MATEMATIQKI VESNIK
originalni nauqni rad
research paper
66, 1 (2014), 19–32
March 2014
GENERALIZED HAUSDORFF OPERATORS
ON WEIGHTED HERZ SPACES
Kuang Jichang
Abstract. In this paper, we introduce new generalized Hausdorff operators. They include
many famous operators as special cases. We obtain necessary and sufficient conditions for these
operators to be bounded on the weighted Herz spaces. The corresponding new operator norm
inequalities are obtained. They are significant improvements and generalizations of many known
results. Several open problems are formulated.
1. Introduction
The classical Hardy operator T0 is defined by
Z
1 x
f (t) dt, x > 0,
T0 (f, x) =
x 0
and the classical Hardy inequality stated in Hardy et al. [4] is
p
kT0 f kp ≤
kf kp , 1 < p < ∞,
p−1
where the best constant factor
operator T0 , that is,
p
p−1
(1.1)
is the best value and it is the norm of the
kT0 k =
p
.
(p − 1)
As pointed out by Kufner et al. [6], the Hardy inequality has a fascinating past
and will have (hopefully) also a fascinating future. These authors of [6] present some
important steps of the development of (1.1), of its early weighted generalizations
and of its various modifications and extensions. Another classical operator is the
Hausdorff operator
Z ∞
ψ(t) ³ x ´
T1 (f, x) =
f
dt, x > 0,
(1.2)
t
t
0
2010 AMS Subject Classification: 26D10, 47A30
Keywords and phrases: Hausdorff operator; weighted Herz space; norm inequality.
19
20
K. Jichang
where ψ is a local integrable function in (0, ∞). This operator and its varieties
have attracted many authors, for example, see [7–10]. With u = xt , (1.2) yields
Z ∞
ψ( ux )
f (u) du, x > 0.
T1 (f, x) =
u
0
Choosing ψ(u) = u−1 ϕE1 (u) and ψ(u) = ϕE2 (u), where E1 = (1, ∞), E2 = (0, 1],
and ϕE denotes the characteristic function of the set E, we obtain the Hardy
operator T0 and the dual Hardy operator (or Cesáro operator) T0∗ defined by
Z ∞
f (u)
∗
T0 (f, x) =
du, x > 0,
u
x
respectively. In addition, T2 = T0 +T0∗ becomes the Calderòn maximal operator [1]:
Z
Z ∞
1 x
f (t)
T2 (f, x) =
f (t) dt +
dt, x > 0.
x 0
t
x
The aim of this paper is to introduce the following new generalized Hausdorff
operator
Z ∞
ψ(t)
T (f, x) =
f (g(t)x) dt, x = (x1 , x2 , . . . , xn ) ∈ Rn ,
(1.3)
t
0
where g(t)x = (g(t)x1 , g(t)x2 , . . . , g(t)xn ), ψ : (0, ∞) → (0, ∞) is a locally integrable function, g : (0, ∞) → (0, ∞) is a monotonic function (increasing or decreasing), and f is a measurable complex valued function on Rn . If g(t) = 1t and n = 1,
then T reduces to the Hausdorff operator T1 .
If we introduce other forms of g and ψ, it is possible to obtain other operators
of interest. For example, if g(t) = t, ψ(t) = tω(t)ϕE (t), where E = (0, 1], ω is
a non-negative weight function, then T reduces to the weighted Hardy-Littlewood
mean operator defined in [12]:
Z 1
T3 (f, x) =
f (tx)ω(t) dt, x = (x1 , x2 , . . . , xn ) ∈ Rn .
(1.4)
0
If g(t) = 1t , ψ(t) = t−(n−1) ω(t)ϕE (t), ω, E are as in (1.4), then T reduces to the
weighted Cesàro mean operator defined in [12]:
Z 1 ³ ´
x −n
T4 (f, x) =
f
t ω(t) dt, x = (x1 , x2 , . . . , xn ) ∈ Rn .
t
0
If g(t) = t, ψ(t) = te−λt , λ > 0, n = 1, x > 0, then T reduces to
Z ∞
Z
1 ∞
1
T5 (f, x) =
f (tx)e−λt dt =
f (u)e−αu du = L(f, α),
x 0
x
R∞ 0
where L(f, α) = 0 f (u)e−αu du is the Laplace transform of f , u = tx, α = λx > 0.
Hence, (1.3) is a significant generalization of many famous operators.
It is well-known that the Herz spaces play an important role in characterizing
the properties of functions and multipliers on the classical Hardy spaces.
In this paper, we obtain necessary and sufficient conditions for the generalized
Hausdorff operator T defined by (1.3) to be bounded on the weighted Herz spaces.
The corresponding new operator norm inequalities are obtained. Several open
problems are formulated.
21
Generalized Hausdorff operators
2. Definitions and statement of the main results
Let k ∈ Z, Bk = {x ∈ Rn : |x| ≤ 2k }, Dk = Bk −Bk−1 and let ϕk = ϕDk denote
the characteristic function of the set Dk . Moreover, for a measurable function f on
Rn and a non-negative weight function ω(x), we write
µZ
¶1/p
p
kf kp,ω =
|f (x)| ω(x) dx
.
Rn
In what follows, if ω ≡ 1, then we will denote Lp (Rn , ω) (in brief Lp (ω)) by Lp (Rn ).
Definition 2.1. (see [11]) Let α ∈ R1 , 0 < p, q < ∞ and ω1 and ω2 be nonnegative weight functions. The homogeneous weighted Herz space K̇qα,p (ω1 , ω2 ) is
defined by
K̇qα,p (ω1 , ω2 ) = {f ∈ Lqloc (Rn − {0}) : kf kK̇qα,p (ω1 ,ω2 ) < ∞},
where
kf kK̇qα,p (ω1 ,ω2 ) =
nP
[ω1 (Bk )]
k∈Z
αp
n
kf ϕk kpq,ω2
o1/p
.
We can similarly define the non-homogeneous weighted Herz spaces Kqα,p (ω1 , ω2 ).
It is easy to see that when ω1 = ω2 = 1, we have
K̇qα,p (1, 1) = K̇qα,p (Rn ),
K̇p(α/p),p (Rn ) = Lp (|x|α dx), K̇p0,p (Rn ) = Lp (Rn ).
Definition 2.2. (see [2]) A non-negative weight function ω satisfies Muckenhoupt’s A∞ condition or ω ∈ A∞ , if there is a constant C independent of the cube
Q in Rn , such that
µ
¶
½
µ
¶ ¾
Z
Z
1
1
1
ω(x) dx exp
log
dx ≤ C, ∀Q ⊂ Rn ,
|Q| Q
|Q| Q
ω(x)
where |Q| is the Lebesgue measure of Q.
Our main results are the following three theorems:
Theorem 2.1. Let α ∈ R1 , 0 < p < ∞, 1 ≤ q < ∞, ω1 ∈ A∞ , and a
non-negative weight function ω2 satisfy
ω2 (tx) = tβ ω2 (x), t > 0, β ∈ R1 , x ∈ Rn
(2.1)
Let ψ : (0, ∞) → (0, ∞) be a locally integrable function having the compact support on (0, ∞); let g : (0, ∞) → (0, ∞) be an increasing function satisfying the
submultiplicative condition
g(uv) ≤ g(u)g(v), u, v > 0.
Let kT k be the norm of the operator T defined by (1.3) and mapping
K̇qα,p (ω1 , ω2 ) → K̇qα,p (ω1 , ω2 ).
22
R∞
0
K. Jichang
(1) If g(t)−(β+n)/q ψ(t)
is a concave function on (0, ∞) and
t
dt
<
∞, then
g(t)−αδ−(β+n)/q ψ(t)
t
Z ∞
ψ(t)
kT k ≤ C(p, α)
g(t)−αδ−(β+n)/q
dt,
t
0
where
(
C(p, α) =
α
C0n 2(1/p)−2 (1 + p)1/p (1 + g(2)|α|δ ),
α
n
C0 2
1−(2/p)
(1 + (1/p))(1 + g(2)
|α|δ
),
(2.2)
0 < p < 1,
1 ≤ p < ∞.
(2.3)
(2) If kT k < ∞, and g is a strictly increasing function on (0, ∞) and the
inverse g −1 of g satisfies g −1 (t) → 0 (t → 0+ ), then
Z ∞
ψ(t)
g(t)−αδ−(β+n)/q
dt ≤ kT k.
t
0
(C0 and δ are constants given in (3.4), see Section 3 below.)
Remark 1. ω2 is an extension of the power weight ω2 (x) = |x|β , (x ∈ Rn ).
We use the following notation:
³ ψ(t) ´
n
KF = f : f ∈ K̇qα,p (ω1 , ω2 ), F (t) = sup |f (g(t)x)|
t
x∈Rn
o
is a concave function on (0, ∞) .
Then KF is a subspace of the space K̇qα,p (ω1 , ω2 ).
Theorem 2.2 Let α ∈ R1 , 0 < p < ∞, 0 < q < 1, g, ψ, ω1 , ω2 be as in
Theorem 2.1, and kT k be the norm of the operator T defined by (1.3), mapping
KF → K̇qα,p (ω1 , ω2 ).
R∞
0
(1) If g(t)−(β+n)/q ψ(t)
is a concave function on (0, ∞) and
t
−αδ−(β+n)/q ψ(t)
g(t)
t dt < ∞, then
Z ∞
ψ(t)
kT k ≤ C(p, q, α, )
g(t)−αδ−(β+n)/q
dt,
t
0
where C(p, q, α)
 α/n (1/p)−(1/q)−2 −1/p
q
(1+q)1/q (p+q)1/p (1+g(2)|α|δ ),

 C0 2
= C0α/n 2(1/q)−2 (1 + q)1/q (1 + g(2)|α|δ ),

 α/n (1/q)−(2/p)−1
C0 2
(1+q)1/q (1+(1/p))(1+g(2)|α|δ ),
(2.4)
0 < p ≤ q < 1,
0 < q < p < 1,
0<q<1 ≤ p<∞.
(2.5)
(2) If kT k < ∞, and g is a strictly increasing function on (0, ∞) and the
inverse g −1 of g satisfies g −1 (t) → 0(t → 0+ ), then
Z ∞
ψ(t)
g(t)−αδ−(β+n)/q
dt ≤ kT k.
t
0
(C0 and δ are constants given in (3.4), see Section 3 below.)
Generalized Hausdorff operators
23
Remark 2. When g is a decreasing function, similar results with g(2−1 ) and
g (t) → ∞(t → ∞) instead of g(2) and g −1 (t) → 0(t → 0) can be obtained from
the previous two theorems.
−1
Theorem 2.3. Let β ∈ R1 , 1 ≤ p < ∞, and g, ψ be two positive measurable
functions defined on (0, ∞), and g be a strictly monotonic function on (0, ∞) and
the inverse g −1 of g satisfies: (1) when g is increasing, g −1 (t) → 0 (t → 0+ );
(2) when g is decreasing, g −1 (t) → ∞ (t → ∞), and a nonnegative weight function
ω satisfies
ω(tx) = tβ ω(x), t > 0, β ∈ R1 , x ∈ Rn .
Then the operator T defined by (1.3), mapping Lp (ω) → Lp (ω), exists as a bounded
operator if and only if
Z ∞
ψ(t)
dt < ∞.
(2.6)
g(t)−(β+n)/p
t
0
Moreover, when (2.6) holds, the operator norm kT k of T on Lp (ω) satisfies
Z ∞
ψ(t)
kT k =
g(t)−(β+n)/p
.
(2.7)
t
0
Remark 3. It follows from (2.7) and (1.3) that
¯p
½Z ¯ Z ∞
¾1/p
¯
¯
ψ(t)
¯
¯ ω(x) dx
f
(g(t)x)
dt
¯
¯
t
Rn
0
¶ ½Z
¾1/p
µZ ∞
ψ(t)
dt ×
|f (x)|p ω(x) dx
, (2.8)
≤
g(t)−(β+n)/p
t
Rn
0
R∞
wherekT k = 0 g(t)−(β+n)/p ψ(t)
t dt is the best possible constant. In particular,
when ψ(t) = tλ ϕE (t), E = (0, 1], g(t) = t, n = 1, and (−∞, ∞) is substituted by
(0, ∞), and λ > 0, β < λp − 1, then by (2.7), we get
p
kT k =
.
λp − (β + 1)
It follows from (2.8) that
¯p
½Z ∞ ¯
½Z ∞
¾1/p
¾1/p
Z x
¯ 1
¯
p
λ−1
p
¯
¯
f (t)t
dt¯ ω(x) dx
≤
|f (x)| ω(x) dx
,
¯ xλ
λp − (β + 1) 0
0
0
(2.9)
p
where kT k = λp−(β+1)
is the best possible constant. If λ = 1, ω(x) = 1, that is,
β = 0, then (2.9) reduces to the Hardy inequality (1.1). If λ = 1, ω(x) = xβ , then
(2.9) reduces to the result of [6, p. 23]. When ψ(t) = t−λ ϕE (t), E = (0, 1], λ ≥ 0,
g(t) = t−1 , n = 1, and (−∞, ∞) is substituted by (0, ∞), and β > λp − 1, then by
p
(2.7), we get kT k = (β+1)−λp
. It follows from (2.8) that
µZ ∞
¶1/p
µZ ∞
¶1/p
Z ∞
1
p
| λ
f (t)tλ−1 dt|p ω(x) dx
|f (x)|p ω(x) dx
≤
,
x x
(β + 1) − λp 0
0
24
K. Jichang
p
where kT k = (β+1)−λp
is the best possible constant. This is the weighted extension
of the dual Hardy inequality in [5,6]. When g(t) = t, ψ(t) = te−λt , λ > 0, n = 1,
and (−∞, ∞) is substituted by (0, ∞), and β < p − 1, x > 0, then by (2.7), we get
Z ∞
Γ(1 − (β + 1)/p)
kT k =
t−(β+1)/p e−λt dt =
.
λ1−(β+1)/p
0
It follows from (2.8) that
½Z ∞ Z ∞
¾1/p
1
|
f (t)e−(λ/x)t dt|p ω(x) dx
x 0
0
Γ(1 − (β + 1)/p)
≤
λ1−(β+1)/p
where kT k =
(β+1)
)
p
(β+1)
1−
p
λ
Γ(1−
½Z
¾1/p
∞
p
|f (x)| ω(x) dx
,
(2.10)
0
is the best possible constant. In particular, when f (x) ≥ 0,
ω(x) = 1, that is, β = 0, then (2.10) reduces to the following Laplace transform
inequality:
¾1/p
½Z ∞ µZ ∞
¶p
−(λ/x)t
−p
f (t)e
dt x dx
0
0
≤
Γ(1 − (1/p))
λ1−(1/p)
µZ
∞
¶1/p
f p (x) dx
.
(2.11)
0
Remark 4. Hardy [4, Theorems 350, 352] proved the following three inequalities:
Let 1 < p < ∞,
1
p
+
1
q
= 1, f (x) ≥ 0, ω(x) = xp−2 . Then
kL(f )kp ≤ Γ(1/p)kf kp,ω ,
1
kL(f )kp,ω ≤ Γ(1 − )kf kp .
p
(2.12)
(2.13)
For 1 < p ≤ 2,
2π 1/q
) kf kp .
(2.14)
q
These inequalities are “the Laplace transforms analogues” of inequalities in the
theory of Fourier series. As pointed out by Hardy [3,4], it is not asserted that the
constant in (2.14) is the best possible and it may be difficult to find the best possible
value. Here we prove that the constants in (2.8)–(2.11) are the best possible. Hence,
our main results are significant generalization of many known results.
kL(f )kq ≤ (
Remark 5. There are some similar results for the non-homogeneous weighted
Herz spaces Kqα,p (ω1 , ω2 ). We omit the details here.
Several open problems. In Theorem 2.3, we solve the best value for C(p, β)
in the following inequality
kT f kp,ω ≤ C(p, β)kf kp,ω ,
Generalized Hausdorff operators
25
R∞
that is, C(p, β) = kT k = 0 g(t)−(β+n)/p ψ(t)
t dt is the best possible constant, but
yet in Theorems 2.1 and 2.2, the best value for C(p, α) and C(p, q, α) in the following
inequalities
kT f kK ≤ C(p, α)kf kK
and
kT f kK ≤ C(p, q, α)kf kKF
are not solved. It is not asserted that the constants C(p, α) and C(p, q, α) in
Theorems 2.1 and 2.2 are the best possible.
3. Proofs of the theorems
We require the following lemmas to prove our results.
Lemma 3.1. Let f be a nonnegative measurable function on [0, b], 0 < b < ∞.
If 1 ≤ p < ∞, then
µZ b
¶p
Z b
(p−1)
f (x) dx ≤ b
f p (x) dx.
(3.1)
0
0
Lemma 3.1 is an immediate consequences of Hölder inequality.
Lemma 3.2. (see [5]) Let f be a nonnegative measurable and concave function
on [a, b], 0 < α ≤ β. Then
½
¾1/β ½
¾1/α
Z
Z
β+1 b
α+1 b
[f (x)]β dx
≤
[f (x)]α dx
.
(3.2)
b−a a
b−a a
Setting a = 0, for α = p, β = 1, that is, 0 < p ≤ 1, we obtain from (3.2)
µZ b
¶p
Z b
p+1
p−1
×
b
f p (x) dx
(3.3).
f (x) dx ≤
2p
0
0
By the properties of A∞ weights, we have
Lemma 3.3 (see [2]) If ω ∈ A∞ , then there exist δ > 0, C0 > 0, such that for
each ball B in Rn and measurable subset E of B,
µ
¶δ
ω(E)
|E|
≤ C0
,
(3.4)
ω(B)
|B|
R
where |E| is the Lebesgue measure of E and ω(E) = E ω(x) dx.
Lemma 3.4 (see [5]) (Cp inequality) Let a1 , a2 , . . . , an be arbitrary real (or
complex) numbers. Then
µ n
¶p
µ n
¶
P
P
|ak | ≤ Cp
|ak |p , 0 < p < ∞,
(3.5)
k=1
k=1
26
K. Jichang
where
½
Cp =
1,
np−1 ,
0 < p < 1,
1 ≤ p < ∞.
In what follows, we shall write simply K̇qα,p (ω1 , ω2 ) to denote K and B(2k ) to
denote Bk = {x ∈ Rn : |x| ≤ 2k }.
Proof of Theorem 2.1. Since ψ has a compact support on (0, ∞), there exists
b > 0, such that supp ψ(t) ⊂ (0, b]. First, we prove (2.2). Using Minkowski’s
inequality for integrals and (2.1), and setting u = g(t)x, we get
¾1/q
Z b ½Z
ψ(t)
q
|f (g(t)x)| ω2 (x) dx
dt
k(T f )ϕk kq,ω2 ≤
t
Dk
0
¾1/q
Z b ½Z
ψ(t)
q
=
|f (u)| ω2 (u) du
g(t)−(β+n)/q
dt.
t
0
2k−1 g(t)<|u|≤2k g(t)
For each t ∈ (0, b), there exists an integer m such that 2m−1 < t ≤ 2m . Setting
A1 = {u ∈ Rn : 2k−1 g(2m−1 ) < |u| ≤ 2k g(2m−1 )}, E1 = B(2k g(2m−1 )),
A2 = {u ∈ Rn : 2k g(2m−1 ) < |u| ≤ 2k g(2m )}, E2 = B(2k g(2m )),
we obtain
Z b ½Z
k(T f )ϕk kq,ω2 ≤
0
Z
|f (u)|q ω2 (u) du
A1
¾1/q
|f (u)|q ω2 (u) du
+
A2
Z
b
≤
0
g(t)−(β+n)/q
ψ(t)
dt
t
(kf ϕA1 kq,ω2 + kf ϕA2 kq,ω2 )g(t)−(β+n)/q
ψ(t)
dt.
t
It follows that
kT f kK ≤
½X
·Z b
¸p ¾1/p
αp
ψ(t)
[ω1 (Bk )] n
(kf ϕA1 kq,ω2 + kf ϕA2 kq,ω2 )g(t)−(β+n)/q
dt
.
t
0
(3.6)
k∈Z
Now, we consider two cases for p:
Case 1. 0 < p < 1. In this case, it follows from (3.6), (3.3) and (3.5) that
½
(1 + p)1/p X
kT f kK ≤
[ω1 (Bk )]αp/n
2b(1/p)−1 k∈Z
µ
¶p ¾1/p
Z b
p
p
−(β+n)p/q ψ(t)
(kf ϕA1 kq,ω2 + kf ϕA2 kq,ω2 )g(t)
×
dt
t
0
¶αp/n
½·Z b X
µ
1
ω1 (Bk )
≤ 2(1/p)−2 (1 + p)1/p b1−( p )
ω1 (E1 )αp/n kf ϕA1 kpq,ω2
ω1 (E1 )
0
k∈Z
27
Generalized Hausdorff operators
−(β+n)p/q
× g(t)
¸1/p ·Z b X
ψ(t) p
(
) dt
+
ω1 (E2 )αp/n kf ϕA2 kpq,ω2
t
0
k∈(Z)
µ
×
By (3.4) and |Bk | =
ω1 (Bk )
ω1 (E2 )
¶αp/n
µ
g(t)
−(β+n)p/q
ψ(t)
t
¶p
¸1/p ¾
dt
.
(3.7)
π n/2
Γ( n
2 +1)
× 2kn , we have
µ
¶δ
ω1 (Bk )
|Bk |
−nδ
≤ C0
= C0 g(2m−1 )
ω1 (E1 )
|E1 |
(3.8)
and
ω1 (Bk )
≤ C0 g(2m )−nδ .
ω1 (E2 )
It follows from (3.7)–(3.9) that
(3.9)
α
kT f kK ≤ C0n 2(1/p)−2 (1 + p)1/p kf kK
Z b
ψ(t)
dt.
×
(g(2m−1 )−αδ + g(2m )−αδ )g(t)−(β+n)/q
t
0
(3.10)
By the submultiplicativity of g, we have g(2m ) ≤ g(2)g(2m−1 ). If α > 0, then
[g(2m−1 )]−αδ ≤ [g(2m )]−αδ [g(2)]αδ .
Since g is a increasing function, thus t ≤ 2m implies that g(t) ≤ g(2m ), and therefore
that g(2m )−αδ ≤ g(t)−αδ . This implies
g(2m−1 )−αδ + g(2m )−αδ ≤ g(2m )−αδ {1 + g(2)αδ } ≤ g(t)−αδ {1 + g(2)αδ }. (3.11)
Similarly, if α < 0, then
g(2m−1 )−αδ + g(2m )−αδ ≤ g(t)−αδ {1 + g(2)−αδ }.
(3.12)
(3.11) and (3.12) imply that
g(2m−1 )−αδ + g(2m )−αδ ≤ g(t)−αδ {1 + g(2)|α|δ }.
(3.13)
Thus, by (3.10) and(3.13), we get
Z
(1+p)1/p (1+g(2)|α|δ )kf kK
∞
ψ(t)
dt.
t
0
(3.14)
Case 2. 1 ≤ p < ∞. In this case, by (3.6), (3.1), (3.3), (3.8) and (3.9), we
similarly obtain
½X
αp
1−(1/p)
[ω1 (Bk )] n
kT f kK ≤ (2b)
α/n (1/p)−2
kTf kK ≤ C0
2
g(t)−αδ−(β+n)/q
k∈Z
µ
¶p ¾1/p
ψ(t)
(kf ϕA1 kpq,ω2 +kf ϕA2 kpq,ω2 )g(t)(−(β+n)/q)p
dt
t
0
Z ∞
α
ψ(t)
≤ C0n 21−(2/p) (1 + (1/p))(1 + g(2)|α|δ )kf kK
g(t)−αδ−(β+n)/q
dt.
t
0
Z
b
×
(3.15)
28
K. Jichang
Hence, by (3.14) and (3.15), we get
Z
kT k ≤ C(p, α)
∞
g(t)−αδ−(β+n)/q
0
ψ(t)
dt,
t
where C(p, α) is defined by (2.3).
To prove the opposite inequality, putting ε ∈ (0, 1), we set ω1 (Bk ) = 2knδ ,
ω2 (x) = |x|β , and
½
0,
|x| ≤ 1,
fε (x) =
|x|−αδ−(β+n)/q−ε , |x| > 1.
Then for k = 0, −1, −2, . . . , kfε ϕk kq,ω2 = 0 and for k ∈ Z+ , we have
¶1/q
µZ
−(αδ+(β+n)/q+ε)q
|x|
ω2 (x) dx
kfε ϕk kq,ω2 =
½
=
2k−1 <|x|≤2k
Z
2π n/2
Γ(n/2)
where
¾1/q
2k
r
−(αδ+ε)q−1
dr
2k−1
= Cn1/q 2−k(αδ+ε) ,
¯
¯
2π n/2 ¯¯ 2(αδ+ε)q − 1 ¯¯
Cn =
.
Γ(n/2) ¯ (αδ + ε)q ¯
It follows that
µZ
½X
∞
ω1 (Bk )αp/n
kfε kK =
2k−1 <|x|≤2k
k=1
= Cn1/q
¶p/q ¾1/p
½X
∞
¾1/p
2−kpε
= Cn1/q
n=1
|x|−(αδ+(β+n)/q+ε)q ω2 (x) dx
2−ε
.
(1 − 2−pε )1/p
(3.16)
1
), we have
Since g(t)|x| > 1 implies that t > g −1 ( |x|
Z ∞
ψ(t)
T (fε , x) = |x|−αδ−(β+n)/q−ε
g(t)−αδ−(β+n)/q−ε
dt.
1
t
−1
g ( |x| )
Let ε (0 < ε < 1) be given. Since g is a strictly increasing function, there exists
m ∈ N, such that 2m−1 ≤ 1ε < 2m and 2−mε → 1 (ε → 0+ ). Note that 2k−1 <
1
1
|x| ≤ 2k , so that if k ≥ m + 1, then g −1 ( |x|
) ≤ g −1 ( 2k−1
) ≤ g −1 ( 21m ) ≤ g −1 (ε).
Thus,
½Z
¾p/q
X
kT fε kpK =
ω1 (Bk )αp/n
[T (fε , x)ϕk (x)]q ω2 (x) dx
|x|>1
k∈Z
=
∞
X
½Z
ω1 (Bk )
k=1
µZ
αp/n
∞
×
1
g −1 ( |x|
)
|x|−(αδ+(β+n)/q+ε)q
2k−1 <|x|≤2k
g(t)−αδ−(β+n)/q−ε
¶q
¾p/q
ψ(t)
dt ω2 (x) dx
t
29
Generalized Hausdorff operators
µZ
¶p
∞
−αδ−(β+n)/q−ε ψ(t)
≥
g(t)
dt
t
g −1 (ε)
½Z
¾p/q
∞
X
αp/n
−(αδ+(β+n)/q+ε)q
×
ω1 (Bk )
|x|
ω2 (x) dx
2k−1 <|x|≤2k
k=m+1
µZ
∞
=
g(t)−αδ−(β+n)/q−ε
g −1 (ε)
¶p
ψ(t)
2−(m+1)pε
dt Cnp/q
.
t
(1 − 2−pε )
Thus, by (3.16) and (3.17), we get
½Z ∞
¾
kT fε kK
−mpε
−αδ−(β+n)/q−ε ψ(t)
kT k ≥
g(t)
≥2
dt .
kfε kK
t
g −1 (ε)
(3.17)
(3.18)
Taking limit as ε → 0 in (3.18), we obtain
Z ∞
ψ(t)
kT k ≥
g(t)−αδ−(β+n)/q
dt.
t
0
This finishes the proof of Theorem 2.1.
Proof of Theorem 2.2. First, we prove (2.4). Using (3.3) and the notations in
the proof of Theorem 2.1, we obtain
½Z µZ b
¶q
¾1/q
ψ(t)
k(T f )ϕk kq,ω2 ≤
( sup |f (g(t)x)|)
dt ω2 (x) dx
t
Dk
0 x∈Rn
½Z µZ b
¶
¾1/q
ψ(t) q
≤ 2−1 (1 + q)1/q b1−(1/q)
( sup |f (g(t)x)|)q (
) dt ω2 (x) dx
t
Dk
0 x∈Rn
≤ 2−1 (1 + q)1/q b1−(1/q)
½Z b µZ
¶
¾1/q
q
−(β+n) ψ(t) q
×
|f (u)| ω2 (u) du g(t)
(
) dt
t
0
2k−1 g(t)<|u|≤2k g(t)
½Z b
¾1/q
−1
1/q 1−(1/q)
q
q
−(β+n) ψ(t) q
≤ 2 (1 + q) b
(kf ϕA1 kq,ω2 +kf ϕA2 kq,ω2 )g(t)
(
) dt
.
t
0
It follows that
½X
−1
1/q 1−(1/q)
ω1 (Bk )αp/n
kT f kK ≤ 2 (1 + q) b
·Z
×
0
k∈Z
b
ψ(t) q
(kf ϕA1 kqq,ω2 +kf ϕA2 kqq,ω2 )g(t)−(β+n) (
)
t
¸p/q ¾1/p
dt
.
(3.19)
Now,we consider three cases:
Case 1. 0 < p ≤ q < 1. In this case, it follows from (3.19), (3.3), (3.5), (3.1),
(3.8) and (3.9) that
kT f kK ≤ 2−(1+(1/q)) q −(1/p) (1 + q)1/q (p + q)1/p b1−(1/p)
½X
¾1/p
Z b
ψ(t) p
×
(kf ϕA1 kpq,ω2 +kf ϕA2 kpq,ω2 )g(t)−(β+n)p/q (
) dt
ω1 (Bk )αp/n
t
0
k∈Z
30
K. Jichang
≤ 2(1/p)−(1/q)−2 q −(1/p) (1 + q)1/q (p + q)1/p kf kK
¾
Z b½
ω1 (Bk ) α/n ω1 (Bk ) α/n
ψ(t)
×
(
) +(
)
g(t)−(β+n)/q
dt
ω
(E
)
ω
(E
)
t
1
1
1
2
0
α/n
≤ C0 2(1/p)−(1/q)−2 q −(1/p) (1 + q)1/q (p + q)1/p kf kK
Z b
ψ(t)
dt
×
(g(2m−1 )−αδ +g(2m )−αδ )g(t)−(β+n)/q
t
0
α/n (1/p)−(1/q)−2 −(1/p)
≤ C0
2
q
(1 + q)1/q (p + q)1/p (1 + g(2)|α|δ )kf kK
Z ∞
ψ(t)
×
g(t)−αδ−(β+n)/q
dt. (3.20)
t
0
Case 2. 0 < q ≤ p < 1. In this case, by (3.19), (3.5), (3.1), (3.8) and (3.9), we
similarly obtain
½
(1 + q)1/q 1−(1/p) X
kT f kK ≤
b
ω1 (Bk )αp/n
2
k∈Z
¾1/p
Z b
q
q p/q
−(β+n)p/q ψ(t) p
×
(kf ϕA1 kq,ω2 +kf ϕA2 k ) g(t)
(
) dt
t
0
µ
¶α/n ¾
Z b½
ω1 (Bk ) α/n
ω1 (Bk )
ψ(t)
(1/q)−2
1/q
≤2
(1 + q) kf kK
(
) +
g(t)−(β+n)/q
dt
ω
(E
)
ω
(E
)
t
1
1
1
2
0
Z ∞
ψ(t)
α/n
dt. (3.21)
≤ C0 2(1/q)−2 (1 + q)1/q (1 + g(2)|α|δ )kf kK
g(t)−αδ−(β+n)/q
t
0
Case 3. 0 < q ≤ 1 ≤ p < ∞. In this case, by (3.19), (3.1), (3.5), (3.3), (3.8)
and (3.9), we obtain
½
(1 + q)1/q b1−(1/p) X
kT f kK ≤
ω1 (Bk )αp/n
2(1/p)+1−(1/q)
k∈Z
¾1/p
Z b
p
p
−(β+n)p/q ψ(t) p
×
(kf ϕA1 kq,ω2 + kf ϕA2 kq,ω2 )g(t)
(
) dt
t
0
α/n (1/q)−(2/p)−1
≤ C0
2
(1 + q)1/q (1 + (1/p))(1 + g(2)|α|δ )kf kK
Z ∞
ψ(t)
×
g(t)−αδ−(β+n)/q
dt.
t
0
Hence, by (3.20)–(3.22), we get
Z
∞
(3.22)
ψ(t)
dt,
t
0
where C(p, q, α) is defined by (2.5). By the same technique used in Theorem 2.1
one can show that the opposite inequality:
Z ∞
ψ(t)
dt.
kT k ≥
g(t)−αδ−(β+n)/q
t
0
This finishes the proof of Theorem 2.2.
kT k ≤ C(p, q, α)
g(t)−αδ−(β+n)/q
31
Generalized Hausdorff operators
Proof of Theorem 2.3. By Minkowski inequality for integrals and setting u =
g(t)x, we get
½Z Z ∞
¾1/p
ψ(t)
kT f kp,ω ≤
dt)p ω(x) dx
(
|f (g(t)x)|
t
Rn 0
¾1/p
Z ∞ ½Z
ψ(t)
≤
|f (g(t)x)|p ω(x) dx
dt
t
0
Rn
¾1/p
Z ∞ ½Z
ψ(t)
p
|f (u)| ω(u) du
g(t)−(β+n)/p
dt
=
t
n
0
R
Z ∞
ψ(t)
= kf kp,ω
dt.
g(t)−(β+n)/p
t
0
It follows that
kT f kp,ω
≤
f 6=0 kf kp,ω
Z
∞
kT k = sup
t−(β+n)/p
0
ψ(t)
dt.
t
To prove the opposite inequality, putting ε ∈ (0, 1), we set ω(x) = |x|β and
½
0,
|x| ≤ 1,
fε (x) =
−(β+n)/p−ε
|x|
, |x| > 1,
thus
Z
p
kfε kp,ω =
|x|−(β+n+εp) ω(x) dx =
|x|>1
2π n/2
Γ(n/2)
Z
∞
r−n−pε rn−1 dr =
1
2π n/2
.
pεΓ(n/2)
1
ε,
1
If g is strictly increasing, putting |x| <
g(t)|x| > 1 implies that t > g −1 ( |x|
)>
−1
g (ε). It follows that
½Z
¾1/p
Z ∞
p
−(β+n)/p−ε ψ(t)
kT fε kp,ω =
(
dt) ω(x) dx
(g(t)|x|)
1
t
|x|>1 g −1 ( |x|
)
µZ ∞
¶
2π n/2 1/p
−(β+n)/p−ε ψ(t)
≥
g(t)
dt {
} .
t
pεΓ(n/2)
g −1 (ε)
This implies
kT fε kp,ω
kT k ≥
≥
kfε kp,ω
Z
∞
g(t)−(β+n)/p−ε
g −1 (ε)
ψ(t)
dt.
t
Taking limits as ε → 0 in (3.23), we get
Z ∞
ψ(t)
kT k ≥
g(t)−(β+n)/p
dt.
t
0
Then by (3.23) and (3.24), we have
Z ∞
ψ(t)
dt.
kT k =
g(t)−(β+n)/p
t
0
The proof for the decreasing case is similar. The theorem is proved.
(3.23)
(3.24)
32
K. Jichang
Acknowledgement. The author would like to thank anonymous referees on
suggestions to improve this text.
REFERENCES
[1] C. Bennett, R.A. Devore, R. Sharpley, Weak L∞ and BMO, Ann. Math. 113 (1981), 601–
611.
[2] J. Garcı́a-Cuerva, J.L. Rubio de Francia, Weighted Norm Inequalities and Related Topics,
North-Holland Publishing, Amsterdam, 1985.
[3] G.H. Hardy, The constants of certain inequalities, J. London Math.Soc. 8 (1933), 601–611.
[4] G.H. Hardy, J.E. Littlewood, G. Polya, Inequalities (2nd edition), Cambridge University
Press, Cambrige, 1967.
[5] J.C. Kuang, Applied Inequalities (4th edition), Shangdong Science Press, Ji’nan, 2010 (in
Chinese).
[6] A. Kufner, L. Maligranda, L.E. Persson, The Hardy inequality—About its history and some
related results, Pilsen, 2007.
[7] A.K. Lerner, E. Liflyand, Multidimensional Hausdorff operators on the real Hardy spaces, J.
Austral. Math. Soc. 83 (2007), 79–86.
[8] E. Liflyand, F. Mórecz, The Hausdorff operator is bounded on real H 1 space, Proc. Amer.
Math. Soc. 128 (2000), 1391–1396.
[9] E. Liflyand, Open problems on Hausdorff operators, in: Complex Analysis and Potential
Theory (Proceedings of the Conference, Istanbul, Turkey, Sept 8–14, 2006).
[10] E. Liflyand, A. Miyachi, Boundedness of the Hausdorff operators in H p spaces, 0 < p < 1,
Studia Math. 194 (2009), 279–292.
[11] S. Lu, D. Yang, The decomposition of the weighted Herz spaces and its applications, Sci. in
China A 38 (1995), 147–158.
[12] J. Xiao, Lp and BMO bounds of weighted Hardy–Littlewood averages, J. Math. Anal. Appl.
262 (2001), 660–666.
(received 21.01.2012; in revised form 18.08.2012; available online 01.11.2012)
Department of Mathematics, Hunan Normal University, Changsha, 410081 P.R.CHINA
E-mail: jc-kuang@hotmail.com