GEORGIAN MATHEMATICAL JOURNAL: Vol. 3, No. 1, 1996, 69-80
LITTLEWOOD–PALEY OPERATORS ON THE
GENERALIZED LIPSCHITZ SPACES
SHANZHEN LU, CHANGMEI TAN, AND KOZO YABUTA
Abstract. Littlewood–Paley operators defined on a new kind of generalized Lipschitz spaces E0α,p are studied. It is proved that the image of a function under the action of these operators is either equal
to infinity almost everywhere or is in E0α,p , where −n < α < 1 and
1 < p < ∞.
1. Introduction
For x ∈ Rn , y > 0, the Poisson kernel is P (x, y) = cn y(y 2 + |x|2 )−(n+1)/2 .
Denote the Poisson integral of f by
Z
f (x, y) = f (z)P (x − z, y) dz.
Rn
We have (see [1])
|∇f (x, y)| ≤ cn
Z
Rn
|f (z)| (y + |x − z|)−(n+1) dz.
(1)
Let us now consider the following two kinds of Littlewood–Paley functions:
 ZZ
‘1/2
S(f )(x) =
y 1−n |∇f (z, y)|2 dz dy
Γ(x)
and
gλ∗ (f )(x) =
n ZZ 
Rn+1
+
‘λn
o1/2
y
.
y 1−n |∇f (z, y)|2 dz dy
y + |x − z|
1991 Mathematics Subject Classification. 42B25.
Key words and phrases. Littlewood–Paley functions, generalized Lipschitz spaces,
Poisson integral, Poisson kernel.
69
c 1996 Plenum Publishing Corporation
1072-947X/96/0100-0069$09.50/0 
70
SHANZHEN LU, CHANGMEI TAN, AND KOZO YABUTA
The generalized Lipschitz space E α,p consists of functions f which are locally
integrable and satisfy the following condition: there exists a constant C such
that for any cube Q
Z
αp
(2)
|f (x) − fQ |p dx ≤ C|Q|1+ n ,
Q
where fQ =
1
Q
R
Q
f (x)dx. Denote the norm of f in E α,p by
ˆ
‰
kf kα,p = inf C 1/p : C satisfies (2) .
Recently, Qiu [2] has obtained the following result.
Theorem A. Let 1 < p < ∞, −n/p ≤ α < 1/2, α =
6 0, and λ >
max(1, 2/p). If f ∈ E α,p and T f is S(f ) or gλ∗ (f ), then either T f (x) = ∞
a.e. or T f (x) < ∞ a.e., and there exists a constant C independent of f
such that
kT f kα,p ≤ Ckf kα,p .
We notice that the range of α in Theorem A seems somewhat rough. It
is natural to consider whether the conclusion of the above theorem holds
for −n < α < 1. The last named author of this paper proved that the
conclusion of Theorem A holds for −n/p < α < 1 (see [3]). In this paper,
with the aid of the idea in [4], we shall introduce a variant of E α,p , E0α,p , and
prove that the conclusion of Theorem A holds for E0α,p with −n < α < 1.
Let us first definie E0α,p .
Definition. A locally integrable function f is called a generalized Lipschitz function of central type if there exists a constant C such that (2) holds
for any cube Q centered at the origin. Moreover, the space consisting of all
generalized Lipschitz functions of central type is denoted by E0α,p . We call
E0α,p the generalized Lipschitz space of central type.
It is easy to see that E α,p ⊂ E0α,p and E0α,p is just the bounded mean
oscillation space of central type, BM O0 in [4]. Let us now formulate our
results.
Theorem 1. Let 1 < p < ∞ and −n < α < 1. If f ∈ E0α,p , then either
S(f )(x) = ∞ a.e. or S(f )(x) < ∞ a.e., and there exists a constant C
independent of f such that
kS(f )kα,p ≤ Ckf kα,p .
LITTLEWOOD–PALEY OPERATORS
71
Theorem 2. Let 1 < p < ∞, −n < α < 1, and λ > max(1, 2/p) + 2/n.
If f ∈ E0α,p , then either gλ∗ (f )(x) = ∞ a.e. or gλ∗ (f )(x) < ∞ a.e., and there
exists a constant C = C(n, α, p, λ) such that
kgλ∗ (f )kα,p ≤ Ckf kα,p .
2. Some Lemmas
Lemma 1. Let 1 < p < ∞, −n < α < 1, α 6= 0, and 0 < d, α < d. If
f ∈ E0α,p and Q is a cube centered at the origin with the edge length r, then
there exists a constant C = C(n, p, α, d) such that for any y > 0
Z
|f (x) − fQ |
dx ≤ Cy −d (y α + rα )kf kα,p .
y n+d + |x|n+d
Rn
(3)
See [1] and [2] for its proof.
Lemma 10 . Let 1 < p < ∞ and d > 0. If f ∈ BM O0 = E00,p and Q
is a cube centered at the origin with the edge length r, then there exists a
constant C = C(n, p, d) such that for any y > 0,
Z
Rn
Œ

|f (x) − fQ |
Œ
−d
Cy
1
+
dx
≤
Πlog2
y n+d + |x|n+d
y ŒŒ‘
Πkf k0,p .
r
Proof. By the known result in [5] we have
Z
Rn
|f (x) − fQ |
dx ≤ Cr−d kf k0,p .
+ |x|n+d
rn+d
Let R be the cube centered at the origin with the edge length y. Then
Z
Rn
+|fR − fQ |
Z
Rn
|f (x) − fQ |
dx ≤
y n+d + |x|n+d
Z
Rn
|f (x) − fR |
dx
y n+d + |x|n+d
dx
dx ≤ Cy −d kf k0,p + Cy −d |fR − fQ |.
y n+d + |x|n+d
Thus it remains to prove
Œ

y ŒŒ‘
Œ
|fR − fQ | ≤ C 1 + Œ log2 Œ kf k0,p .
r
72
SHANZHEN LU, CHANGMEI TAN, AND KOZO YABUTA
Let y > r, and let k satisfy 2k ≤ y < 2k+1 r. Then k ≤ log2
|fR − fQ | ≤ |fR − fQk | +
≤ 2n
k
X
j=1
y
r
and
|fQj − fQj−1 |
k
‘1/p X
 1 Z
+
2n kf k0,p
|f (x) − fR |p dx
|R|
j=1
R
≤ 2n (1 + k)kf k0,p

y‘
≤ C 1 + log2
kf k0,p ,
r
where Qk is the concentric extension of Q by 2k times.
When y < r, by exchanging y and r, we shall get the same estimate as
above with log2 yr = | log2 yr |.
Let χE be the characteristic function of E. For a cube Q in Rn and d > 0
let dQ be the concentric extension of Q by d times.
Lemma 2. Suppose that 1 < p < ∞, −n < α < 1, and f ∈ E0α,p . Let
Q be a cube centered at the origin with the edge length r, and hQ (x) =
]χQc (x). If there is x0 ∈ dQ such that S(hQ )(x0 ) < ∞, where
[f (x) −
√fQ−1
d = (8 n) , then there exists a constant C = C(n, α, p) such that
S(hQ )(x) < ∞,
∀x ∈ dQ
and
|S(hQ )(x) − S(hQ )(x0 )| < Crα kf kα,p ,
∀x ∈ dQ.
Proof. Let us first consider the case of α 6= 0. Fix x ∈ dQ. Set
ˆ
‰
Γ− (x) = (z, y) ∈ Γ(x) : y ≤ dr
and
Then
where
ˆ
‰
Γ+ (x) = (z, y) ∈ Γ(x) : y > dr .
S(hQ )(x) ≤ S − + S + ,
S− =
x ∈ dQ,
 ZZ
y 1−n |∇hQ (z, y)|2 dz dy
 ZZ
y 1−n |∇hQ (z, y)|2 dz dy
Γ− (x)
and
S+ =
Γ+ (x)
‘1/2
‘1/2
.
LITTLEWOOD–PALEY OPERATORS
73
Estimating S − as in [2], we have
S − ≤ Crα kf kα,p .
(4)
For S + we have
 ZZ
‘1/2
+
S =
y 1−n |∇hQ (x + z, y)|2 dz dy
Γ+ (0)
≤
 ZZ
y 1−n |∇hQ (x0 + z, y)|2 dz dy
Γ+ (0)
+
 ZZ
y 1−n |∇hQ (x + z, y) − ∇hQ (x0 + z, y)|2 dz dy
Γ+ (0)
0
≤ S(hQ )(x ) +
×
Z
Qc
‘1/2
n ZZ
‘1/2
y 1−n
Γ+ (0)
|∇P (x + z − t, y)∇P (x0 + z + t, y)| |f (t) − fQ |dt
‘2
dz dy
o1/2
.
Note that
|∇P (x, y) − ∇P (x0 , y)| =
where
where
Thus
∂
∂xn+1
=
∂
∂y .
 n+1
‘1/2
X ŒŒ ∂
∂
p(x, y) −
P (x0 , y)
,
Œ
∂xj
∂xj
j=1
By the mean value theorem we have
Œ
Œ
∂
Œ ∂
Œ
p(x, y) −
P (x0 , y)Œ
Œ
∂xj
∂xj
Œ
Œ ∂
Œ
Œ
P (x, y)Œ
= Œ∇
|x − x0 |, 0 < θj < 1,
∂xj
x+θj (x−x0 )
Œ ∂
Œ
C
Œ
Œ
P (x, y)Œ ≤
.
Œ∇
∂xj
(y + |x|)n+2
|∇P (x + z − t, y) − ∇P (x0 + z − t, y)|
n n+1
X€
−2(n+2) o1/2
y + |x + z − t + θj (x − x0 )|
≤ C|x − x0 |
.
j=1
(5)
74
SHANZHEN LU, CHANGMEI TAN, AND KOZO YABUTA
Since (x, y) ∈ Γ+ (0), x, x0 ∈ dQ, and t 6∈ Q, we have |t| > r/2, |z| < y,
|x| < r/16 < |t|/8, and |x − x0 | < r/8 < |t|/4. Thus,
|t| ≤ |x + z − t + θj (x − x0 )| + |x| + |z| + |x − x0 |
≤ |x + z − t + θj (x − x0 )| + |t|/8 + y + |t|/4
and

5 €
y + |t| ≤ |x + z − t + θj (x − x0 )| + y,
16
where 1 ≤ j ≤ n| + 1. Therefore
|∇P (x + z − t, y) − ∇P (x0 + z − t, y)| ≤
Cr
.
(y + |t|)n+2
(6)
Using (6) and (3), we obtain
n ZZ
h Z r|f (t) − f | i2
o1/2
Q
y 1−n
dzdy
S + ≤ S(hQ )(x0 ) + C
dt
(y + |t|)n+2
Qc
Γ+ (0)
≤ S(hQ )(x0 ) + C
n ZZ
Γ+ (0)
o1/2
‚
ƒ2
y 1−n r2 y −2 (y α + rα )kf kα,p dzdy
0
≤ S(hQ )(x ) + Crkf kα,p
0
n Z∞ Z
y 1−n y −4 (y 2α + r2α )dzdy
dr |z|<y
α
o1/2
≤ S(hQ )(x ) + Cr kf kα,p .
(7)
Combining (4) with (7) we have
S(hQ )(x) ≤ S(hQ )(x0 ) + Crα kf kα,p .
Thus S(hQ )(x) < ∞. Exchanging x and x0 , we obtain
|S(hQ )(x) − S(hQ )(x0 )| ≤ Crα kf kα,p .
Hence the proof of Lemma 2 is complete for the case of α 6= 0.
When α = 0, by using Lemma 10 instead of Lemma 1 we obtain
o1/2
h Z r|f (t) − f | i2
n ZZ
Q
dt dzdy
S + ≤ S(hQ )(x0 ) + C
y 1−n
n+2
(y + |t|)
Qc
Γ+ (0)
≤ S(hQ )(x0 ) + C
n ZZ
Γ+ (0)
0
o1/2
Œ
‚
€
ƒ2
y Œ
y 1−n r2 y −2 1 + Œ log2 Œ kf k0,p dzdy
r
≤ S(hQ )(x ) + Crkf k0,p
n Z∞ Z
dr |z|<y
o1/2
Œ
€
y Œ 2
y −3−n 1 + Œ log2 Œ dzdy
r
LITTLEWOOD–PALEY OPERATORS
0
α
≤ S(hQ )(x ) + Cr kf k0,p
n Z∞
≤ S(hQ )(x0 ) + Crα kf k0,p .
1
€
2 o1/2
u−3 1 + | log2 u| du
75
(8)
Now, it is easy to see that the conclusion of the lemma for α = 0 follows
from (8) and (4) with α = 0.
Lemma 3. Under the hypothesis of Lemma 2, if there is x0 ∈ dQ such
that gλ∗ (hQ )(x0 ) < ∞, where λ > max(1, 2/p) + 2/n, then there exists a
constant C = C(n, α, λ, p) such that gλ∗ (hQ )(x) < ∞ and
|gλ∗ (hQ )(x) − gλ∗ (x0 ) ≤ Crα kf kα,p ,
∀x ∈ dQ.
Proof. We only consider the case of α =
6 0. As in Lemma 2, the proof in
the case α = 0 is similar. Let
‰
ˆ
Jk = (z, y) ∈ Rn+1
: |z| < 2k−2 r, 0 < y < 2k−2 r , k ≥ 0.
+
For fixed x ∈ dQ we have
where
−
G =
 ZZ 
J0
and
+
G =
 ZZ
\J0
Rn+1
+
gλ∗ (hQ )(x) ≤ G− + G+ ,
‘1/2
y ‘λn 1−n
|∇hQ (x + z, y)|2 dz dy
y
y + |z|

‘1/2
y ‘λn 1−n
.
y
|∇hQ (x + z, y)|2 dz dy
y + |z|
Note that if (z, y) ∈ J0 , x ∈ dQ, and t 6= Q, then |z| < r/4, |x| < r/16, and
|t| > r/2. Thus
5
|t| ≤ |t − x − z| + |x| + |z| ≤ |x + z − t| + |t|
8
and
1
(r + |t|) ≤ |x + z − t| + y.
16
By (1) and Lemma 1 we get
n Z Z  y ‘λn
o1/2
hZ
i2
r|f (t) − fQ |
G− ≤ C
dt dzdy
y 1−n
n+1
y + |z|
(y + |x + z − t|)
Qc
J0
≤C
n ZZ 
J0
y
y + |z|
‘λn
o1/2
h Z r|f (t) − f | o2
Q
dzdy
y 1−n
dt
(r + |t|)n+1
Qc
76
SHANZHEN LU, CHANGMEI TAN, AND KOZO YABUTA
≤C
n Z∞ Z
dr |z|<y
≤ Cr
α−1

o1/2
ƒ2
y ‘λn 1−n ‚ α−1
kf kα,p dzdy
y
r
y + |z|
kf kα,p
≤ Crα kf kα,p .
 Zr
y 1−n rn dy
0
‘1/2
To estimate C + we observe that
n Z Z  y ‘λn
o1/2
G+ ≤
y 1−n |∇hQ (x0 + z, y)|2 dzdy
y + |z|
Rn+1
\J0
+
+
n ZZ
n+1
R+
\J0

y ‘λn 1−n
y
|∇hQ (x + z, y) −
y + |z|
− ∇hQ (x0 + z, y)|2 dzdy
≤ gλ∗ (hQ )(x0 ) + D,
where
D=
n ZZ
\J0
Rn+1
+

o1/2
y ‘λn 1−n
|∇hQ (x + z, y) −
y
y + |z|
o1/2
− ∇hQ (x0 + z, y)|2 dzdy
ZZ
∞
nX
y λn+1−n
≤C
(2k r)−λn
k=1
×
hZ
Qc
≤C
Here
Ak =
i2
o1/2
|∇P (x + z − t, y) − ∇P (x0 + z − t, y)| |f (t) − fQ |dt dzdy
∞
nX
k=1
ZZ
Jk \Jk−1
×
h Z
Jk \Jk−1
c
Qk+1
o1/2
(2k r)−λn (Ak + Bk )
.
y λn+1−n
i2
|∇P (x + z − t, y) − ∇P (x0 + z − t, y)| |f (t) − fQ |dt dzdy,
LITTLEWOOD–PALEY OPERATORS
ZZ
Bk =
y λn+1−n
Jk \Jk−1
×
h
77
Z
Qk+1 \Q
i2
|∇P (x + z − t, y) − ∇P (x0 + z − t, y)| |f (t) − fQ |dt dzdy,
and Qk+1 = 2k+1 Q. Without loss of generality we may assume that
max(1, 2/p) + 2/n < λ < 3 + 2/n. By the easy inequality (see [3])
|∇P (x, y) − ∇P (x0 , y)|

‘
1
1
≤ C|x − x0 |
+
, ∀x, x0 ∈ Rn , y > 0,
n+2
n+2
0
(y + |x|)
(y + |x |)
together with the Minkowski inequality for integrals, we have
Bk ≤ Cr
2
ZZ
y
λn+1−n
Jk \Jk−1
n
Z
Qk+1 \Q

|f (t) − fQ |
‘ o2
1
+
dt dz dy
(y + |x0 + z − t|)n+2
Z∞ Z
n Z

2
≤ Cr
y λn+1−n
|f (t) − fQ |
0 Rn
Qk+1
1
(y + |x + z − t|)n+2
1
(y + |x + z − t|)n+2
‘ o2
1
+
dt dz dy
(y + |x0 + z − t|)n+2
Zh Z
‘1/2 i2
 Z∞
y λn+1−n
2
dy
dt dz
|f (t)−fQ |
≤ Cr
(y+|x + z − t|)2(n+2)
Rn Qk+1
|f (t)−fQ |
Zh Z
Rn Qk+1

|f (t)−fQ |
|z +x−t|(3n−λn+2)/2
Zh Z

|f (t) − fQ |
|z+x0 −t|(3n−λn+2)/2
+ Cr2
Rn Qk+1
= Cr
+Cr
2
2
0
Zh Z
Rn Qk+1
= Cr2
Z  Z
Rn
Qk+1
 Z∞
0
(y+|x0
‘1/2 i2
y λn+1−n
dy
dt dz
2(n+2)
+ z − t|)
Z∞
0
‘1/2 i2
y λn+1−n
dy
dt dz
(1+y)2(n+2)
Z∞
0
‘1/2 i2
y λn+1−n
dy
dt dz
(1+y)2(n+2)
‘2
|f (t) − fQ |
du.
dt
|u − t|n−[(λn−1)−2]/2
78
SHANZHEN LU, CHANGMEI TAN, AND KOZO YABUTA
Using the Hardy–Littlewood–Sobolev theorem on fractional integration with
γ = [(λ − 1)n − 2]/2, q = 2, and 1/s = 1/q + γ/n = λ/2 − 1/n (see [6]), we
obtain
 Z
‘2/s
2
Bk ≤ Cr
|f (z) − fQ |s dz
.
Qk+1
Since λ ≥ 2/p + 2/n, then p ≥ s. Thus
Bk ≤ Cr
2
 Z
|f (z) − fQ |p dz
Qk+1
≤ Cr2
n Z
‘2/p
|f (z) − fQ |p dz
Qk+1
|Qk+1 |2(1/s−1/p)
‘1/p
o2
+ |Qk+1 |1/p |fQk+1 − fQ | |Qk+1 |2(1/s−1/p)
ˆ
≤ Cr2 |Qk+1 |1/p+α/n kf kα,p
‰2
+ |Qk+1 |1/p (2k r)α kf kα,p |Qk+1 |2(1/s+1/p)
≤ Cr2 (2k r)2α (2k r)λn−2 kf kα,p
≤ C(2k r)λn (22k(α−1) r2α kf kα,p .
To estimate Ak we observe that if (z, y) ∈ Jk \Jk−1 and t 6∈ Qk+1 , then
|t| > 2k r, k ≥ 1, and |z| < 2k−2 r < |t|/4. Thus,
|t| ≤ |x + z − t + θj (x − x0 )| + |x| + |z| + |x − x0 |
5
≤ |x + z − t + θ(x − x0 )| + |t|.
16
By Lemma 1 we have
Ak ≤ C
ZZ
y λn+1−n
Qck+1
Jk \Jk+1
≤ Cr2
h Z
ZZ
Jk \Jk+1
i2
r|f (t) − fQ |
dt
dz dy
(2k r + |t|)n+2
ˆ
‰2
y λn+1−n (2k r)−2 [(2k r)α + rα ]kf kα,p dz dy
k
2
k
≤ Cr (2 r)
2α
k
−4+2α
≤ Cr (2 r)
kf kα,p
λn 2k(α−1)
2
Z2
r
Z
0 |z|<2k r
kf kα,p .
y λn+1−n dz dy
LITTLEWOOD–PALEY OPERATORS
79
Combining the estimate of Ak with that of Bk , we obtain
D≤C
∞
nX
(2k r)−λn (2k r)λn r2α 22k(α−1) kf kα,p
k=1
o1/2
≤ Crα kf kα,p .
Therefore
gλ∗ (hQ )(x) ≤ gλ∗ (hQ )(x0 ) + Crα kf kα,p .
As in the last part of the proof of Lemma 2, we have
(
|gλ hQ )(x) − gλ∗ (hQ )(x0 ) ≤ Crα kf kα,p .
3. The Proofs of the Theorems
Let T be one of the Littlewood–Paley functions as in Section 1. Suppose
∆
that T f (x) 6= ∞ a.e. Then |E| = |{x : T f (x) < ∞}| > 0. Thus there is
a cube Q0 centered at the origin such that |Q0 ∩ E| > 0. Set Q = (1/d)Q0
(then Q0 = dQ). We write f as
f (x) = fQ + [f (x) − fQ ]χQ (x) + [f (x) − fQ ]χQc (x)
∆
= fQ + gQ (x) + hQ (x).
Since
T f (x) ≤ T gQ (x) + T hQ (x)
(9)
T hQ (x) ≤ T f (x) + T gQ (x),
(10)
and
it is easy to see that the inequality
‘1/p
Z
kgQ kp =
|f (t) − fQ |p dt
≤ C|Q|1/p+α/n kf kα,p
(11)
Q
p
implies that gQ ∈ L . Then it follows from the Lp -boundedness of the
Littlewood–Paley operator that T gQ (x) < ∞ a.e.. Since |Q0 ∩ E| > 0, there
is x0 ∈ Q0 ∩ E ⊂ dQ such that T f (x0 ) < ∞ and T gQ (x0 ) < ∞. By (10) and
Lemmas 2 and 3, we have T hQ (x0 ) < ∞ and
T hQ (x) < ∞,
∀x ∈ dQ = Q0 .
By (9) we obtain
T f (x) < ∞ a.e. x ∈ Q0 .
Finally, let the edge length of Q0 tend to ∞; we have T f (x) < ∞ a.e.,
x ∈ Rn .
80
SHANZHEN LU, CHANGMEI TAN, AND KOZO YABUTA
Let Q0 be a cube centered at the origin, and Q = (1/d)Q0 . Choose
x ∈ dQ so that T hQ (x0 ) < ∞. Then it follows from (11) and Lemmas 2
and 3 that
Z
‘1/p
|T f (x) − T hQ (x0 )|p dx
0
Q0
≤
Z
Q0
‘1/p  Z
‘1/p
|T gQ (x)|p dx
+
|T hQ (x) − T hQ (x0 )|p dx
Q0
≤ CkgQ kp + C|Q0 |1/p rα kf kα,p
≤ C|Q0 |1/p+α/n kf kα,p .
This completes the proof of the theorems.
Acknowledgement
The work is supported by the NSFC (Tian Yuan). Shanzhen Lu would
like to thank Kozo Yabuta for inviting him to visit Nara during his sabbatiacal in Japan.
References
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Soc. 99(1987), 657–666.
2. S. G. Qiu, Boundedness of Littlewood–Paley operators and Marcinkiewicz integral on E α,p . J. Math. Res. Exposition 12(1992), 41–50.
3. K. Yabuta, Boundedness of Littlewood–Paley operators. Preprint.
4. S. Z. Lu and D. C. Yang, The Littlewood–Paley function and
ϕ-transform characterizations of a new Hardy space HK2 associated with
the Herz space. Studia Math. 101(1992), 285–298.
5. F. B. Fabes, R. L. Janson, and U. Neri, Spaces of harmonic functions
representable by Poisson integrals of functions in BMO and Lp,λ . Indiana
Univ. Math. J. 25(1976), 159–170.
6. E. M. Stein, Singular integrals and differentiability properties of functions. Princeton University Press, Princeton, 1970.
(Received 04.05.1994)
Authors’ addresses:
Shanzhen Lu and Changmei Tan
Department of Mathematics
Beijing Normal University
Beijing 100875, P. R. China
Kozo Yabuta
Department of Mathematics
Nara Women’s University
Nara 630, Japan