Bull. Malays. Math. Sci. Soc. DOI 10.1007/s40840-017-0444-5
Weighted p-Adic Hardy Operators and Their Commutators on p-Adic Central Morrey Spaces Qing Yan Wu1 · Zun Wei Fu1
Received: 15 November 2013 / Revised: 28 April 2014 © Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2017
Abstract In this paper, we establish necessary and sufficient conditions for boundedness of weighted p-adic Hardy operators on p-adic Morrey spaces, p-adic central Morrey spaces and p-adic λ-central BMO spaces, respectively, and obtain their sharp bounds. We also give the characterization of weight functions for which the commutators generated by weighted p-adic Hardy operators and λ-central BMO functions are bounded on the p-adic central Morrey spaces. This result is different from that on Euclidean spaces due to the special structure of p-adic integers. Keywords Weighted p-adic Hardy operator · p-Adic central Morrey space · p-Adic λ-central BMO space Mathematics Subject Classification 11F85 · 47A63 · 47G10
1 Introduction Let ω : [0, 1] → [0, ∞) be a function. The weighted Hardy operator Hω [6] is defined by Communicated by Mohammad Sal Moslehian. This work was partially supported by NSF of China (Grant Nos. 11271175, 11671185 and 11301248), NSF of Shandong Province (Grant No. ZR2012AQ026) and AMEP of Linyi University.
B
Zun Wei Fu
[email protected] Qing Yan Wu
[email protected]
1
Department of Mathematics, Linyi University, Linyi 276005, Shandong, People’s Republic of China
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Q. Y. Wu, Z. W. Fu
Hω f (x) :=
1
f (t x)ω(t)dt,
0
for all measurable complex-valued functions f on Rn and x ∈ Rn . Xiao [32] gave the characterization of ω for which Hω is bounded on either L p (Rn ), 1 ≤ p ≤ ∞, or BMO(Rn ). Meanwhile, the corresponding operator norms were worked out. In [12], Fu et al. gave the characterization of ω to ensure that Hω is bounded on central Morrey spaces and λ-central BMO spaces; they also obtained the corresponding operator norms. It is clear that if ω ≡ 1 and n = 1, then Hω is precisely reduced to the classical Hardy operators H defined by H f (x) =
1 x
x
f (t)dt, x = 0,
0
which is one of the fundamental integral averaging operators in real analysis. A celebrated operator norm estimate states that, for 1 < q < ∞, the sharp norm of H from L q (R) to L q (R) is q/(q − 1), see [14]. If n = 1 and ω(t) = (1 − t)α−1 / (α), 0 < α < 1, then for all x > 0, Hω f (x) = x −α Iα f (x), where Iα is Riemann–Liouville fractional integral defined by 1 Iα f (x) = (α)
0
x
f (t) dt, x > 0, (x − t)1−α
for all locally integrable functions f on (0, ∞). For n ≥ 2, if ω(t) = nt n−1 and f is a radical function, then Hω is just reduced to the n-dimensional Hardy operator H defined by 1 H f (x) = vn |x|n
|y|<|x|
f (y)dy,
where vn is the volume of the unit sphere S n−1 . See [33] for more details. In 1995, Christ and Grafakos [9] obtained that the precise norm of H from L q (Rn ) to L q (Rn ) is also q/(q − 1), 1 < q < ∞. More recently, Fu et al. [11] obtained the precise norm of m-linear Hardy operators on weighted Lebesgue spaces and central Morrey spaces. In recent years, the field of p-adic numbers has been widely used in theoretical and mathematical physics (cf. [3,5,15–17,20,26–30]). And harmonic analysis on p-adic field has been drawn more and more concern ([4,7,8,18,19,22–25] and references therein). For a prime number p, let Q p be the field of p-adic numbers. It is defined as the completion of the field of rational numbers Q with respect to the non-Archimedean p-adic norm | · | p . This norm is defined as follows: |0| p = 0; if any nonzero rational number x is represented as x = p γ mn , where γ is an integer and integers m, n are
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Weighted p-Adic Hardy Operators and Their Commutators…
indivisible by p, then |x| p = p −γ . It is easy to see that the norm satisfies the following properties: |x y| p = |x| p |y| p , |x + y| p ≤ max |x| p , |y| p . Moreover, if |x| p = |y| p , then |x ± y| p = max{|x| p , |y| p }. It is well known that Q p is a typical model of non-Archimedean local fields. From the standard p-adic analysis [28], we see that any nonzero p-adic number x ∈ Q p can be uniquely represented in the canonical series x = pγ
∞
a j p j , γ = γ (x) ∈ Z,
(1.1)
j=0
where a j are integers, 0 ≤ a j ≤ p − 1, a0 = 0. The space Qnp consists of points x = (x1 , x2 , · · · , xn ), where x j ∈ Q p , j = 1, 2, · · · , n. The p-adic norm on Qnp is |x| p := max |x j | p , x ∈ Qnp . 1≤ j≤n
Denote by Bγ (a) = {x ∈ Qnp : |x − a| p ≤ p γ } the ball with center at a ∈ Qnp and radius p γ , and by Sγ (a) := {x ∈ Qnp : |x − a| p = p γ } the sphere with center at a ∈ Qnp and radius p γ , γ ∈ Z. It is clear that Sγ (a) = Bγ (a)\Bγ −1 (a) and Bγ (a) =
Sk (a).
(1.2)
k≤γ
We set Bγ (0) = Bγ and Sγ (0) = Sγ . Let Z p = {x ∈ |x| p ≤ 1} be the class of all p-adic integers in Q p , and denote Z∗p = Z p \{0} Since Qnp is a locally compact commutative group under addition, it follows from the standard analysis that there exists a Haar measure dx on Qnp , which is unique up to a positive constant factor and is translation invariant. We normalize the measure dx by the equality B0 (0)
dx = |B0 (0)| H = 1,
where |E| H denotes the Haar measure of a measurable subset E of Qnp . By simple calculation, we can obtain that |Bγ (a)| H = p γ n ,
|Sγ (a)| H = p γ n (1 − p −n )
for any a ∈ Qnp . For a more complete introduction to the p-adic field, one can refer to [25] or [28].
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Q. Y. Wu, Z. W. Fu p
by
On p-adic field, Rim and Lee [22] defined the weighted p-adic Hardy operator Hψ
p
Hψ f (x) =
Z∗p
f (t x)ψ(t)dt,
(1.3)
where ψ is a nonnegative function defined on Z∗p , and gave the characterization of p p p functions ψ for which Hψ are bounded on L r (Qn ), 1 ≤ r ≤ ∞, and on BMO(Qn ). Also, in each case, they obtained the corresponding operator norms. Obviously, if ψ ≡ 1 and n = 1, then Hω is just reduced to the p-adic Hardy operator H p on Q p , which is defined by 1 H f (x) = |x| p
p
|t| p ≤|x| p
f (t)dt, x = 0. p
Let 0 < α < 1. We define the p-adic Riemann–Liouville fractional integral Rα by Rαp f (x) =
1 − p −α 1 − p α−1
f (y) |y| p <|x| p
|x − y|1−α p
dy.
α−1 ), then For n = 1, if we take ψ(t) = (1 − p −α )|1 − t|α−1 p χ B0 \S0 (t)/(1 − p p Hψ f (x) = |x|−α p Rα f (x). p
−1 For n ≥ 2, if we take ψ(t) = (1 − p −n )|t|n−1 p /(1 − p ), and f satisfies f (x) = −1 f (|x| p ), then p
Hψ f (x) = H p f (x), p
where H p is the p-adic Hardy operator on Qn defined by 1 H p f (x) = f (t)dt, x ∈ Qnp \{0}, |x|np B(0,|x| p ) here B(0, |x| p ) is a ball in Qnp with center at 0 ∈ Qnp and radius |x| p . In fact, by definition, we have 1 p H f (x) = f (t)dt = f (|x|−1 p y)dy |x|np B(0,|x| p ) B(0,1) 0 = f (x|y|−1 )dy = f (x|y|−1 p p )dy B(0,1)
=
0 k=−∞
123
k=−∞ Sk
f ( p −k x) p kn (1 − p −n )
Weighted p-Adic Hardy Operators and Their Commutators…
0 1 − p −n n−1 f (|t|−1 p x)|t| p dt k 1 − p −1 |t| = p p k=−∞ −n 1− p = f (t x) |t|n−1 p dt −1 ∗ 1 − p Zp
=
(1.4)
Fu et al. [13] established the sharp estimates of p-adic Hardy operators on p-adic weighted Lebesgue spaces. Wu et al. [31] obtained the sharp bounds of p-adic Hardy operators on p-adic central Morrey spaces and p-adic λ-central BMO spaces. They also obtained the boundedness for commutators of p-adic Hardy operators on these spaces. The main purpose of this paper is to make clear the mapping properties of weighted p-adic Hardy operators as well as their commutators on p-adic Morrey, central Morrey and λ-central BMO spaces. Morrey [21] introduced the L q,λ (Rn ) spaces to study the local behavior of solutions to second-order elliptic partial differential equations. The p-adic Morrey space is defined as follows. Definition 1.1 Let 1 ≤ q < ∞ and λ ≥ − q1 . The p-adic Morrey space L q,λ (Qnp ) is defined by L q,λ (Qnp ) =
q f ∈ L loc (Qnp ) : f L q,λ (Qnp ) < ∞ ,
where
f L q,λ (Qnp ) =
sup
a∈Qnp ,γ ∈Z
1
1
q
| f (x)| dx q
1+λq
|Bγ (a)| p
Bγ (a)
.
Remark 1.2 It is clear that L q,−1/q (Qnp ) = L q (Qnp ), L q,0 (Qnp ) = L ∞ (Qnp ). Alvarez, Guzmán–Partida and Lakey [1] studied the relationship between central BMO spaces and Morrey spaces. Furthermore, they introduced λ-central BMO spaces and central Morrey spaces, respectively. Next, we introduce their p-adic versions. Definition 1.3 Let λ ∈ R and 1 < q < ∞. The p-adic central Morrey space B˙ q,λ (Qnp ) is defined by
f B˙ q,λ (Qn ) := sup p
γ ∈Z
q
| f (x)| d x q
1+λq
|Bγ | H
1
1
< ∞,
(1.5)
Bγ
where Bγ = Bγ (0). Remark 1.4 It is clear that L q,λ (Qnp ) ⊂ B˙ q,λ (Qnp ),
1
q,− q B˙ (Qnp ) = L q (Qnp ).
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Q. Y. Wu, Z. W. Fu
When λ < −1/q, the space B˙ q,λ (Qnp ) reduces to {0}; therefore, we can only consider the case λ ≥ −1/q. If 1 ≤ q1 < q2 < ∞, by Hölder’s inequality B˙ q2 ,λ (Qnp ) ⊂ B˙ q1 ,λ (Qnp ) for λ ∈ R. Definition 1.5 Let λ < condition
1 n
and 1 < q < ∞. The space CMOq,λ (Qnp ) is defined by the
f CMOq,λ (Qn ) := sup p
γ ∈Z
1
1
q
| f (x) − f Bγ | d x q
1+λq
|Bγ | H
Bγ
< ∞.
(1.6)
Remark 1.6 When λ = 0, the space CMOq,λ (Qnp ) is just CMOq (Qnp ), which is defined in [13]. If 1 ≤ q1 < q2 < ∞, by Hölder’s inequality, CMOq2 ,λ (Qnp ) ⊂ CMOq1 ,λ (Qnp ) for λ ∈ R. By the standard proof as that in Rn , we can see that
f CMOq,λ (Qn ) ∼ sup inf p
γ ∈Z c∈C
1 1+λq
|Bγ | H
1
q
| f (x) − c| dx q
.
Bγ
Remark 1.7 The Formulas (1.5) and (1.6) yield that B˙ q,λ (Qnp ) is a Banach space continuously included in CMOq,λ (Qnp ). The outline of this paper is as follows. In Sect. 2, we establish the necessary and sufficient conditions for boundedness of p-adic Hardy operators on p-adic Morrey spaces, p-adic central Morrey spaces and p-adic λ-central BMO spaces, respectively, and obtain their corresponding operator norms. In Sect. 3, we give the characterization of weight functions for which the commutators generated by weighted p-adic Hardy operators and p-adic central BMO functions are bounded on p-adic central Morrey spaces. Throughout this paper, the letter C will be used to denote various constants, and the various uses of the letter do not, however, denote the same constant.
2 Sharp Estimates of Weighted p-Adic Hardy Operator We get the following sufficient and necessary conditions of weight functions, for which the weighted p-adic Hardy operators are bounded on p-adic Morrey, central Morrey and λ-central BMO spaces. p
Theorem 2.1 Let 1 < q < ∞ and −1/q < λ ≤ 0. Then Hψ is bounded on L q,λ (Qnp ) if and only if
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Weighted p-Adic Hardy Operators and Their Commutators…
Z∗p
|t|nλ p ψ(t)dt < ∞.
(2.1)
Moreover,
p
Hψ L q,λ (Qnp )→L q,λ (Qnp ) =
Z∗p
|t|nλ p ψ(t)dt.
Corollary 2.2 Let 1 < q < ∞, −1/q < λ ≤ 0 and 0 < α < 1. Then 1 − p −1 , 1 − p −(1+λ) (1 − p −α )(1 − p −1 ) . = (1 − p α−1 )( p 1+λ − 1)
H p L q,λ (Q p )→L q,λ (Q p ) =
Rαp L q,λ (Q p )→L q,λ (|x|−αq dx) p
Moreover, write Lq,λ (Qnp ) = { f : f ∈ L q,λ (Qnp ) and satisfies f (x) = f (|x|−1 p )}. Then
H p Lq,λ (Qnp )→L q,λ (Qnp ) =
1 − p −n . 1 − p −n(1+λ)
p Theorem 2.3 Let 1 < q < ∞ and −1/q < λ ≤ 0. Then Hψ is bounded on B˙ q,λ (Qnp ) if and only if (2.1) holds. Moreover, p |t|nλ
Hψ B˙ q,λ (Qn )→ B˙ q,λ (Qn ) = p ψ(t)dt. p
p
Z∗p
Corollary 2.4 Let 1 < q < ∞, −1/q < λ ≤ 0 and 0 < α < 1. Then 1 − p −1 , 1 − p −(1+λ) (1 − p −α )(1 − p −1 ) . = (1 − p α−1 )( p 1+λ − 1)
H p B˙ q,λ (Q p )→ B˙ q,λ (Q p ) =
Rαp B˙ q,λ (Q p )→ B˙ q,λ (|x|−αq d x) p
Moreover, set B˙ q,λ (Qnp ) = Then
f : f ∈ B˙ q,λ (Qnp ) and satisfies f (x) = f (|x|−1 p ) .
H p B˙ q,λ (Qn )→ B˙ q,λ (Qn ) = p
p
1 − p −n . 1 − p −n(1+λ) p
Theorem 2.5 Let 1 < q < ∞ and 0 ≤ λ < 1/n. Then Hψ is bounded on CMOq,λ (Qnp ) if and only if (2.1) holds. Moreover, p
Hψ CMOq,λ (Qn )→CMOq,λ (Qn ) p p
=
Z∗p
|t|nλ p ψ(t)dt.
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Q. Y. Wu, Z. W. Fu
Corollary 2.6 Let 1 < q < ∞. (I). If 0 ≤ λ < 1, then 1 − p −1 , 1 − p −(1+λ) (1 − p −α )(1 − p −1 ) = . (1 − p α−1 )( p 1+λ − 1)
H p CMOq,λ (Q p )→CMOq,λ (Q p ) =
Rαp CMOq,λ (Q p )→CMOq,λ (|x|−αq d x) p
(II). If 0 ≤ λ < 1/n and set CMOq,λ (Qnp ) = f (x) = f (|x|−1 p ) , then
f : f ∈ CMOq,λ (Qnp ) and satisfies
H p CMOq,λ (Qn )→CMOq,λ (Qn ) = p
p
1 − p −n . 1 − p −n(1+λ)
Proof of Theorem 2.1 Suppose that (2.1) holds. Let γ ∈ Z and denote t Bγ (a) = B(ta, |t| p p γ ). Using Minkowski’s inequality and change of variable, we have
1+λq
|Bγ (a)| H ≤
Z∗p
=
1
1
q
Bγ (a)
p |Hψ
1
1
q
|Bγ (a)| H
Bγ (a)
1+λq
≤ f L q,λ (Qnp )
ψ(t)dt
1
1 |t Bγ (a)| H
Z∗p
| f (t x)| dx q
1+λq
f (x)|q dx
Z∗p
t Bγ (a)
q
| f (y)|q dy
|t|λn p ψ(t)dt
|t|nλ p ψ(t)dt.
p
Thus, Hψ is bounded on L q,λ (Qnp ) and
p
Hψ L q,λ (Qnp )→L q,λ (Qnp ) ≤
Z∗p
|t|nλ p ψ(t)dt.
p
On the other hand, assume that Hψ is bounded on L q,λ (Qnp ). Take n f 0 (x) = |x|nλ p , x ∈ Qp.
Now we show that f 0 ∈ L q,λ (Qnp ).
123
(2.2)
Weighted p-Adic Hardy Operators and Their Commutators…
(I). If |a| p > p γ and x ∈ Bγ (a), then |x| p = max{|x − a| p , |a| p } > p γ . Since − q1 ≤ λ < 0, we have
1 1+λq
|Bγ (a)| H
nλq |x| p dx
Bγ (a)
<
1 1+λq
|Bγ (a)| H
Bγ (a)
p γ nλq dx = 1.
(II). If |a| p ≤ p γ and x ∈ Bγ (a), then |x| p ≤ max{|x − a| p , |a| p } ≤ p γ ; therefore, x ∈ Bγ . Recall that two balls in Qnp are either disjoint or one is contained in the other (cf. P.21 in [2]). So we have Bγ (a) = Bγ ; thus,
1 1+λq
|Bγ (a)| H
Bγ (a)
1
nλq
|x| p dx =
=p
nλq
|x| p dx
1+λq
|Bγ | H
Bγ
γ
−γ n(1+λq)
p knλq dx
k=−∞ Sk
=
1 − p −n . 1 − p −n(1+λq)
From the above discussion, we can see that f 0 ∈ L q,λ (Qnp ). It is clear that
p
Hψ f 0 (x) =
Z∗p
nλ |t x|nλ p ψ(t)dt = |x| p
= f 0 (x)
Z∗p
Z∗p
|t|nλ p ψ(t)dt (2.3)
|t|nλ p ψ(t)dt.
Therefore,
p
Hψ L q,λ (Qnp )→L q,λ (Qnp ) ≥
Z∗p
|t|nλ p ψ(t)dt.
(2.4)
Consequently, Z∗p
|t|nλ p ψ(t)dt < ∞.
And (2.2) and (2.4) yield the desired result.
Proof of Theorem 2.3 Suppose that (2.1) holds. For any γ ∈ Z, by Minkowski’s inequality and change of variable, we have
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Q. Y. Wu, Z. W. Fu
1+λq
|Bγ | H =
Z∗p
1
1
Bγ
f (x)| dx
p
Z∗p
1+λq
1+λq p )| H
1 q
B(0, p γ |t|
p)
ψ(t)dt
Bγ
| f (y)| dy q
p γ |t|
≤ f B˙ q,λ (Qn )
q
| f (t x)| dx q
|Bγ | H
Z∗p
1
1
1 |B(0,
≤
q
q
p |Hψ
|t|λn p ψ(t)dt
|t|nλ p ψ(t)dt.
p Therefore, Hψ is bounded on B˙ q,λ (Qnp ) and p
Hψ B˙ q,λ (Qn )→ B˙ q,λ (Qn ) p p
≤
Z∗p
|t|nλ p ψ(t)dt.
(2.5)
p On the other hand, suppose that Hψ is bounded on B˙ q,λ (Qnp ). Take f 0 (x) = |x|nλ p , then
1 1+λq |Bγ | H
| f 0 (x)|q dx = p −nγ (1+λq) Bγ
γ
p nkλq dx
k=−∞ Sk
= (1 − p −n ) p −nγ (1+λq)
γ
(2.6)
p nk(1+λq)
k=−∞
=
1 − p −n , 1 − p −n(1+λq)
where the series converge due to λ > −1/q. Thus, f 0 ∈ B˙ q,λ (Qnp ). Then by (2.3), we can see that p |t|nλ (2.7) p ψ(t)dt ≤ Hψ B˙ q,λ (Qn )→ B˙ q,λ (Qn ) < ∞. p
Z∗p
p
And (2.5) and (2.7) yield the desired result.
Proof of Theorem 2.5 Suppose that (2.1) holds and f ∈ CMOq,λ (Qnp ). Let γ ∈ Z and denote t Bγ = B(0, |t| p p γ ); by Fubini theorem and change of variable, we have p (Hψ
f ) Bγ
1 = f (t x)ψ(t)dt dx |Bγ | H Bγ Z∗p
1 = f (t x)dx ψ(t)dt Z∗p |Bγ | H Bγ f t Bγ ψ(t)dt. = Z∗p
123
(2.8)
Weighted p-Adic Hardy Operators and Their Commutators…
Using Minkowski’s inequality, we get
1+λq
|Bγ | H =
1
1
q
p
Bγ
1 1+λq
|Bγ | H
≤
Z∗p
=
Bγ
1 1+λq
p
|Hψ f (x) − (Hψ f ) Bγ |q dx
|Bγ | H
q 1 q ( f (t x) − f t Bγ )ψ(t)dt dx Z∗p
1 q f (t x) − f t B q dx ψ(t)dt
1 1+λq
|t Bγ | H
Z∗p
γ
Bγ
≤ f CMOq,λ (Qn ) p
t Bγ
Z∗p
f (y) − f t B q dy γ
1 q
|t|nλ p ψ(t)dt
|t|nλ p ψ(t)dt.
p
Therefore, Hψ is bounded on CMOq,λ (Qnp ) and p
Hψ CMOq,λ (Qn )→CMOq,λ (Qn ) p p
≤
|t|nλ p ψ(t)dt.
Z∗p
(2.9)
p
Conversely, if Hψ is bounded on CMOq,λ (Qnp ), take f 0 (x) = |x|nλ p ; from (2.6) we can see that f 0 ∈ B˙ q,λ (Qnp ). Recall that B˙ q,λ (Qnp ) is continuously embedded in CMOq,λ (Qnp ). Therefore, f 0 ∈ CMOq,λ (Qnp ). By (2.3) and (2.8), we get
1+λq
|Bγ | H =
1
1
q
Bγ
1 1+λq
|Bγ | H
p p |Hψ f 0 (x) − (Hψ f 0 ) Bγ |q dx
Bγ
f 0 − ( f 0 ) B q dx γ
1 q Z∗p
|t|nλ p ψ(t)dt.
Therefore,
p
Hψ f 0 CMOq,λ (Qn ) = f 0 CMOq,λ (Qn ) p
p
Z∗p
|t|nλ p ψ(t)dt,
which implies that p
Hψ CMOq,λ (Qn )→CMOq,λ (Qn ) p p
≥
Z∗p
|t|nλ p ψ(t)dt,
(2.10)
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Q. Y. Wu, Z. W. Fu
and Z∗p
|t|nλ p ψ(t)dt < ∞.
This completes the theorem.
3 Characterizations of Weight Functions Via Commutators Recently, commutators of operators have been paid much attention due to their important applications. For example, some function spaces can be characterized in terms of commutators [10]. In this section, we consider the boundedness for commutators generated by H p and λ-central BMO functions on p-adic central Morrey spaces. Definition 3.1 The commutator between a function b that is locally integrable on Qnp p and the weighted p-adic Hardy operator Hψ is defined by p,b
p
p
Hψ f = bHψ f − Hψ (b f ),
(3.1)
for some suitable functions f . We establish the following sufficient and necessary condition for weight functions to ensure that the commutators generated by weighted p-adic Hardy operators and p-adic central BMO functions are bounded on p-adic central Morrey spaces. Theorem 3.2 Let 1 < q < q1 < ∞, 1/q = 1/q1 + 1/q2 and −1/q1 ≤ λ < 0. Assume that ψ is a positive integrable function on Z∗p . Then for any b ∈ CMOq2 (Qnp ), p,b the commutator H is bounded from B˙ q1 ,λ (Qnp ) to B˙ q,λ (Qnp ) if and only if ψ
Z∗p
ψ(t)|t|nλ p log p
1 dt < ∞. |t| p
(3.2)
Remark 3.3 Since ψ : Z∗p → [0, +∞) is integrable, and log p |t|1p ≥ 1 for |t| p ≤ p −1 , we have nλ ψ(t)|t|nλ dt = ψ(t)|t| dt + ψ(t)|t|nλ p p p dt Z∗p
0<|t| p ≤ p −1
|t| p =1
1 dt + ψ(t)dt ≤ |t| p 0<|t| p ≤ p −1 Z∗p 1 ψ(t)|t|nλ log dt + ψ(t)dt < ∞, = p p |t| p Z∗p Z∗p
ψ(t)|t|nλ p log p
if 3.2 holds.
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Weighted p-Adic Hardy Operators and Their Commutators…
Corollary 3.4 Let 1 < q < q1 < ∞, 1/q = 1/q1 + 1/q2 , −1/q1 ≤ λ < 0 and 0 < α < 1. Then for any b ∈ CMOq2 (Q p ) (I) the commutator H p,b is bounded from B˙ q1 ,λ (Q p ) to B˙ q,λ (Q p ). p,b (II) the commutator Rα is bounded from B˙ q1 ,λ (Q p ) to B˙ q,λ (|x|−αq dx). For any q n b ∈ CMO 2 (Q p ), (III) the commutator H p,b is bounded from B˙ q1 ,λ (Qnp ) to B˙ q,λ (Qnp ), where B˙ q1 ,λ (Qnp ) is defined in Corollary 2.4. When b ∈ CMOq,λ (Qnp ) with λ = 0, we have the following conclusion. Theorem 3.5 Let 1 < q < q1 < ∞, 1/q = 1/q1 + 1/q2 , −1/q < λ < 0, −1/q1 < λ1 < 0, 0 < λ2 < n1 and λ = λ1 + λ2 . If Z∗p
1 ψ(t)|t|nλ p dt < ∞,
(3.3)
then for any b ∈ CMOq2 ,λ2 (Qnp ), the corresponding commutator Hψ from B˙ q1 ,λ1 (Qnp ) to B˙ q,λ (Qnp ).
p,b
is bounded
We have obtained the values of Z∗ ψ(t)|t|nλ p dt in Corollaries 2.2, 2.4 and 2.6, Thus p by Theorem 3.5, we obtain the following result. Corollary 3.6 Let 1 < q < q1 < ∞, 1/q = 1/q1 + 1/q2 , −1/q < λ < 0, −1/q1 < λ1 < 0. (I) If 0 < λ2 < 1, λ = λ1 + λ2 and 0 < α < 1, then for any b ∈ CMOq2 ,λ2 (Q p ), (i) the commutator H p,b is bounded from B˙ q1 ,λ1 (Q p ) to B˙ q,λ (Q p ). p,b (ii) the commutator Rα is bounded from B˙ q1 ,λ1 (Q p ) to B˙ q,λ (|x|−αq dx). (II) If 0 < λ2 < n1 and λ = λ1 +λ2 , then for any b ∈ CMOq2 ,λ2 (Qnp ), the commutator H p,b is bounded from B˙ q1 ,λ1 (Qnp ) to B˙ q,λ (Qnp ), where B˙ q1 ,λ (Qnp ) is defined in Corollary 2.4. Before proving these theorems, we need the following result. One can refer to (Lemma 15 in [31]) for another version; here, we give a more accurate estimation. Lemma 3.7 Suppose that b ∈ CMOq,λ (Qnp ) and j, k ∈ Z. (I). If λ > 0, then |b B j − b Bk | ≤
p n (1 + p −|k− j|nλ )
b CMOq,λ (Qn ) max |B j |λH , |Bk |λH . −nλ p 1− p
(II). If λ = 0, then |b B j − b Bk | ≤ p n | j − k| b CMOq (Qnp ) .
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Q. Y. Wu, Z. W. Fu
Proof Without loss of generality, we may assume that k > j. Recall that b Bi = 1 |Bi | H Bi b(x)dx. By Hölder’s inequality, we have |b Bi+1 − b Bi | ≤ ≤ ≤
1 |Bi | H 1 |Bi | H
Bi
1 |b(x) − b Bi+1 |dx |Bi | H Bi+1 1 q 1− 1 |b(x) − b Bi+1 |q dx |Bi+1 | H q
|b(x) − b Bi+1 |dx ≤
Bi+1 1+λ |Bi+1 | H
b CMOq,λ (Qn ) = p n |Bi+1 |λH b CMOq,λ (Qn ) .
|Bi | H
p
p
Therefore, if λ > 0, then |b B j − b Bk | ≤
k−1
|b Bi+1 − b Bi | ≤ p n b CMOq,λ (Qn )
k−1
p
i= j
|Bi+1 |λH
i= j
p n 1 + p −(k− j)nλ =
b CMOq,λ (Qn ) |Bk |λH . p 1 − p −nλ If λ = 0, then |b B j − b Bk | ≤
k−1
|b Bi+1 − b Bi | = (k − j) p n b CMOq (Qnp ) .
i= j
Proof of Theorem 3.2 Let γ ∈ Z and denote t Bγ = B(0, |t| p p γ ). Assume that (3.2) holds; by definition, we have
1 q 1 p,b q |H f (x)| dx ψ 1+λq Bγ |Bγ | H
q q1 1 ≤ |(b(x) − b(t x)) f (t x)|ψ(t)dt dx 1+λq Bγ Z∗p |Bγ | H
q q1 1 ≤ |(b(x) − b Bγ ) f (t x)|ψ(t)dt dx 1+λq Bγ Z∗p |Bγ | H
q q1 1 + |(b Bγ − bt Bγ ) f (t x)|ψ(t)dt dx 1+λq Bγ Z∗p |Bγ | H
q q1 1 + |(b(t x) − bt Bγ ) f (t x)|ψ(t)dt dx 1+λq Bγ Z∗p |Bγ | H := I1 + I2 + I3 . (3.4)
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Weighted p-Adic Hardy Operators and Their Commutators…
In the following, we will estimate I1 , I2 , and I3 , respectively. For I1 , by Minkowski’s inequality and Hölder’s inequality (1/q = 1/q1 + 1/q2 ), we get
I1 ≤
≤
1+λq
− 1 −λ |Bγ | H q
q
|(b(x) − b Bγ ) f (t x)| dx
ψ(t)dt
q
|Bγ | H
Z∗p
1
1
Bγ
Z∗p
Bγ
≤ b CMOq2 (Qnp )
|b(x) − b Bγ |q2 dx
p
| f (t x)|q1 dx Bγ
Z∗p
q1
ψ(t)dt
1
q1
| f (y)| dy q1
1+λq1
≤ b CMOq2 (Qnp ) f B˙ q1 ,λ (Qn )
1
1 |t Bγ | H
Z∗p
1 q2
t Bγ
|t|nλ p ψ(t)dt
|t|nλ p ψ(t)dt.
(3.5)
Similarly, we have
I3 ≤
≤
Z∗p
×
q
|(b(t x) − bt Bγ ) f (t x)| dx q
1+λq
Z∗p
1
1 |Bγ | H
1 |t Bγ | H
Bγ
1
t Bγ
|b(y) − bt Bγ |q2 dy q1
| f (y)|q1 dy
1+λq |t Bγ | H 1
q2
(3.6)
1
1
ψ(t)dt
t Bγ
|t|nλ p ψ(t)dt
≤ b CMOq2 (Qnp ) f B˙ q1 ,λ (Qn ) p
Z∗p
|t|nλ p ψ(t)dt.
For I2 , by Minkowski’s inequality and Hölder’s inequality (1/q = 1/q1 + 1/q2 ), we have
I2 ≤
Z∗p
≤
Z∗p
=
Z∗p
1+λq
|Bγ | H
Bγ
|Bγ | H
| f (t x)|q1 dx Bγ
| f (y)| dy q1
1+λq |t Bγ | H 1
Z∗p
t Bγ
q1
|b Bγ − bt Bγ |ψ(t)dt
1
1
p
|b Bγ − bt Bγ |ψ(t)dt
1
1
≤ f B˙ q1 ,λ (Qn )
q
| f (t x)| dx q
1+λq1
1
1
q1
|t|nλ p |b Bγ − bt Bγ |ψ(t)dt
|t|nλ p |b Bγ − bt Bγ |ψ(t)dt.
(3.7)
123
Q. Y. Wu, Z. W. Fu
Note that t ∈ Z∗p ; thus, |t| p ≤ 1. By Lemma 3.7 for λ = 0, we get
|b Bγ − bt Bγ | =
γ −1
1 |t| p
|b Bk+1 − b Bk | ≤ p n b CMOq2 (Qnp ) log p
k=γ +log p |t| p
Therefore, I2 ≤ p n b CMOq2 (Qnp ) f B˙ q1 ,λ (Qn ) p
Z∗p
ψ(t)|t|nλ p log p
1 dt. |t| p
(3.8)
By (3.2) and Remark 3.3 and then combining with the inequalities (3.5)–(3.8), we obtain p,b
Hψ B˙ q,λ (Qn )→ B˙ q,λ (Qn ) p p ≤ C b CMOq2 (Qnp ) ψ(t)|t|nλ + p Z∗p
Z∗p
ψ(t)|t|nλ p log p
1 dt |t| p
< ∞.
p,b On the other hand, suppose that Hψ is bounded from B˙ q,λ (Qnp ) to B˙ q,λ (Qnp ) and q n 2 b ∈ CMO (Q p ), we will show that (3.2) holds. In fact, take b0 (x) = log p |x| p , x ∈ Qnp . From Lemma 2.1 in [22], we can see that b0 ∈ BMO(Qnp ). By Corollary 5.17 in [18], · BMO(Qnp ) and · BMOq (Qnp ) are equivalent. Therefore, b0 (x) ∈ BMOq2 (Qnp ) ⊂ CMOq2 (Qnp ). By assumption, we have
p,b0
Hψ
B˙ q,λ (Qn )→ B˙ q,λ (Qn ) < ∞. p
p
(3.9)
˙ q,λ (Qnp ), and We will also take f 0 (x) = |x|nλ p , from (2.6) we can see that f 0 ∈ B
f 0 B˙ q,λ (Qn ) = p
1− p −n . 1− p −n(1+λq)
p,b Hψ 0 f 0 (x)
Since
= =
Z∗p Z∗p
(b0 (x) − b0 (t x)) f 0 (t x)ψ(t)dt (log p |x| p − log p |t x| p )|t x|nλ p ψ(t)dt
= f 0 (x)
123
Z∗p
ψ(t)|t|nλ p log p
1 dt. |t| p
Weighted p-Adic Hardy Operators and Their Commutators…
Using Hölder’s inequality (1 = q/q1 + q/q2 ), we have p,b0
Hψ
f 0 B˙ q,λ (Qn ) p
1 q 1 p,b0 q = sup |H f (x)| 0 ψ 1+λq Bγ γ ∈Z |Bγ | H
1 q 1 1 q = sup | f (x)| ψ(t)|t|nλ dt 0 p log p 1+λq |t| p Bγ Z∗p γ ∈Z |Bγ | H 1 − p −n 1 = ψ(t)|t|nλ dt p log p −n(1+λq) |t| p 1− p Z∗p 1 1 − p −n(1+λq1 ) 1 − p −n = × ψ(t)|t|nλ dt p log p −n(1+λq) −n(1+λq ) 1 |t| p 1− p 1− p Z∗p 1 ψ(t)|t|nλ dt. = Cq,q1 f 0 B˙ q1 ,λ (Qn ) p log p p ∗ |t| p Zp
Therefore, p,b0
Hψ
B˙ q,λ (Qn )→ B˙ q,λ (Qn ) ≥ Cq,q1 p
p
Z∗p
ψ(t)|t|nλ p log p
1 dt. |t| p
Then by (3.9), we obtain Z∗p
ψ(t)|t|nλ p log p
1 dt < ∞. |t| p
The proof is complete. Proof of Corollary 3.4 (1) When ψ ≡ 1 and n = 1, we have Hψ f = H p f . Since Z∗p
|t|λp1
∞
1 log p dt = |t| p k=0
S−k
|t|λp1 log p
= (1 − p −1 )
∞
1 dt |t| p
kp −k(1+λ1 ) < ∞.
k=0
We can get Corollary 3.4 (I) directly from Theorem 3.2. α−1 ), then (2) For n = 1, if we take ψ(t) = (1 − p −α )|1 − t|α−1 p χ B0 \S0 (t)/(1 − p p p Hψ f (x) = |x|−α p Rα f (x). At this time
123
Q. Y. Wu, Z. W. Fu
Z∗p
ψ(t)|t|λp1 log p
1 1 − p −α dt = |t| p 1 − p α−1 1 − p −α = 1 − p α−1
0<|t| p <1
0<|t| p <1
|t|λp1 |1 − t|α−1 log p p |t|λp1 log p
1 dt |t| p
1 dt |t| p
∞
=
(1 − p −α )(1 − p −1 ) −k(1+λ1 ) kp < ∞. 1 − p α−1 k=1
Then Corollary 3.4 (II) follows from Theorem 3.2. −1 (3) For n ≥ 2, if we take ψ(t) = (1 − p −n )|t|n−1 p /(1 − p ), and f satisfies p −1 p f (x) = f (|x| p ), then Hψ f (x) = H f (x), and Z∗p
ψ(t)|t|λp1 log p
1 1 − p −n dt = |t| p 1 − p −1
Z∗p
= (1 − p −n )
1 )n−1 log |t|(1+λ p p
∞
1 dt |t| p
kp −k(1+λ1 )n < ∞.
k=0
Therefore, Corollary 3.4 (III) holds. Proof of Theorem 3.5 As in the proof of Theorem 3.2, we can write
1
1 1+λq
|Bγ | H
q
Bγ
p,b |Hψ
f (x)| dx
= I1 + I2 + I3 ,
q
where I1 , I2 , I3 are the ones in (3.4). By the similar estimates to (3.5) and (3.6), we have I1 ≤ b CMOq2 ,λ2 (Qn ) f B˙ q1 ,λ1 (Qn ) p
p
I3 ≤ b CMOq2 ,λ2 (Qn ) f B˙ q1 ,λ1 (Qn ) p
p
Z∗p Z∗p
1 |t|nλ p ψ(t)dt,
1 |t|nλ p ψ(t)dt.
For I2 , like (3.7), we have I2 ≤
1 |Bγ |λH2
f B˙ q1 ,λ1 (Qn ) p
Z∗p
1 |t|nλ p |b Bγ − bt Bγ |ψ(t)dt.
By Lemma 3.7 and the fact that if t ∈ Z∗p then log p |t| p is a nonnegative integer, we get
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Weighted p-Adic Hardy Operators and Their Commutators…
p n (1 + p nλ2 log p |t| p )
b CMOq2 ,λ2 (Qn ) |Bγ |λH2 p 1 − p −nλ2 n 2p ≤
b CMOq2 ,λ2 (Qn ) |Bγ |λH2 p 1 − p −nλ2
|b Bγ − bt Bγ | = |b Bγ − b Bγ +log p |t| p | ≤
Consequently, 2 pn I2 ≤
b CMOq2 ,λ2 (Qn ) f B˙ q1 ,λ1 (Qn ) p p 1 − p −nλ2
Z∗p
1 ψ(t)|t|nλ p dt.
The estimates of I1 , I2 , I3 imply that
p,b
Hψ f B˙ q,λ (Qn ) ≤ C b CMOq2 ,λ2 (Qn ) f B˙ q1 ,λ1 (Qn ) p
Theorem 3.5 is proved.
p
p
Z∗p
1 ψ(t)|t|nλ p dt.
Acknowledgements The authors would like to express their sincere gratitude to the anonymous referees for their valuable comments and suggestions.
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