in Lp,q.
Lemma 1.1. Let 0 < p <∞, and 0 < q ≤ ∞. Assume that the non-negative
sequence {µk} satisfies {2
kµk} ∈ ℓ
q. Further suppose that the non-negative
function ϕ verifies the following property: there exists 0 < ε < 1 such that,
given an arbitrary integer k0, we have ϕ ≤ ψk0 + ηk0, where ψk0 is essentially
bounded and satisfies ‖ψk0‖∞ ≤ c 2
k0, and
2k0εpm(ηk0 , 2
k0) ≤ c
[2kεµk]
Then, ϕ ∈ Lp,q, and ‖ϕ‖p,q ≤ c ‖{2
kµk}‖ℓq.
Proof. It clearly suffices to verify that ‖{2k |{ϕ > γ 2k}|1/p}‖ℓq <∞, where γ
is an arbitrary positive constant. Now, given k0, let ψk0 and ηk0 be as above,
and put γ = c+ 1, where c is the constant in the above inequalities; for this
choice of γ, {ϕ > γ 2k0} ⊆ {ηk0 > 2
When q = ∞, we have
2k0εm(ηk0 , 2
k0)1/p ≤ c
[2−k(1−ε) 2k µk]
≤ c 2−k0(1−ε) sup
[ 2k µk] .
Thus, 2k0 m(ηk0 , 2
k0)1/p ≤ supk≥k0[ 2
k µk] , and, consequently,
2k0 m(ϕ, γ 2k0)1/p ≤ c ‖{2kµk}‖ℓ∞ , all k0.
When 0 < q < ∞, let 1 − ε = 2δ and rewrite the right-hand side above
[2k(1−δ)µk]
When p < q, by Hölder’s inequality with exponent r = q/p and its conjugate
r′, this expression is dominated by
2k δpr
)1/r′(
2k(1−δ)µk
]rp )1/r
≤ c 2−k0 δp
2k(1−δ)µk
]q )p/q
and, when 0 < q ≤ p, r < 1, and we get a similar bound by simply observing
that it does not exceed
2−k0δp
[2k(1−δ)µk]
≤ 2−k0δp
2k(1−δ)µk
]q )p/q
Whence, continuing with the estimate, we have
2k0εpm(ηk0 , 2
k0) ≤ c 2−k0δp
2k(1−δ)µk
]q )p/q
which yields, since 1− ε = 2 δ,
2k0 m(ϕ, γ 2k0)1/p ≤ c 2k0 δ
2k(1−δ)µk
]q )1/q
Thus, raising to the q and summing, we get
2k0 m(ϕ, γ 2k0)1/p
2k0 δ q
2k(1−δ)µk
which, upon changing the order of summation in the right-hand side of the
above inequality, is bounded by
2k(1−δ)µk
k0=−∞
2k0 δ q
The reader will have no difficulty in verifying that, for Lemma 1.1 to hold,
it suffices that ψx0 satisfies
m(ψx0 , 2
k0)1/p ≤ c µk0 , all k0 .
This holds, for instance, when ‖ψx0‖
r ≤ c 2
, 0 < r < ∞. In fact, the
assumptions of Lemma 1.1 correspond to the limiting case of this inequality
as r → ∞.
Another useful condition is given by our next result, the proof is left to
the reader.
Lemma 1.2. Let 0 < p < ∞, and let the non-negative sequence {µk} be
such that {2kµk} ∈ ℓ
q, 0 < q ≤ ∞. Further, suppose that the non-negative
function ϕ satisfies the following property: there exists 0 < ε < 1 such that,
given an arbitrary integer k0, we have ϕ ≤ ψk0+ηk0, where ψk0 and ηk0 satisfy
2k0pm(ψk0 , 2
k0)ε ≤ c
2kµεk
, 0 < ε < min(1, q/p) ,
2k0ε|{ηk0 > 2
k0}| ≤ c
2kεµk
Then, ϕ ∈ Lp,q, and ‖ϕ‖p,q ≤ c ‖{2
kµk}‖ℓq.
We will also require some basic concepts from the theory of real interpo-
lation. Let A0, A1, be a compatible couple of quasinormed Banach spaces,
i.e., both A0 and A1 are continuously embedded in a larger topological vector
space. The Peetre K functional of f ∈ A0 + A1 at t > 0 is defined by
K(t, f ;A0, A1) = inf
f=f0+f1
‖f0‖0 + t ‖f1‖1 ,
where f = f0 + f1, f0 ∈ A0 and f1 ∈ A1.
In the particular case of the Lq spaces, the K functional can be computed
by Holmstedt’s formula, see [12]. Specifically, for 0 < q0 < q1 ≤ ∞, let α be
given by 1/α = 1/q0 − 1/q1. Then,
K(t, f ;Lq0, Lq1) ∼
f ∗(s)q0ds
)1/q0
f ∗(s)q1ds
)1/q1
The intermediate space (A0, A1)η, q, 0 < η < 1, 0 < q < ∞, consists of
those f ’s in A0 + A1 with
‖f‖(A0,A1)η, q =
t−ηK(t, f ;A0, A1)
]q dt
‖f‖(A0,A1)η,∞ = sup
t−ηK(t, f ;A0, A1)
<∞ , q = ∞ .
Finally, for the Lq and Lp,q spaces, we have the following result. Let
0 < q1 < q < q2 ≤ ∞, and suppose that 1/q = (1 − η)/q1 + η/q2. Then,
Lq = (Lq1 , Lq2)η,q, and, L
1,q = (L1,q1, L1,q2)η,q, see [4].
2 The Hardy-Lorentz spaces Hp,q
In this paper we adopt the atomic characterization of the Hardy spaces Hp,
0 < p ≤ 1. Recall that a compactly supported function a with [n(1/p− 1)]
vanishing moments is an Hp atom with defining interval I (of course, I is
a cube in Rn), if supp(a) ⊆ I, and |I|1/p |a(x)| ≤ 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,
p < ∞, and the convergence is
in the sense of distributions as well as in Hp. Furthermore,
‖f‖Hp ∼ inf
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 .
C. Fefferman, Rivière and Sagher identified the intermediate spaces be-
tween the Hardy space Hp0, 0 < p0 < 1, and L
∞, as
(Hp0, L∞)η,q = H
p,q, 1/p = (1− η)/p0 , 0 < q ≤ ∞ ,
where Hp,q consists of those distributions f whose radial maximal function
Mf(x) = supt>0 |(f ∗ ϕt)(x)| belongs to L
p,q. Here ϕ is a compactly sup-
ported, smooth function with nonvanishing integral, see [10]. R. Fefferman
and Soria studied in detail the space H1,∞, which they called Weak H1, see
[11].
Just as in the case of Hp, Hp,q can be characterized in a number of
different ways, including in terms of non-tangential maximal functions and
Lusin functions. In what follows we will calculate the quasinorm of f in Hp,q
by the means of the expression
2km(Mf, 2k)1/p
, 0 < p ≤ 1, 0 < q ≤ ∞ ,
where Mf is an appropriate maximal function of f .
Passing to the atomic decomposition of Hp,q, the proof is divided in two
parts. First, we construct an essentially optimal atomic decomposition; Par-
ilov has obtained independently this result for H1,q when 1 ≤ q, see [14].
Also, R. Fefferman and Soria gave the atomic decomposition of Weak H1,
see [11], and Alvarez the atomic decomposition of Weak Hp, 0 < p < 1, see
Theorem 2.1. Let f ∈ Hp,q, 0 < p ≤ 1, 0 < q ≤ ∞. Then f has an atomic
decomposition f =
j,k λj,kaj,k, where the aj,k’s are H
p atoms with defining
intervals Ij,k that have bounded overlap uniformly for each k, the sequence
{λj,k} satisfies
j |λj,k|
< ∞, and the convergence is in the
sense of distributions. Furthermore,
j |λj,k|
∼ ‖f‖Hp,q .
Proof. The idea of constructing an atomic decomposition using Calderón’s
reproducing formula is well understood, so we will only sketch it here, for
further details, see [5] and [18]. Let Nf(x) = sup{|(f ∗ ψt)(y)| : |x− y| < t}
denote the non-tangential maximal function of f with respect to a suitable
smooth function ψ with nonvanishing integral. One considers the open sets
Ok = {Nf > 2
k}, all integers k, and builds the atoms with defining interval
associated to the intervals, actually cubes, of the Whitney decomposition
of Ok, and hence satisfying all the required properties. More precisely, one
constructs a sequence of bounded functions fk with norm not exceeding c 2
for each k, and such that f −
|k|≤n fk → 0 as n→ ∞ in the sense of distri-
butions. These functions have the further property that fk(x) =
j αj,k(x) ,
where |αj,k(x)| ≤ c 2
k, c is a constant, each αj,k has vanishing moments up
to order [n(1/p − 1)] and is supported in Ij,k - roughly one of the Whitney
cubes -, where the Ij,k’s have bounded overlaps for each k, uniformly in k. It
only remains now to scale αj,k,
αj,k(x) = λj,k aj,k(x) ,
and balance the contribution of each term to the sum. Let λj,k = 2
k|Ij,k|
Then, aj,k(x) is essentially an H
p atom with defining interval Ij,k, and one
j |λj,k|
∼ 2k |Ok|
1/p. Thus,
|λj,k|
)1/p∥
2k |Ok|
∼ ‖f‖Hp,q , 0 < q ≤ ∞ . �
As an application of this atomic decomposition, the reader should have
no difficulty in showing directly the C. Fefferman, Rivière, Sagher character-
ization of Hp,q, see [10].
Another interesting application of this decomposition is to Hp,q − Lp,∞
estimates for Calderón-Zygmund singular integral operators T , p < q ≤ ∞.
This approach combines the concept of p-quasi local operator of Weisz, see
[17], with the idea of variable dilations of R. Fefferman and Soria, see [11].
Intuitively, since Hörmander’s condition implies that T maps H1 into L1, say,
for T to be defined in H1,s, 1 < s ≤ ∞, some strengthening of this condition
is required. This is accomplished by the variable dilations. Moreover, since
we will include p < 1 in our discussion, as p gets smaller, more regularity of
the kernel of T will be required. This justifies the following definition.
Given 0 < p ≤ 1, let N = [n(1/p − 1)], and, associated to the kernel
k(x, y) of a Calderón-Zygmund singular integral operator T , consider the
modulus of continuity ωp given by
ωp(δ) = sup
Rn\(2/δ)I
| k(x, y)−
|α|≤N
(y − yI)
αkα(x, yI)| dy
where 0 < δ ≤ 1, and the sup is taken over the collection of arbitrary intervals
I of Rn centered at yI . Here, for a multi-index α = (α1, . . . , αn),
kα(x, yI) =
Dαk(x, y)
ωp(δ) controls the behavior of T on atoms. More precisely, if a is an H
p atom
with defining interval I, and 0 < δ < 1, observe that
T (a)(x) =
[k(x, y)−
|α|≤N
(y − yI)
αkα(x, yI)] a(y) dy ,
and, consequently,
Rn\(2/δ)I
|T (a)(x)|p dx ≤ ωp(δ) .
We are now ready to prove the Hp,q − Lp,∞ estimate for a Calderón-
Zygmund singular integral operator T with kernel k(x, y).
Theorem 2.2. Let 0 < p ≤ 1, and p < q ≤ ∞. Assume that a Calderón-
Zygmund singular integral operator T is of weak-type (r, r) for some 1 < r <
∞, and that the modulus of continuity ωp of the kernel k satisfies a Dini
condition of order q/(q − p), namely,
Ap,q =
ωp(δ)
q/(q−p)dδ
](q−p)/q
Then T maps Hp,q continuously into Lp,∞, and ‖Tf‖p,∞ ≤ cA
p,q ‖f‖Hp,q .
Proof. We need to show that
2k0pm(Tf, 2k0) ≤ c ‖f‖
Hp,q , all k0 .
Let f =
j λj,kaj,k , be the atomic decomposition of f given in Theorem
2.1, and set f1 =
j λj,kaj,k, and f2 = f − f1. Further, let µk =
j |λj,k|
, and recall that ‖{µk}‖ℓq ∼ ‖f‖Hp,q .
Since ‖f1‖
r ≤ c 2
k0(r−p)‖f‖
Hp,∞, we have
2pk0m(Tf1, 2
k0) ≤ c ‖f‖
Hp,∞ .
Next, put I∗j,k = 2
1/n(3/2)p(k−k0)/nIj,k, and let
I∗j,k .
Since |I∗j,k| = 2(3/2)
p(k−k0)|Ij,k| ∼ 2
−k0p(3/4)p(k−k0)|λj,k|
p, we get
|Ω| ≤
|I∗j,k| ≤ c 2
(3/4)p(k−k0)
|λj,k|
≤ c 2−k0p
≤ c 2−k0p‖f‖
Hp,∞ .
Also, since 0 < p ≤ 1, it readily follows that
|T (f2)(x)|
|λj,k|
p|T (aj,k)(x)|
and, by Tonelli and the estimate for T (a), we have
|T (f2)(x)|
p dx ≤
|λj,k|
Rn\I∗
|T (aj,k)(x)|
)p(k−k0)/n
)pk/n)q/(q−p))(q−p)/q
ωp(δ)
q/(q−p)dδ
](q−p)/q
Hp,q .
This bound gives at once
2pk0 |{x /∈ Ω : |T (f2)(x)| > 2
k0}| ≤ cAp,q ‖f‖
Hp,q ,
which implies that
2pk0m(Tf2, 2
k0−1) ≤ 2pk0
|Ω|+ |{x /∈ Ω : |T (f2)(x)| > 2
k0−1}|
≤ c ‖f‖
Hp,∞ + cAp,q ‖f‖
Hp,q .
Finally,
2k0pm(Tf, 2k0) ≤ 2k0pm(Tf1, 2
k0−1) + 2k0pm(Tf2, 2
k0−1)
≤ c ‖f‖
Hp,∞ + cAp,q ‖f‖
Hp,q ,
and, since ‖f‖Hp,∞ ≤ c ‖f‖Hp,q for all q, we have finished. �
We pass now to the converse of Theorem 2.1. It is apparent that a
condition that relates the coefficients λj with the corresponding atoms aj
involved in an atomic decomposition of the form
j λjaj(x) is relevant here.
More precisely, if Ij denotes the supporting interval of aj , let
Ik = {j : 2
k ≤ |λj|/|Ij|
1/p < 2k+1} ,
and, for λ = {λj}, put
‖λ‖[p,q] =
]q/p)1/q
We then have,
Theorem 2.3. Let 0 < p ≤ 1, 0 < q ≤ ∞, and let f be a distribution given
by f =
j λj aj(x) , where the aj’s are H
p atoms, and the convergence is in
the sense of distributions. Further, assume that the family {Ij} consisting of
the supports of the aj’s has bounded overlap at each level Ik uniformly in k,
and ‖λ‖[p,q] <∞. Then, f ∈ H
p,q, and ‖f‖Hp,q ≤ c ‖λ‖[p,q].
Proof. Let Mf(x) = supt>0 |(f ∗ ψt)(x)| denote the radial maximal function
of f with respect to a suitable smooth function ψ with support contained in
{|x| ≤ 1} and nonvanishing integral. We will verify that Mf satisfies the
conditions of Lemma 1.1 and is thus in Lp,q.
Fix an integer k0 and let
g(x) =
λjaj(x) .
Since ‖Mg‖∞ ≤ ‖g‖∞ it suffices to estimate |g(x)|. Let C be the bounded
overlap constant for the family of the supports of the aj’s. Then, for j ∈ Ik,
|λj| |aj(x)| =
|Ij|1/p
|λj | |Ij|
1/p |aj(x)| ≤ 2
kχIj (x) ,
and, consequently,
|g(x)| ≤
χIj(x) ≤ C 2
Next, let
h(x) =
λjaj(x) .
Since aj has N = [n(1/p − 1)] vanishing moments, it is not hard to see
that, if Ij is the defining interval of aj and Ij is centered at xj , and γ =
(n+N+1)/n > 1/p, then, with c independent of j, ϕj(x) =Maj(x) satisfies
ϕj(x) ≤ c
|Ij |
γ−1/p
(|Ij|+ |x− xj |n)γ
Thus, if 1/γ < εp < 1,
Mh(x)εp ≤ c
j∈Ik,k≥k0
(|λj| |Ij|
γ−1/p)εp
(|Ij |+ |x− xj |n)γεp
which, upon integration, yields
Mh(x)εp dx ≤ c
j∈Ik,k≥k0
(|λj| |Ij|
γ−1/p)εp
(|Ij |+ |x− xj |n)γεp
The integrals in the right-hand side above are of order |Ij|
1−γεp and, conse-
quently, by Chebychev’s inequality,
2k0εp|{Mh > 2k0}| ≤ c
j∈Ik,k≥k0
εp |Ij|
1−ε ≤ c
|Ij| .
Thus, Lemma 1.1 applies with ϕ = Mf , ψk0 = Mg, ηk0 = Mh, and µk =
, and we get
2km(Mf, 2k)1/p
)1/p}∥
which, since
|Ij| ∼
, j ∈ Ik ,
is bounded by c ‖λ‖[p,q], 0 < q ≤ ∞. �
The next result is of interest because it applies to arbitrary decomposi-
tions in Hp,q. The proof relies on Lemma 1.2, and is left to the reader.
Theorem 2.4. Let 0 < p ≤ 1, 0 < q ≤ ∞, and let f be a distribution given
by f =
j λj aj(x) , where the aj’s are H
p atoms, and the convergence is
in the sense of distributions. Further, assume that ‖λ‖[η,q] < ∞ for some
0 < η < min(p, q). Then, f ∈ Hp,q, and ‖f‖Hp,q ≤ c ‖λ‖[η,q].
2.1 Interpolation between Hardy-Lorentz spaces
We are now ready to identify the intermediate spaces of a couple of Hardy-
Lorentz spaces with the same first index p ≤ 1.
Theorem 2.5. Let 0 < p ≤ 1. Given 0 < q1 < q < q2 ≤ ∞, define 0 < η < 1
by the relation 1/q = (1− η)/q1 + η/q2. Then, with equivalent quasinorms,
Hp,q = (Hp,q1, Hp,q2)η,q .
Proof. Since the non-tangential maximal function Nf of a distribution f in
Hp,q1 is in Lp,q1, and that of f in Hp,q2 is in Lp,q2, we have
K(t, Nf ;Lp,q1, Lp,q2) ≤ cK(t, f ;Hp,q1, Hp,q2) .
Thus,
‖Nf‖p,q ∼ ‖Nf‖(Lp,q1 ,Lp,q2)η,q ≤ c ‖f‖(Hp,q1 ,Hp,q2 )η,q ,
and (Hp,q1, Hp,q2)η,q →֒ H
To show the other embedding, with the notation in the proof of Theorem
2.1, write f =
j λj,kaj,k , and recall that for every integer k, the level
set Ik = {j : |λj,k|/|Ij,k|
1/p ∼ 2k} contains exclusively the sequence {λj,k}.
Let µ
|λj,k|
p. By construction,
k ∼ ‖f‖
Hp,q . Now, rearrange
{µk} into {µ
l }, and, for each l ≥ 1, let kl be such that µkl = µ
l . For
l0 ≥ 1, let Kl0 = {k1, . . . , kl0}, and put f1,l0 =
k∈Kl0
j λj,kaj,k and f2,l0 =
f − f1,l0 . Then, by Theorem 2.2, f1,l0 ∈ H
p,q1, f2,l0 ∈ H
p,q2, and, with the
usual interpretation for q2 = ∞,
‖f1,l0‖Hp,q1 ≤ c
)1/q1
, ‖f2,l0‖Hp,q2 ≤ c
)1/q2
So, for t > 0 and every positive integer l0, we have
K(t, f ;Hp,q1, Hp,q2) ≤ c
)1/q1
)1/q2
Now, by Homstedt’s formula, there is a choice of l0 such that the right-hand
side above ∼ K(t, {µk}; ℓ
q1, ℓq2), and, consequently,
K(t, f ;Hp,q1, Hp,q2) ≤ cK(t, {µk}; ℓ
q1, ℓq2) .
Thus,
‖f‖(Hp,q1 ,Hp,q2 )η,q ≤ c ‖{µk}‖(ℓq1 ,ℓq2)η,q
≤ c ‖{µk}‖ℓq ≤ c ‖f‖Hp,q ,
and Hp,q →֒ (Hp,q1, Hp,q2)η,q. �
The reader will have no difficulty in verifying that Theorem 2.5 gives that
if T is a continuous, sublinear map fromH1 into L1, and fromH1,∞ into L1,∞,
then ‖Tf‖1,q ≤ c ‖f‖H1,q for 1 < q < ∞. This observation has numerous
applications. For instance, consider the Calderón-Zygmund singular integral
operators with variable kernel defined by
TΩ(f)(x) = p.v.
Ω(x, x−y)
|x− y|n
f(y) dy .
Under appropriate growth and smoothness assumptions on Ω, TΩ maps H
continuously into L1, see [6], and H1,∞ continuously into L1,∞, see [8]. Thus,
if Ω satisfies the assumptions of both of these results, TΩ maps H
1,q contin-
uously into L1,q for 1 < q < ∞. A similar result follows by invoking the
characterization of H1,q given by C. Fefferman, Rivière and Sagher. How-
ever, in this case the Hp−Lp estimate requires additional smoothness of Ω, as
shown, for instance, in [6]. Similar considerations apply to the Marcinkiewicz
integral, see [9], and [7].
Finally, when p < 1, our results cover, for instance, the δ-CZ operators
satisfying T ∗(1) = 0 discussed by Alvarez and Milman, see [3]. These oper-
ators, as well as a more general related class introduced in [15], preserve Hp
and Hp,∞ for n/(n + δ) < p ≤ 1, and, consequently, by Theorem 2.5, they
also preserve Hp,q for p in that same range, and q > p.
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DEPARTMENT OF MATHEMATICS, INDIANA UNIVERSITY,
BLOOMINGTON, IN 47405
E-mail: wabusham@indiana.edu, torchins@indiana.edu
	The Lorentz spaces
	The Hardy-Lorentz spaces Hp,q
	Interpolation between Hardy-Lorentz spaces
