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The p -adic weighted Hardy–Cesàro operator and an application to discrete Hardy inequalities
J. Math. Anal. Appl. 409 (2014) 868–879
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The p-adic weighted Hardy–Cesàro operator and an application to discrete Hardy inequalities Ha Duy Hung Hanoi National University of Education, Hanoi, Viet Nam
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Article history: Received 22 April 2013 Available online 30 July 2013 Submitted by Jie Xiao
This paper aims to investigate the boundedness of the p-adic analog of the weighted Hardy–Cesàro operator Uψ,s : f → Z⋆ f (s(t )·) ψ(t )dt on weighted Lebesgue spaces and p
Keywords: Weighted Hardy–Cesàro operator Discrete Hardy inequalities BMO Commutator p-adic analysis
weighted BMO spaces. In each case, we obtain the corresponding operator norms |Uψ,s |. In particular, these results have a surprising relevance to discrete Hardy inequalities on the real field. We prove a reverse BMO–Hardy inequality on ψ and give a necessary condition so that the commutator of Uψ,s is bounded on Lrω Qnp with symbols in BMOω Qnp . © 2013 Elsevier Inc. All rights reserved.
1. Introduction Theories of functions from Qnp into R or C play an important role in p-adic quantum mechanics, the theory of p-adic probability in which real-valued random variables have to be considered to solve covariance problems [19,20]. Studies of the p-adic Hardy operators and p-adic Hausdorff operators are also useful for p-adic analysis [3,4,7,13,18,21–23]. In this paper, we shall give an interesting relation between p-adic Hardy operators and discrete Hardy inequalities on the real field. Hausdorff operators appeared long ago, aiming to solve certain classical problems in analysis. The modern theory of Hausdorff operators started with the work of Siskakis in complex analysis setting and with the work of Georgakis [9], Liflyand and Moricz [17] in the Fourier transform setting. One of the most general definitions of the Hausdorff operator is
H Φ ,A f ( x ) =
Rn
Φ (u)f (xA(u)) du,
n
where Φ is a measurable function on Rn and A = A(u) = aij (u) i,j=1 is the n × n matrix with the entries aij (u) being measurable functions of u. The properties of these operators were further studied in [11,12,10,17,1,16] and the references therein. In 1984, Carton-Lebrun and Fosset [2] considered a Hausdorff operator of a special kind, which is called the weighted Hardy operator Uψ :
Uψ f (x) =
1
f (tx)ψ(t )dt ,
x ∈ Rn .
(1)
0
The authors showed the boundedness of Uψ on Lebesgue spaces and BMO(Rn ) space. In 2001, J. Xiao [24] obtained that Uψ is bounded on Lp (Rn ) if and only if 1
t −n/p ψ(t )dt < ∞.
0
E-mail address: [email protected]. 0022-247X/$ – see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jmaa.2013.07.056
H.D. Hung / J. Math. Anal. Appl. 409 (2014) 868–879
869
Meanwhile, the corresponding operator norm was worked out. The result seems to be of interest as it is related closely to the classical Hardy integral inequality. Also, Xiao obtained the BMO(Rn )-bounds of Uψ , which sharpened and extended the main result of Carton-Lebrun and Fosset in [2]. In 2006, Rim and Lee [18] proved analogous results to those of Xiao on p-adic fields. They introduced the p-adic form of Uψ as follows: Uψ f (x) =
Z⋆p
f (tx)ψ(t )dt ,
x ∈ Qnp .
(2)
Here, Qp is the field of all p-adic numbers and Z⋆p = {x ∈ Qp : 0 < |x|p ≤ 1}. In 2010, Volosivets [21] introduced p-adic analogs of the Hausdorff operator and gave sufficient conditions for the boundedness of p-adic Hausdorff operator in p-adic Hardy and BMO spaces. Also, in [22,23], the author studied the boundedness of Uψ as in (2) in BMO-type p-adic spaces, Morrey p-adic spaces and Herz p-adic spaces. In 2012, Chuong and the author of this paper considered a weighted Hardy–Cesàro operator [5], a more general form of Uψ in the real case as follows. Definition 1.1. Let ψ : [0, 1] → [0, ∞) and s : [0, 1] → R be measurable functions. The weighted Hardy–Cesàro operator Uψ,s , associated to the parameter curve s(x, t ) := s(t )x, is defined by Uψ,s f (x) =
1
f (s(t )x) ψ(t )dt ,
(3)
0
for all measurable complex valued functions f on Rn . With certain conditions on functions s and ω, the authors of [5] proved that Uψ,s is bounded on weighted Lebesgue spaces and weighted BMO spaces. The corresponding operator norms are worked out too. The authors also give a necessary condition on the weight function ψ for the boundedness of the commutators of operator Uψ,s on Lrω (Rn ) with symbols in BMOω (Rn ). The form of the Hardy–Cesàro operator in p-adic analysis is Uψ,s f (x) =
Z⋆p
f (s(t )x) ψ(t )dt ,
where s : Z⋆p → Qp and ψ : Z⋆p → [0; ∞) are measurable functions. This paper aims to study the bounds of the p-adic Hardy–Cesàro operator on weighted Lebesgue spaces and weighted BMO spaces. The main motivations for our results are the works in [3–5,8,24,18,21–23]. We prove the continuities of the weighted p-adic Hardy–Cesàro operator Uψ,s on weighted Lebesgue spaces and weighted BMO spaces. The norms of the operator are worked out too. Here we note that our results are somewhat different from those in [21–23], since they hold also when s(t ) ̸= t and for certain weighted functional spaces. It is interesting to note that the results allow us to obtain a discrete Hardy inequality over the real field, and thus there is a relation between the behavior of the p-adic Hardy–Cesàro operator Uψ,s in functional spaces and discrete Hardy inequalities over the real field. We also obtain a reverse BMO–Hardy inequality and give a necessary condition on weight functions such that the commutator of Uψ,s with symbols in weighted BMO is bounded on weighted Lebesgue spaces. 2. Preliminaries Let p be a prime in Z, and let r ∈ Q⋆ . Write r = pγ ba , where a and b are integers not divisible by p. Define the p-adic absolute value |·|p on Q by |r |p = p−γ and |0|p = 0. The absolute value |·|p gives a metric on Q defined by dp (x, y) = |x − y|p . We denote by Qp the completion of Q with respect to the metric d. Qp with natural operations and topology induced by the metric dp is a locally compact, non-discrete, complete and totally disconnected field. A non-zero element x of Qp is uniquely represented as a canonical form x = pγ x0 + x1 p + x2 p2 + · · · , where xj ∈ Z/pZ and x0 ̸= 0. We then have |x|p = p−γ . Define Zp = {x ∈ Qp : |x|p ≤ 1} and Z⋆p = Zp \ {0}. Qnp = Qp × · · · × Qp contains all n-tuples of Qp . The norm on Qnp is |x|p = max1≤k≤n |xk | for x = (x1 , . . . , xn ) ∈ Qnp . The space Qnp is a complete metric locally compact and totally disconnected space. For each a ∈ Qp and x = (x1 , . . . , xn ) ∈ Qnp , we denote ax = (ax1 , . . . , axn ). For γ ∈ Z, we denote Bγ as a γ -ball of Qnp with center at 0, containing all x with |x|p ≤ pγ , and with Sγ = Bγ \ Bγ −1 its boundary. Also, for a ∈ Qnp , Bγ (a) consists of all x with x − a ∈ Bγ , and Sγ (a) consists of all x with x − a ∈ Sγ . Since Qnp is a locally compact commutative group with respect to addition, there exists the Haar measure dx on the additive group of Qnp normalized by
B0
dx = 1. Then d(ax) = |a|np dx for all a ∈ Q⋆p , |Bγ (x)| = pnγ , and Sγ (x) =
pnγ 1 − p−n . Let ω be any weight function on Qnp , that is, a non-negative locally integrable function from Qnp into R. Let Lrω Qnp (1 ≤ r < ∞) be the space of complex-valued functions f on Qnp such that
1/r
∥f ∥Lrω (Qnp ) =
|f (x)| ω(x)dx r
Qnp
< ∞.
870
H.D. Hung / J. Math. Anal. Appl. 409 (2014) 868–879
∞
Also, Lω
n
Qp consists of the set of all measurable real-valued functions f on Qnp satisfying
n := inf C > 0 : ω x ∈ Qnp : |f (x)| > C = 0 < ∞. ∥f ∥L∞ ω (Qp ) Here, ω(A) := A ω(x)dx for any measurable subset A of Qnp .
Some often used computational principles are worth mentioning at the outset. First, if f ∈ L1ω Qp , we can write
Qnp
f (x)ω(x)dx =
γ ∈Z
f (y)ω(y)dy. Sγ
Second, we also often use the fact that
Qnp
f (ax) dx =
1
|a|np
Qnp
f (x)dx,
if a ∈ Qnp \ {0} and f ∈ L1 (Qnp ).
The p-adic BMOω Qnp space of all functions f on Qnp which are bounded mean oscillations with weight ω is
1
∥f ∥BMOω (Qnp ) = sup
ω(B)
B
|f (x) − fB,ω |ω(x)dx < ∞,
where the supremum is taken over all balls B of Qnp . Here, ω(B) = weight ω: fB,ω =
1
ω(B)
(4)
B
B
ω(x)dx, and fB,ω is the mean value of f on B with
f (x)ω(x)dx.
(5)
B
We note that there are other p-adic BMO spaces which are useful in studying harmonic analysis by way of Brownian motion (see [14]). Also, in [14], Krantz determined the relation with ordinary continuous space BMO, and it is surprising that the results are related to information about the distribution of primes. Let s : Z⋆p → Qp and ψ : Z⋆p → R+ be measurable functions, and let ω : Qnp → R+ be a locally integrable function. For a function f on Qnp , we define the weighted p-adic Hardy–Cesàro operator Uψ,s on Qnp as Uψ,s f (x) =
Z⋆p
f (s(t )x) ψ(t )dt .
(6)
We shall consider the class of weights Wα , which consists of all non-negative locally integrable functions ω on Qnp such that
ω(tx) = |t |αp ω(x) for all x ∈ Qnp and t ∈ Q⋆p and 0 < α > −n.
S0
ω(x)dx < ∞. It is easy to see that ω(x) = |x|αp is in Wα if and only if
3. Main results β
Theorem 3.1. Suppose that r ∈ [1; ∞], α ∈ R, and ω ∈ Wα . Let s : Z⋆p → Qp be a measurable function such that |s(t )|p ≥ |t |p for some real number β and almost everywhere t ∈ Z⋆p . Then, Uψ,s is bounded on Lrω Qp if and only if
− n+α r
Z⋆p
|s(t )|p
n
ψ(t )dt < ∞.
Moreover,
∥Uψ,s ∥ ( ) n ω Qp
Lr
( )=
n ω Qp
→Lr
− n+α r
Z⋆p
|s(t )|p
ψ(t )dt .
(7)
When s(t ) = t and ω(x) ≡ 1, we obtain Theorem 1.1 in [18]. The boundedness of Hardy operators can be deduced from the so-called Hardy inequalities in both their discrete and continuous forms. Here we shall show that the discrete form of Hardy inequalities can be deduced from the p-adic continuous form of Hardy operators. Indeed, from Theorem 3.1, we can deduce the following surprising inequality. Corollary 3.2. Let (xj )j∈Z and (yk )k≥0 be two non-negative sequences. For any non-negative integer β , and for any 1 ≤ r < ∞, the following Hardy inequality holds:
∞ j∈Z
k=0
r 1/r xj+β k yk
1/r
≤
j∈Z
xrj
∞ k=0
yk
.
(8)
H.D. Hung / J. Math. Anal. Appl. 409 (2014) 868–879
871
To obtain a discrete Hardy inequality from the boundedness of p-adic Hardy operators, our main idea is using special radical functions and computational principles for Haar integrals. Thus, this method can be applied to derive a wide class of discrete Hardy-type inequalities. The boundedness of Uψ,s on weighted BMO is studied in the following theorem. Theorem 3.3. Let r ∈ [1; ∞] be a real number, and let ω belong to W = such that s(t ) ̸= 0 almost everywhere on Z⋆p .
α>−n
Wα . Let s : Z⋆p → Qp be a measurable function
(i) Uψ,s : BMOω Qnp → BMOω Qnp exists as a bounded operator if and only if
Z⋆p
ψ(t )dt < ∞.
(9)
(ii) If one of the following conditions holds: ◦ n = 1 and both integrals Z⋆ log |s(t )|p ψ(t )dt and Z⋆ ψ(t )dt are finite, or p
◦ n > 1 and
Z⋆p
p
ψ(t )dt < ∞,
then Uψ,s is bounded on BMOω Qnp , and
∥Uψ,s ∥
( )=
( )
BMOω Qnp
→BMOω Qnp
Z⋆p
ψ(t )dt .
(10)
Up to now, in the one-dimensional case, it remains unknown if the condition
Z⋆p
ψ(t )dt < ∞ is enough or not to
imply (10). In the last argument in the proof of Theorem 2 in [18], the authors use the wrong argument: they suppose that ψ( t ) dt < ∞ implies that log |t |p ψ(t )dt < ∞. Indeed, we can find an easy counterexample, as the following. Let ⋆ Z Z⋆ p
p
ψ(t ) =
1
|t |p (log |t |p )
2
. Then an easy computation gives us
Z⋆p
ψ(t )dt < ∞, but
Z⋆p
log |t |p ψ(t )dt = ∞.
As we shall see in the proof of Theorem 3.1, for the case n = 1 it seems hard to compute the norm of Uψ,s if we have only the condition that Z⋆ ψ(t )dt is finite. In the proof, we use a test function f1 (x) = log |x|p , which is a homomorphism p
from Q⋆p into R. Let us denote by hom Q⋆p , R the set of all homomorphisms from multiplication group (Q⋆p , ·) to addition group (R, +). It is known that Qp is a direct product of three subgroups: the infinite cyclic subgroup generated by p, the cyclic subgroup of order p − 1 consisting of non-zero elements which are generated by ζ , a primitive root of unity of order p − 1 and the group 1 + pZp of p-adic integers congruent to 1. Thus, it is easy to see that all homomorphisms from Q⋆p into R must have the following form:
f pj ζ k u = ja + g (u),
where g is a homomorphism from 1 + pZp into R. Using the usual arguments as for the Cauchy functional equation, by Zorn’s lemma (see [15]), we see that there are infinitely many such g. Let BMO ω,hom consist of all f ∈ BMOω Q⋆p such that f ∈ hom Q⋆p , R . Then log |x|p is in BMO ω,hom (see Lemma 6.1). It seems hard to find other non-trivial examples. Using the proof of Theorem 3.1, we obtain the following. Theorem 3.4. Let r , ω(x), s(t ) be as in Theorem 3.3. If there exists a g ∈ BMO ω,hom such that both integrals and
Z⋆p
ψ(t )dt are finite, then (10) holds for n = 1.
Z⋆p
g (s(t )) ψ(t )dt
One of the ways to derive (10) is to find the reverse inequality, that is, ∥Uψ,s f ∥BMO ≥ C ∥f ∥BMO for f in some class functions. In Euclid’s case, Xiao [25] showed that the reverse BMO–Hardy inequality
∥Pf ∥BMO(R+ ) ≥
1 17
∥f ∥BMO(R+ ) x
is valid for each positive decreasing function f ∈ BMO(R+ ) on R+ . Here, Pf (x) = 1x 0 f (t )dt. Since a decreasing function on Qnp makes no sense, it is interesting to ask if there exists such a reverse inequality on p-adic fields. To state our result, let us denote Sp by the set of all non-negative sequences (ak )k≥0 such that a
• supk≥0 ka+k 1 = p (1 − a0 ) + a0 . ak p • ∞ k=0 pk = p−1 . It is obvious to see that (1, 1, 1, . . . , 1, . . .) is in Sp . By a simple argument based on continuity, we can show that Sp contains infinitely many such sequences. Our result on the reverse BMO–Hardy inequality is the following.
872
H.D. Hung / J. Math. Anal. Appl. 409 (2014) 868–879
Theorem 3.5. Let ψ be a function on Z⋆p such that ψ(t ) = φ |t |p and the sequence φ p−k inequality holds:
∥Uψ f ∥BMO(Qp ) ≥
(p − 1)φ(1) 7p − (p − 1)φ(1)
k≥0
∈ Sp . Then, the reverse BMO
∥f ∥BMO(Qp )
(11)
for all decreasing radical functions f (x) = g (|x|p ). Here, Uψ is defined as in (2). (p−1)φ(1)
(p−1)φ(1)
Since φ(1) ≤ p−1 , 7p−(p−1)φ(1) < 1. We conjecture that the constant 7p−(p−1)φ(1) can be replaced by 1. We note that, in Theorem 4 in [23], the author also proved an interesting reverse BMO inequality for the modified Hardy–Littlewood operator B f (x) := |x1| |t | ≤|x| f (t )dt. p
p
p
p
More recently, great attention was paid to the study of commutators of operators. A well-known result of Coifman, Rochberg, and Weiss [6] states that the commutator Tb f = bTf − T (bf ) (where T is a Calderón–Zygmund singular integral operator) is bounded on Lp (Rn ), 1 < p < ∞, if and only if b ∈ BMO(Rn ). Many results have been generalized to commutators of other operators, not only Calderón–Zygmund singular integral operators. The definition of the weighted Hardy–Cesàro commutator is as follows: b Uψ, s f := bUψ,s f − Uψ,s (bf ).
In the real case, Chuong and the author of this paper [5] gave a necessary condition on ψ so that the commutator of Hardy–Cesàro operator is bounded on weighted Lebesgue spaces with symbols in weighted BMO spaces. This result can be extended to the p-adic case as follows. Theorem 3.6. Let r ∈ [1; ∞] be a real number, and let ω belong to W = α>−n Wα , which has doubling property. n Let b r s : Z⋆p → Qp be a measurable function such that s(t ) ̸= 0 almost everywhere on Z⋆p . Assume that Uψ, is bounded on L ω Qp for s
any b ∈ BMOω Qnp . β
(a) If |s(t )|p ≥ |t |p for almost everywhere t ∈ Z⋆p for some β > 0, then
− n+α r · log |s(t )|p · ψ(t )dt < ∞. |s(t )|p Z⋆p
(12)
(b) If 0 < |s(t )|p ≤ |t |θp for almost everywhere t ∈ Z⋆p for some positive integer θ > 0, then
0<|t |p ≤ 1p
b ψ(t )dt ≤ ∥Uψ, s ∥Lrω (Qnp )→Lrω (Qnp ) .
(13)
As far as we know, the problem of characterizing the weight functions so that the converse of Theorem 3.6 holds is still open, even in Euclid’s case. 4. Proof of Theorem 3.1 Since the case r = ∞ is trivial, it suffices to consider r ∈ [1; ∞). Suppose that
Z⋆p
− n+α r
|s(t )|p
ψ(t )dt < ∞. For each
f ∈ Lrω Qp , since s(t ) ̸= 0 almost everywhere, ω is homogeneous of order α ; then, applying Minkowski’s inequality for integrals (see [12, page 14]), we obtain
n
r 1/r ∥Uψ,s f ∥Lrω (Qnp ) = f (s(t ) · x) ψ(t )dt ω(x)dx Qnp Z⋆p 1/r r |f (s(t ) · x)| ω(x)dx ≤ ψ(t )dt
Z⋆p
Qnp
1/r
=
Z⋆p
|f (y)| |s(t )|
Qnp
= ∥f ∥ ( ) · n ω Qp
Lr
−α−n
r
− n+α r
Z⋆p
|s(t )|p
ω(y)dy
ψ(t )dt < ∞.
ψ(t )dt
H.D. Hung / J. Math. Anal. Appl. 409 (2014) 868–879
873
(Here we use the change of variable y = s(t )x.) Thus Uψ,s maps boundedly Lrω Qp into itself and
∥Uψ,s ∥Lrω (Qnp )→Lrω (Qnp ) ≤
− n+α r
|s(t )|p
Z⋆p
n
ψ(t )dt .
(14)
Conversely, suppose that Uψ,s is a bounded operator on Lrω Qnp . Then there exists a constant C > 0 such that
for all f ∈ Lrω Qnp .
∥Uψ,s f ∥Lrω (Qnp ) ≤ C ∥f ∥Lrω (Qnp ) ,
For any positive integer γ , we put if |x|p < 1
0
fγ (x) =
− n+α r −
|x|p
1
γ2
(15)
if |x|p ≥ 1.
Then
∥fγ ∥rLr (Qn ) = ω
p
=
=
−n−α−
|x|p
r
γ2
ω(x)dx
| x| p ≥1 ∞
p
−k n+α+
k=0
Sk
∞
−k n+α+
r γ2
p
r
ω(x)dx
γ2
· pkn+kα
1
=
ω(y)dy S0
k=0
1−p
ω(y)dy.
·
r
−
γ2
S0
Thus fγ ∈ Lrω Qnp for each γ . Then we have
− n+α r −
Uψ,s fγ (x) = |x|p
1
− n+α r −
γ2
S (x)
|s(t )|p
1
γ2
ψ(t )dt , β
where S (x) = {t ∈ Z⋆p : |s(t ) · x|p ≥ 1}. For almost everywhere t ∈ Z⋆p , since |s(t )|p ≥ |t |p , we have that |t |p ≥ implies that t ∈ S (x). Consequently, we have
r Lrω Qnp
∥Uψ,s fγ ∥ ( ) ≥
−n−α−
|x|p
r
γ2
S (x)
|x|p ≥1
−n−α−
|x|p
≥ |x|p ≥pγ r Lrω Qnp
− n+α r −
= ∥fγ ∥ ( ) · p
−r /γ
|s(t )|p
ψ(t )dt
r
γ2
r
1
γ2
ω(x)dx
ω(x)dx − n+α r −
1≥|t |p ≥p−γ /β
|s(t )|p
− n+α r −
· 1≥|t |p ≥p−γ /β
1 1/β
|x|p
|s(t )|p
ψ(t )dt
r
1
γ2
r
1
γ2
ψ(t )dt
,
which implies that − n+α r −
1≥|t |p ≥p−γ /β
|s(t )|p
1
γ2
ψ(t )dt ≤ p1/γ ·
∥Uψ,s fγ ∥Lrω (Qnp ) ∥fγ ∥Lrω (Qnp )
≤ C · p1/γ .
Letting γ to infinity, then, by Fatou’s lemma, we obtain
− n+α r
Z⋆p
|s(t )|p
ψ(t )dt ≤ C < +∞.
As we have shown, Uψ,s is bounded on Lrω Qnp , and hence
− n+α r
Z⋆p
|s(t )|p
ψ(t )dt ≤ ∥Uψ,s ∥Lrω (Qnp )→Lrω (Qnp ) .
From (14) and (16), we have
∥Uψ,s ∥Lrω (Qnp )→Lrω (Qnp ) =
− n+α r
Z⋆p
|s(t )|p
ψ(t )dt .
(16)
874
H.D. Hung / J. Math. Anal. Appl. 409 (2014) 868–879
5. Proof of Corollary 3.2
First, we observe that inequality (8) is trivial if
< ∞ and
r j∈Z xj
be assumed that
y k < ∞.
∞
k=0
j∈Z
xrj = ∞ or
∞
k=0
yk = ∞. Thus, without loss of generality, it may
For each t ∈ Z⋆p and x ∈ Qnp , we put s(t ) = t β , ψ(t ) = φ(|t |p ), and ω(x) = |x|αp . We also choose f (x) = g (|x|p ). Here, α > −n, and functions φ and g will be chosen later. We have Uψ,s f (x) =
=
g |t |βp |x|p φ |t |p dt
Z⋆p
g pβ k |x|p φ pk pkn
1−
k≤0
=
−β k
g p
|x|p φ
1
p
pk
k≥0
−kn
1
pn
1−
1
pn
.
Thus,
∥Uψ,s f ∥rLr (Qn ) = ω
p
=
Qnp
Uψ,s f (x)r ω(x)dx
pα j
Uψ,s f (x)r ω(x)dx Sγ
j∈Z
r ∞ j−β k 1 1 jn 1 −kn αj 1− n . 1− n p p g p = φ k p k=0 p p p j∈Z r ∞ 1 1 1 −kn g pj−β k φ p(n+α)j 1 − = p 1 − . k=0 pk pn pn j∈Z Similar calculations give us
∥f ∥rLr (Qn ) = ω
p
Qnp
g |x|p
r
|x|αp dx =
g p−j r j∈Z
1−
p(n+α)j
1 pn
,
and
− n+α r
Z⋆p
|s(t )|p
ψ(t )dt =
k≤0
−β n+α r
|t |p
φ pk dt
Sk
φ p−k 1 1− n . = k(n−β n+α p r ) k≥0 p For each j ∈ Z and each k = 0, −1, −2, . . ., we put g pj =
x−j p
n+α r j
and φ pk = y−k · p−kn+kβ
n+α r
.
It follows that
1/r 1/r ∥f ∥Lrω (Qnp ) = 1 − n · xrj < +∞, p j∈Z 1 − n+α |s(t )|p r ψ(t )dt = 1 − n · yk < +∞,
1
p
Z⋆p
k≥0
and
r ∞ 1 1 1 r −kn (n+α)j j−β k ∥Uψ,s f ∥Lr (Qn ) = p g p φ k p 1− n 1− n ω p k=0 p p p j∈Z r 1+r ∞ 1 n+α = p(α+n)j xβ k−j yk p− r j 1 − n k=0 p j∈Z
(17)
H.D. Hung / J. Math. Anal. Appl. 409 (2014) 868–879
=
1−
=
1 +r ∞ · x β k −j y k n
1 p
j∈Z
875
r
k=0
r 1 +r ∞ 1 1− n · x β k +j y k . p
j∈Z
k=0
Applying Theorem 3.1, Uψ,s is bounded on Lrω Qnp , and
∥Uψ,s f ∥Lrω (Qnp ) ≤ ∥f ∥Lrω (Qnp ) ·
− n+α r
Z⋆p
|s(t )|p
ψ(t )dt .
(18)
Combining the above equalities together with (18), we arrive immediately at (8). 6. Proof of Theorem 3.3 Lemma 6.1. If ω belongs to W =
α>−n
Wα , then log |x|p ∈ BMOω Qnp .
Note that, in Euclid’s case, ω needs to have the doubling property, but we can remove this condition in the p-adic case. Proof. Suppose that ω ∈ Wα for some α > −n. We just need to show there exists a positive constant C such that, for all x0 ∈ Qnp and γ ∈ Z, we can find a constant cγ ,x0 such that Ax0 ,γ =
1
ω Bγ (x0 )
Bγ (x0 )
log |x|p − cγ ,x ω(x)dx ≤ C . 0
By using the homogeneity of ω, if we use the change of variable z = p−γ x and replace x0 by p−γ x0 , we may assume that γ = 0. Thus, it suffices to check the alternative assertion that there is a constant C > 0 such that, for any x0 ∈ Qnp ,
1
ω (B0 (x0 ))
B0 (x0 )
log |x|p ω(x)dx ≤ C ,
for |x0 |p ≤ 1,
and
1
ω (B0 (x0 ))
B0 (x0 )
log |x|p − log |x0 |p ω(x)dx ≤ C ,
for |x0 |p ≥ p.
If |x0 | ≥ p then for each x ∈ B0 (x0 ), |x|p = |x − x0 + x0 |p = |x0 |p . Thus
1
ω (B0 (x0 ))
B0 (x0 )
log |x|p − log |x0 |p ω(x)dx = 0.
If |x0 |p ≤ 1, then B0 (x0 ) = B0 . Hence
1
ω (B0 (x0 ))
B0 (x0 )
log |x|p ω(x)dx =
1
ω (B0 )
log |x|p ω(x)dx B0
(pn+α − 1) log p γ = ω (S0 ) pn+α ω (S0 ) p(n+α)γ γ ≥0 =: C < ∞. (i) If Uψ,s is bounded on BMOω Qnp , then Uψ,s f0 ∈ BMOω Qnp , where f0 (x) = 1 for all x ∈ Qnp . This implies that Uψ,s f0 (x)
has a finite value at almost everywhere x ∈ Qnp . Hence some α > −n. Let f be in BMOω
Uψ,s f
B,ω
= =
1
ω(B) Z⋆p
Qnp
Z⋆p
ψ(t )dt < ∞. Now, suppose that (9) holds and that ω ∈ Wα for
, and let B be any ball of Qnp . By using Fubini’s theorem, we have
B
Z⋆p
1
ω(B)
f (s(t ) · x) ψ(t )dt f (y) · |s(t )|
−n−α
s(t )·B
ω(x)dx
ω(y)dy ψ(t )dt =
Z⋆p
fs(t )·B,ω ψ(t )dt .
876
H.D. Hung / J. Math. Anal. Appl. 409 (2014) 868–879
Thus 1
ω(B)
Uψ,s f (x) − Uψ,s f B,ω ω(x)dx ≤ B
=
1
ω(B)
Z⋆p
1
ω(B)
Z⋆p
=
B
=
B
ω(B)
Z⋆p
f (s(t ) · x) − fs(t )·B,ω ω(x)dx ψ(t )dt f (y) − fs(t )·B,ω ω(y)|s(t )|−n−α dy ψ(t )dt
1
f (s(t ) · x) − fs(t )·B,ω ψ(t )dt ω(x)dx
s(t )·B
1
f (y) − fs(t )·B,ω ω(y)dy ψ(t )dt
ω(s(t ) · B) s(t )·B ψ(t )dt . ≤ ∥f ∥BMOω (Qnp ) · Z⋆p
Z⋆p
Hence, ∥Uψ,s f ∥BMOω (Qnp ) ≤ ∥f ∥BMOω (Qnp ) ·
∥Uψ,s ∥BMOω (Qnp )→BMOω (Qnp ) ≤
Z⋆p
ψ(t )dt, so Uψ,s is bounded on BMOω Qnp , and in this case
ψ(t )dt .
Z⋆p
(ii) First, we assume that n = 1 and both integrals
∥Uψ,s ∥BMOω (Qp )→BMOω (Qp ) ≥ Uψ,s f1 (x) = f1 (x)
Z⋆p
Z⋆p
(19)
Z⋆p
log |s(t )|p ψ(t )dt and
ψ(t )dt. Let f1 (x) = log |x|p . Then f1 ∈ BMOω
ψ(t )dt +
Z⋆p
ψ(t )dt are finite. We need to show that
Z⋆p n Qp and
∥f1 ∥BMOω (Qp ) > 0. Also, we get
log |s(t )|p ψ(t )dt .
(20)
Since the second integral of (20) has to be finite, by taking the BMO norm on both sides of (20), we have
∥f1 ∥BMOω (Qp )
Z⋆p
ψ(t )dt = ∥f1 (·)
= ∥f1 (·)
Z⋆p
Z⋆p
ψ(t )dt ∥BMOω (Qp ) ψ(t )dt +
Z⋆p
log |s(t )|p ψ(t )dt ∥BMOω (Qp )
= ∥Uψ,s f1 ∥BMOω (Qp )
≤ ∥Uψ,s ∥BMOω (Qp )→BMOω (Qp ) ∥f1 ∥BMOω (Qp ) .
From this, we get
Z⋆p
ψ(t )dt ≤ ∥Uψ,s ∥BMOω (Qp )→BMOω (Qp ) .
Now let us assume that n > 1 and
Z⋆p
(21)
ψ(t )dt < ∞. For each x = (x1 , . . . , xn ) ∈ Qnp , let
|x1 |p , if x ̸= 0 f2 (x) = |x| 0, p if x = 0. Since f2 is not constant and f2 ∈ L∞ Qnp , we have f2 ∈ BMOω Qnp and ∥f2 ∥BMOω (Qnp ) > 0. From f2 (tx) = f2 (x) for any t ∈ Q⋆p , we obtain
Uψ,s f2 (x) = f2 (x)
Z⋆p
ψ(t )dt .
Taking the BMO norm on both sides, we get
∥f2 ∥
( )
BMOω Qnp
Z⋆p
ψ(t )dt = ∥Uψ,s f2 ∥BMOω (Qnp ) ≤ ∥Uψ,s ∥BMOω (Qnp )→BMOω (Qnp ) ∥f2 ∥BMOω (Qnp ) .
So
Z⋆p
ψ(t )dt ≤ ∥Uψ,s ∥BMOω (Qnp )→BMOω (Qnp ) . This completes the proof.
H.D. Hung / J. Math. Anal. Appl. 409 (2014) 868–879
877
7. Proof of Theorem 3.5 First of all, since ψ(t ) = φ |t |p , we have
Z⋆p
ψ(t )dt =
Z⋆p
φ |t |p dt = γ ≤0
=
1
1−
∞ φ
p
1 pγ
= 1.
pγ
γ =0
φ (pγ ) dt Sγ
So, by Theorem 3.3, Uψ f ∈ BMO(Qp ) for all f ∈ BMO(Qp ). Moreover, from the definition of Sp , we have
ψ(t )dt =
1 α =1− , φ |t |p dt = φ (1) · 1 −
S0
p
S0
p
where
α = sup γ ≥0
φ
φ
1 pγ +1
1 pγ
= sup t ∈Z⋆p
ψ(pt ) . ψ(t )
Assume that f (x) = g |x|p , where g is decreasing. Then it is obvious that Uψ f (x) ≥ f (x) for all x. By definition,
Uψ f p−1 x =
f tp−1 x ψ(t )dt
Z⋆p
= ≤ ≤
f tp−1 x ψ(t )dt + f p−1 x
(B−1 )⋆
1 p
α p
Z⋆p
ψ(t )dt S0
f (tx) ψ(pt )dt +
Uψ f (x) +
1−
α
1−
p
α p
f (x)
f (x) .
Thus Uψ f p−1 x ≤
α p
Uψ f (x) +
1−
α
p
f (x) ≤ Uψ f (x).
(22)
Put Vf (x) = αp Uψ f (x) + 1 − αp f (x). Once observing the elementary fact that, if a ≤ c ≤ b, then |c | ≤ max{|a|, |b|}, with the help of (22), we obtain
Vf (x) − Uψ f Bγ (x) ≤ max Uψ f (x) − Uψ f Bγ (x) , Uψ f p−1 x − Uψ f Bγ (x) . The second one is estimated as
Uψ f p−1 x − Uψ f Bγ (x) ≤ Uψ f ≤ Uψ f ∥Vf ∥BMO(Qp ) ≤ 2 sup γ ∈Z
1 pγ
Bγ (x)
+ U f − U f ψ p−1 Bγ (x) ψ Bγ (x) p−1 Bγ (x) −1 p x − Uψ f p−1 B (x) + 2∥Uψ f ∥BMO(Qp ) . γ p −1 x − U ψ f
Uψ f p−1 x − Uψ f Bγ (x) dx ≤ 6∥Uψ f ∥BMO(Qp ) .
On the other hand,
α α ∥Vf ∥BMO(Qp ) ≥ 1 − ∥f ∥BMO(Qp ) − ∥Uψ f ∥BMO(Qp ) . p
p
It follows that
∥Uψ f ∥BMO(Qp ) ≥
p−α 6p + α
which completes the proof.
∥f ∥BMO(Qp ) =
(p − 1)φ(1) 7p − (p − 1)φ(1)
∥f ∥BMO(Qp ) ,
878
H.D. Hung / J. Math. Anal. Appl. 409 (2014) 868–879
8. Proof of Theorem 3.6 β
b r n (a) Suppose that |s(t )|p ≥ |t |p for almost everywhere t ∈ Z⋆p for some β > 0, and that Uψ, s is bounded on Lω Qp for
any b ∈ BMOω Qp . From Lemma 6.1, we get b(x) = log |x|p ∈ BMOω Qp . For any γ ∈ Z+ , let fγ be as in (15). Then
n
− n+α r −
b Uψ, s fγ (x) = −|x|p
1
− n+α r −
γ2
S (x)
|s(t )|p
n
1
γ2
log |s(t )|p · ψ(t )dt ,
where S (x) = {t ∈ Z⋆p | so that |s(t ) · x|p > 1}. We have b
r 1 − n+α 1 r − γ2 |x|p ω(x) · log ψ(t )dt dx |s(t )|p S (x) |s(t )|p Qnp r 1 −n−α− r2 − n+α r − γ2 γ ω(x) · log |s(t )|p ψ(t )dt dx |x|p |s(t )|p 1≥|t |p ≥min{1/|x|1/β ,1} Qnp r 1 − n+α+ r2 − n+α r − γ2 γ · log |s(t )|p · ψ(t )dt ω(x)dx · |x|p |s(t )|p 1≥|t |p >p−γ /β |x|p >pγ r 1 − n+α+ r2 − n+α r r − γ2 γ |s(t )|p |y|p ω(y)dy p γ · · log |s(t )|p · ψ(t )dt −γ /β 1≥|t |p >p |y|>1 r 1 1 − n+α r − γ2 ∥fγ ∥rLr (Qn ) · p γ · log |s(t )|p · ψ(t )dt . |s(t )|p ω p −γ /β 1≥|t |p >p
r Lrω Qnp
∥Uψ,s fγ ∥ ( ) = ≥
≥
=
=
−n−α−
r
γ2
n > 0, so ω (Qp ) 1 − n+α 1 r − γ2 b pγ |s(t )|p · log |s(t )|p · ψ(t )dt ≤ ∥Uψ, s ∥Lrω (Qnp )→Lrω (Qnp ) . 1≥|t |p >p−γ /β
As in the proof of Theorem 3.1, ∥fγ ∥rLr
Letting γ to infinity, then
− n+α b r |s(t )|p · log |s(t )|p · ψ(t )dt ≤ ∥Uψ, s ∥Lrω (Qnp )→Lrω (Qnp ) < +∞. Z⋆p (b) Now we set b(x) = f (x) = χB0 (x), the characteristic function of the ball B0 = {x ∈ Qnp : |x|p ≤ 1}. Note that f ∈ Lrω Qnp , since α > −n and b ∈ BMOω (Qnp ). We have
b Uψ, s f (x) =
Z⋆p
(b(x) − b (s(t ) · x)) f (s(t ) · x) ψ(t )dt
= (b(x) − 1) ·
S (x)
ψ(t )dt ,
where S (x) = {t ∈ Z⋆p : |s(t ) · x|p < 1}. Since |s(t )|p ≤ |t |θp for almost everywhere t in Z⋆p , we can find a set E of measure zero such that S (x) ⊃ {t ∈ Z⋆p : |t |p <
1 1/θ
} \ E. Thus r b r ∥Uψ, ω(x) · ψ(t )dt dx s f ∥Lrω (Qnp ) ≥ θ − 1 1<|x|p ≤p 0<|t |p ≤p r ≥ ω(x)dx · ψ(t )dt 1<|t |p ≤p−1 1<|x|p ≤pθ r (n+α)θ r = ∥f ∥Lr (Qn ) · p −1 · ψ(t )dt . ω p 0<|t |p ≤p−1 |x|p
Therefore, we get
0<|t |p ≤p−1
ψ(t )dt ≤
C
p(n+α)θ
b where C = ∥Uψ, s ∥Lrω (Qnp )→Lrω (Qnp ) .
−1
1/r < ∞,
H.D. Hung / J. Math. Anal. Appl. 409 (2014) 868–879
879
Acknowledgments I would like to thank the referee for his or her helpful suggestions to improve this paper. I am also very grateful to Professor Nguyen Minh Chuong for his constant help and useful suggestions. This work is supported by a grant from the Viet Nam NAFOSTED (National Foundation for Science and Technology Development). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25]
K.F. Andersen, Boundedness of Hausdorff operator on Lp (Rn ), H 1 (Rn ) and BMO(Rn ), Acta Sci. Math. (Szeged) 69 (2003) 409–418. C. Carton-Lebrun, M. Fosset, Moyennes et quotients de Taylor dans BMO, Bull. Soc. Roy. Sci. Liège 53 (2) (1984) 85–87. N.M. Chuong, H.D. Hung, Maximal functions and weighted norm inequalities on local fields, Appl. Comput. Harmon. Anal. 29 (2010) 272–286. N.M. Chuong, H.D. Hung, A Muckenhoupt’s weight problem and vector valued maximal inequalities over local fields, p-adic numbers, Ultrametric Anal. Appl. 2 (2010) 305–321. N.M. Chuong, H.D. Hung, Weighted Lp and weighted BMO—bounds for a new generalized weighted Hardy–Cesàro operator, arXiv:1208.5242 [math.CA], August (2012), Integral Transforms Spec. Funct. (submitted for publication). R.R. Coifman, R. Rochberg, G. Weiss, Factorization theorems for Hardy spaces in several variables, Ann. of Math. (2) 103 (1976) 611–635. MR0412721 (54:843). D.E. Edmunds, W.D. Evans, Hardy Operators, Function Spaces and Embeddings, Springer-Verlag, Berlin, 2004. Z.W. Fu, Z.G. Liu, S.Z. Lu, Commutators of weighted Hardy operators on Rn , Proc. Amer. Math. Soc. 137 (10) (2009) 3319–3328. C. Georgakis, The Hausdorff mean of a Fourier–Stieltjes transform, Proc. Amer. Math. Soc. 116 (1992) 465–471. R.R. Goldberg, Convolutions and general transforms on Lp , Duke Math. J. 27 (1960) 251–259. G.H. Hardy, Divergent Series, Clarendon, Oxford, 1949. MR30620 (11,25a). G.H. Hardy, J.E. Littlewood, G. Pólya, Inequalities, second ed., Cambridge University Press, 1952. MR0046395 (13,727e). S.V. Kozyrev, A.Yu. Khrennikov, Pseudodifferential operators on ultrametric spaces and ultrametric wavelets, Izv. Ross. Akad. Nauk Ser. Mat. 69 (5) (2005) 133–148. MR 2179417. S.G. Krantz, On functions in p-adic BMO and the distribution of prime integers, J. Math. Anal. Appl. 326 (2007) 1437–1444. M. Kuczma, An Introduction to the Theory of Functional Equations and Inequalities, Pánstwowe Wydawnictwo Naukowe, Uniwersytet Ślaski, 1985. A.K. Lerner, E. Liflyand, Multidimensional Hausdorff operators on the real Hardy space, J. Aust. Math. Soc 83 (2007) 79–86. E. Liflyand, F. Moricz, The Hausdorff operator is bounded on the real Hardy space H 1 (R), Proc. Amer. Math. Soc. 128 (2000) 1391–1396. K.S. Rim, J. Lee, Estimates of weighted Hardy–Littlewood averages on the p-adic vector space, J. Math. Anal. Appl. 324 (2006) 1470–1477. Mitchell Taibleson, Fourier Analysis on Local Fields, Princeton University Press, Princeton, 1975. V.S. Vladimirov, I.V. Volovich, E.I. Zelenov, p-adic Analysis and Mathematical Physics, World Scientific, 1994. S.S. Volosivets, Multidimensional Hausdorff operator on p-adic field, p-adic numbers, Ultrametric Anal. Appl. 2 (2010) 252–259. S.S. Volosivets, Hausdorff operator of special kind on p-adic field and BMO-type spaces, p-adic numbers, Ultrametric Anal. Appl. 3 (2011) 149–156. S.S. Volosivets, Hausdorff operator of special kind in Morrey and Herz p-adic spaces, p-adic numbers, Ultrametric Anal. Appl. 4 (2012) 222–230. J. Xiao, Lp and BMO bounds of weighted Hardy–Littlewood Averages, J. Math. Anal. Appl. 262 (2001) 660–666. MR1859331 (2003a:47103). J. Xiao, A reverse BMO-Hardy inequality, Real Anal. Exchange 25 (2) (1999) 673–678.
Contents lists available at ScienceDirect
Journal of Mathematical Analysis and Applications journal homepage: www.elsevier.com/locate/jmaa
The p-adic weighted Hardy–Cesàro operator and an application to discrete Hardy inequalities Ha Duy Hung Hanoi National University of Education, Hanoi, Viet Nam
article
abstract
info
Article history: Received 22 April 2013 Available online 30 July 2013 Submitted by Jie Xiao
This paper aims to investigate the boundedness of the p-adic analog of the weighted Hardy–Cesàro operator Uψ,s : f → Z⋆ f (s(t )·) ψ(t )dt on weighted Lebesgue spaces and p
Keywords: Weighted Hardy–Cesàro operator Discrete Hardy inequalities BMO Commutator p-adic analysis
weighted BMO spaces. In each case, we obtain the corresponding operator norms |Uψ,s |. In particular, these results have a surprising relevance to discrete Hardy inequalities on the real field. We prove a reverse BMO–Hardy inequality on ψ and give a necessary condition so that the commutator of Uψ,s is bounded on Lrω Qnp with symbols in BMOω Qnp . © 2013 Elsevier Inc. All rights reserved.
1. Introduction Theories of functions from Qnp into R or C play an important role in p-adic quantum mechanics, the theory of p-adic probability in which real-valued random variables have to be considered to solve covariance problems [19,20]. Studies of the p-adic Hardy operators and p-adic Hausdorff operators are also useful for p-adic analysis [3,4,7,13,18,21–23]. In this paper, we shall give an interesting relation between p-adic Hardy operators and discrete Hardy inequalities on the real field. Hausdorff operators appeared long ago, aiming to solve certain classical problems in analysis. The modern theory of Hausdorff operators started with the work of Siskakis in complex analysis setting and with the work of Georgakis [9], Liflyand and Moricz [17] in the Fourier transform setting. One of the most general definitions of the Hausdorff operator is
H Φ ,A f ( x ) =
Rn
Φ (u)f (xA(u)) du,
n
where Φ is a measurable function on Rn and A = A(u) = aij (u) i,j=1 is the n × n matrix with the entries aij (u) being measurable functions of u. The properties of these operators were further studied in [11,12,10,17,1,16] and the references therein. In 1984, Carton-Lebrun and Fosset [2] considered a Hausdorff operator of a special kind, which is called the weighted Hardy operator Uψ :
Uψ f (x) =
1
f (tx)ψ(t )dt ,
x ∈ Rn .
(1)
0
The authors showed the boundedness of Uψ on Lebesgue spaces and BMO(Rn ) space. In 2001, J. Xiao [24] obtained that Uψ is bounded on Lp (Rn ) if and only if 1
t −n/p ψ(t )dt < ∞.
0
E-mail address: [email protected]. 0022-247X/$ – see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jmaa.2013.07.056
H.D. Hung / J. Math. Anal. Appl. 409 (2014) 868–879
869
Meanwhile, the corresponding operator norm was worked out. The result seems to be of interest as it is related closely to the classical Hardy integral inequality. Also, Xiao obtained the BMO(Rn )-bounds of Uψ , which sharpened and extended the main result of Carton-Lebrun and Fosset in [2]. In 2006, Rim and Lee [18] proved analogous results to those of Xiao on p-adic fields. They introduced the p-adic form of Uψ as follows: Uψ f (x) =
Z⋆p
f (tx)ψ(t )dt ,
x ∈ Qnp .
(2)
Here, Qp is the field of all p-adic numbers and Z⋆p = {x ∈ Qp : 0 < |x|p ≤ 1}. In 2010, Volosivets [21] introduced p-adic analogs of the Hausdorff operator and gave sufficient conditions for the boundedness of p-adic Hausdorff operator in p-adic Hardy and BMO spaces. Also, in [22,23], the author studied the boundedness of Uψ as in (2) in BMO-type p-adic spaces, Morrey p-adic spaces and Herz p-adic spaces. In 2012, Chuong and the author of this paper considered a weighted Hardy–Cesàro operator [5], a more general form of Uψ in the real case as follows. Definition 1.1. Let ψ : [0, 1] → [0, ∞) and s : [0, 1] → R be measurable functions. The weighted Hardy–Cesàro operator Uψ,s , associated to the parameter curve s(x, t ) := s(t )x, is defined by Uψ,s f (x) =
1
f (s(t )x) ψ(t )dt ,
(3)
0
for all measurable complex valued functions f on Rn . With certain conditions on functions s and ω, the authors of [5] proved that Uψ,s is bounded on weighted Lebesgue spaces and weighted BMO spaces. The corresponding operator norms are worked out too. The authors also give a necessary condition on the weight function ψ for the boundedness of the commutators of operator Uψ,s on Lrω (Rn ) with symbols in BMOω (Rn ). The form of the Hardy–Cesàro operator in p-adic analysis is Uψ,s f (x) =
Z⋆p
f (s(t )x) ψ(t )dt ,
where s : Z⋆p → Qp and ψ : Z⋆p → [0; ∞) are measurable functions. This paper aims to study the bounds of the p-adic Hardy–Cesàro operator on weighted Lebesgue spaces and weighted BMO spaces. The main motivations for our results are the works in [3–5,8,24,18,21–23]. We prove the continuities of the weighted p-adic Hardy–Cesàro operator Uψ,s on weighted Lebesgue spaces and weighted BMO spaces. The norms of the operator are worked out too. Here we note that our results are somewhat different from those in [21–23], since they hold also when s(t ) ̸= t and for certain weighted functional spaces. It is interesting to note that the results allow us to obtain a discrete Hardy inequality over the real field, and thus there is a relation between the behavior of the p-adic Hardy–Cesàro operator Uψ,s in functional spaces and discrete Hardy inequalities over the real field. We also obtain a reverse BMO–Hardy inequality and give a necessary condition on weight functions such that the commutator of Uψ,s with symbols in weighted BMO is bounded on weighted Lebesgue spaces. 2. Preliminaries Let p be a prime in Z, and let r ∈ Q⋆ . Write r = pγ ba , where a and b are integers not divisible by p. Define the p-adic absolute value |·|p on Q by |r |p = p−γ and |0|p = 0. The absolute value |·|p gives a metric on Q defined by dp (x, y) = |x − y|p . We denote by Qp the completion of Q with respect to the metric d. Qp with natural operations and topology induced by the metric dp is a locally compact, non-discrete, complete and totally disconnected field. A non-zero element x of Qp is uniquely represented as a canonical form x = pγ x0 + x1 p + x2 p2 + · · · , where xj ∈ Z/pZ and x0 ̸= 0. We then have |x|p = p−γ . Define Zp = {x ∈ Qp : |x|p ≤ 1} and Z⋆p = Zp \ {0}. Qnp = Qp × · · · × Qp contains all n-tuples of Qp . The norm on Qnp is |x|p = max1≤k≤n |xk | for x = (x1 , . . . , xn ) ∈ Qnp . The space Qnp is a complete metric locally compact and totally disconnected space. For each a ∈ Qp and x = (x1 , . . . , xn ) ∈ Qnp , we denote ax = (ax1 , . . . , axn ). For γ ∈ Z, we denote Bγ as a γ -ball of Qnp with center at 0, containing all x with |x|p ≤ pγ , and with Sγ = Bγ \ Bγ −1 its boundary. Also, for a ∈ Qnp , Bγ (a) consists of all x with x − a ∈ Bγ , and Sγ (a) consists of all x with x − a ∈ Sγ . Since Qnp is a locally compact commutative group with respect to addition, there exists the Haar measure dx on the additive group of Qnp normalized by
B0
dx = 1. Then d(ax) = |a|np dx for all a ∈ Q⋆p , |Bγ (x)| = pnγ , and Sγ (x) =
pnγ 1 − p−n . Let ω be any weight function on Qnp , that is, a non-negative locally integrable function from Qnp into R. Let Lrω Qnp (1 ≤ r < ∞) be the space of complex-valued functions f on Qnp such that
1/r
∥f ∥Lrω (Qnp ) =
|f (x)| ω(x)dx r
Qnp
< ∞.
870
H.D. Hung / J. Math. Anal. Appl. 409 (2014) 868–879
∞
Also, Lω
n
Qp consists of the set of all measurable real-valued functions f on Qnp satisfying
n := inf C > 0 : ω x ∈ Qnp : |f (x)| > C = 0 < ∞. ∥f ∥L∞ ω (Qp ) Here, ω(A) := A ω(x)dx for any measurable subset A of Qnp .
Some often used computational principles are worth mentioning at the outset. First, if f ∈ L1ω Qp , we can write
Qnp
f (x)ω(x)dx =
γ ∈Z
f (y)ω(y)dy. Sγ
Second, we also often use the fact that
Qnp
f (ax) dx =
1
|a|np
Qnp
f (x)dx,
if a ∈ Qnp \ {0} and f ∈ L1 (Qnp ).
The p-adic BMOω Qnp space of all functions f on Qnp which are bounded mean oscillations with weight ω is
1
∥f ∥BMOω (Qnp ) = sup
ω(B)
B
|f (x) − fB,ω |ω(x)dx < ∞,
where the supremum is taken over all balls B of Qnp . Here, ω(B) = weight ω: fB,ω =
1
ω(B)
(4)
B
B
ω(x)dx, and fB,ω is the mean value of f on B with
f (x)ω(x)dx.
(5)
B
We note that there are other p-adic BMO spaces which are useful in studying harmonic analysis by way of Brownian motion (see [14]). Also, in [14], Krantz determined the relation with ordinary continuous space BMO, and it is surprising that the results are related to information about the distribution of primes. Let s : Z⋆p → Qp and ψ : Z⋆p → R+ be measurable functions, and let ω : Qnp → R+ be a locally integrable function. For a function f on Qnp , we define the weighted p-adic Hardy–Cesàro operator Uψ,s on Qnp as Uψ,s f (x) =
Z⋆p
f (s(t )x) ψ(t )dt .
(6)
We shall consider the class of weights Wα , which consists of all non-negative locally integrable functions ω on Qnp such that
ω(tx) = |t |αp ω(x) for all x ∈ Qnp and t ∈ Q⋆p and 0 < α > −n.
S0
ω(x)dx < ∞. It is easy to see that ω(x) = |x|αp is in Wα if and only if
3. Main results β
Theorem 3.1. Suppose that r ∈ [1; ∞], α ∈ R, and ω ∈ Wα . Let s : Z⋆p → Qp be a measurable function such that |s(t )|p ≥ |t |p for some real number β and almost everywhere t ∈ Z⋆p . Then, Uψ,s is bounded on Lrω Qp if and only if
− n+α r
Z⋆p
|s(t )|p
n
ψ(t )dt < ∞.
Moreover,
∥Uψ,s ∥ ( ) n ω Qp
Lr
( )=
n ω Qp
→Lr
− n+α r
Z⋆p
|s(t )|p
ψ(t )dt .
(7)
When s(t ) = t and ω(x) ≡ 1, we obtain Theorem 1.1 in [18]. The boundedness of Hardy operators can be deduced from the so-called Hardy inequalities in both their discrete and continuous forms. Here we shall show that the discrete form of Hardy inequalities can be deduced from the p-adic continuous form of Hardy operators. Indeed, from Theorem 3.1, we can deduce the following surprising inequality. Corollary 3.2. Let (xj )j∈Z and (yk )k≥0 be two non-negative sequences. For any non-negative integer β , and for any 1 ≤ r < ∞, the following Hardy inequality holds:
∞ j∈Z
k=0
r 1/r xj+β k yk
1/r
≤
j∈Z
xrj
∞ k=0
yk
.
(8)
H.D. Hung / J. Math. Anal. Appl. 409 (2014) 868–879
871
To obtain a discrete Hardy inequality from the boundedness of p-adic Hardy operators, our main idea is using special radical functions and computational principles for Haar integrals. Thus, this method can be applied to derive a wide class of discrete Hardy-type inequalities. The boundedness of Uψ,s on weighted BMO is studied in the following theorem. Theorem 3.3. Let r ∈ [1; ∞] be a real number, and let ω belong to W = such that s(t ) ̸= 0 almost everywhere on Z⋆p .
α>−n
Wα . Let s : Z⋆p → Qp be a measurable function
(i) Uψ,s : BMOω Qnp → BMOω Qnp exists as a bounded operator if and only if
Z⋆p
ψ(t )dt < ∞.
(9)
(ii) If one of the following conditions holds: ◦ n = 1 and both integrals Z⋆ log |s(t )|p ψ(t )dt and Z⋆ ψ(t )dt are finite, or p
◦ n > 1 and
Z⋆p
p
ψ(t )dt < ∞,
then Uψ,s is bounded on BMOω Qnp , and
∥Uψ,s ∥
( )=
( )
BMOω Qnp
→BMOω Qnp
Z⋆p
ψ(t )dt .
(10)
Up to now, in the one-dimensional case, it remains unknown if the condition
Z⋆p
ψ(t )dt < ∞ is enough or not to
imply (10). In the last argument in the proof of Theorem 2 in [18], the authors use the wrong argument: they suppose that ψ( t ) dt < ∞ implies that log |t |p ψ(t )dt < ∞. Indeed, we can find an easy counterexample, as the following. Let ⋆ Z Z⋆ p
p
ψ(t ) =
1
|t |p (log |t |p )
2
. Then an easy computation gives us
Z⋆p
ψ(t )dt < ∞, but
Z⋆p
log |t |p ψ(t )dt = ∞.
As we shall see in the proof of Theorem 3.1, for the case n = 1 it seems hard to compute the norm of Uψ,s if we have only the condition that Z⋆ ψ(t )dt is finite. In the proof, we use a test function f1 (x) = log |x|p , which is a homomorphism p
from Q⋆p into R. Let us denote by hom Q⋆p , R the set of all homomorphisms from multiplication group (Q⋆p , ·) to addition group (R, +). It is known that Qp is a direct product of three subgroups: the infinite cyclic subgroup generated by p, the cyclic subgroup of order p − 1 consisting of non-zero elements which are generated by ζ , a primitive root of unity of order p − 1 and the group 1 + pZp of p-adic integers congruent to 1. Thus, it is easy to see that all homomorphisms from Q⋆p into R must have the following form:
f pj ζ k u = ja + g (u),
where g is a homomorphism from 1 + pZp into R. Using the usual arguments as for the Cauchy functional equation, by Zorn’s lemma (see [15]), we see that there are infinitely many such g. Let BMO ω,hom consist of all f ∈ BMOω Q⋆p such that f ∈ hom Q⋆p , R . Then log |x|p is in BMO ω,hom (see Lemma 6.1). It seems hard to find other non-trivial examples. Using the proof of Theorem 3.1, we obtain the following. Theorem 3.4. Let r , ω(x), s(t ) be as in Theorem 3.3. If there exists a g ∈ BMO ω,hom such that both integrals and
Z⋆p
ψ(t )dt are finite, then (10) holds for n = 1.
Z⋆p
g (s(t )) ψ(t )dt
One of the ways to derive (10) is to find the reverse inequality, that is, ∥Uψ,s f ∥BMO ≥ C ∥f ∥BMO for f in some class functions. In Euclid’s case, Xiao [25] showed that the reverse BMO–Hardy inequality
∥Pf ∥BMO(R+ ) ≥
1 17
∥f ∥BMO(R+ ) x
is valid for each positive decreasing function f ∈ BMO(R+ ) on R+ . Here, Pf (x) = 1x 0 f (t )dt. Since a decreasing function on Qnp makes no sense, it is interesting to ask if there exists such a reverse inequality on p-adic fields. To state our result, let us denote Sp by the set of all non-negative sequences (ak )k≥0 such that a
• supk≥0 ka+k 1 = p (1 − a0 ) + a0 . ak p • ∞ k=0 pk = p−1 . It is obvious to see that (1, 1, 1, . . . , 1, . . .) is in Sp . By a simple argument based on continuity, we can show that Sp contains infinitely many such sequences. Our result on the reverse BMO–Hardy inequality is the following.
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H.D. Hung / J. Math. Anal. Appl. 409 (2014) 868–879
Theorem 3.5. Let ψ be a function on Z⋆p such that ψ(t ) = φ |t |p and the sequence φ p−k inequality holds:
∥Uψ f ∥BMO(Qp ) ≥
(p − 1)φ(1) 7p − (p − 1)φ(1)
k≥0
∈ Sp . Then, the reverse BMO
∥f ∥BMO(Qp )
(11)
for all decreasing radical functions f (x) = g (|x|p ). Here, Uψ is defined as in (2). (p−1)φ(1)
(p−1)φ(1)
Since φ(1) ≤ p−1 , 7p−(p−1)φ(1) < 1. We conjecture that the constant 7p−(p−1)φ(1) can be replaced by 1. We note that, in Theorem 4 in [23], the author also proved an interesting reverse BMO inequality for the modified Hardy–Littlewood operator B f (x) := |x1| |t | ≤|x| f (t )dt. p
p
p
p
More recently, great attention was paid to the study of commutators of operators. A well-known result of Coifman, Rochberg, and Weiss [6] states that the commutator Tb f = bTf − T (bf ) (where T is a Calderón–Zygmund singular integral operator) is bounded on Lp (Rn ), 1 < p < ∞, if and only if b ∈ BMO(Rn ). Many results have been generalized to commutators of other operators, not only Calderón–Zygmund singular integral operators. The definition of the weighted Hardy–Cesàro commutator is as follows: b Uψ, s f := bUψ,s f − Uψ,s (bf ).
In the real case, Chuong and the author of this paper [5] gave a necessary condition on ψ so that the commutator of Hardy–Cesàro operator is bounded on weighted Lebesgue spaces with symbols in weighted BMO spaces. This result can be extended to the p-adic case as follows. Theorem 3.6. Let r ∈ [1; ∞] be a real number, and let ω belong to W = α>−n Wα , which has doubling property. n Let b r s : Z⋆p → Qp be a measurable function such that s(t ) ̸= 0 almost everywhere on Z⋆p . Assume that Uψ, is bounded on L ω Qp for s
any b ∈ BMOω Qnp . β
(a) If |s(t )|p ≥ |t |p for almost everywhere t ∈ Z⋆p for some β > 0, then
− n+α r · log |s(t )|p · ψ(t )dt < ∞. |s(t )|p Z⋆p
(12)
(b) If 0 < |s(t )|p ≤ |t |θp for almost everywhere t ∈ Z⋆p for some positive integer θ > 0, then
0<|t |p ≤ 1p
b ψ(t )dt ≤ ∥Uψ, s ∥Lrω (Qnp )→Lrω (Qnp ) .
(13)
As far as we know, the problem of characterizing the weight functions so that the converse of Theorem 3.6 holds is still open, even in Euclid’s case. 4. Proof of Theorem 3.1 Since the case r = ∞ is trivial, it suffices to consider r ∈ [1; ∞). Suppose that
Z⋆p
− n+α r
|s(t )|p
ψ(t )dt < ∞. For each
f ∈ Lrω Qp , since s(t ) ̸= 0 almost everywhere, ω is homogeneous of order α ; then, applying Minkowski’s inequality for integrals (see [12, page 14]), we obtain
n
r 1/r ∥Uψ,s f ∥Lrω (Qnp ) = f (s(t ) · x) ψ(t )dt ω(x)dx Qnp Z⋆p 1/r r |f (s(t ) · x)| ω(x)dx ≤ ψ(t )dt
Z⋆p
Qnp
1/r
=
Z⋆p
|f (y)| |s(t )|
Qnp
= ∥f ∥ ( ) · n ω Qp
Lr
−α−n
r
− n+α r
Z⋆p
|s(t )|p
ω(y)dy
ψ(t )dt < ∞.
ψ(t )dt
H.D. Hung / J. Math. Anal. Appl. 409 (2014) 868–879
873
(Here we use the change of variable y = s(t )x.) Thus Uψ,s maps boundedly Lrω Qp into itself and
∥Uψ,s ∥Lrω (Qnp )→Lrω (Qnp ) ≤
− n+α r
|s(t )|p
Z⋆p
n
ψ(t )dt .
(14)
Conversely, suppose that Uψ,s is a bounded operator on Lrω Qnp . Then there exists a constant C > 0 such that
for all f ∈ Lrω Qnp .
∥Uψ,s f ∥Lrω (Qnp ) ≤ C ∥f ∥Lrω (Qnp ) ,
For any positive integer γ , we put if |x|p < 1
0
fγ (x) =
− n+α r −
|x|p
1
γ2
(15)
if |x|p ≥ 1.
Then
∥fγ ∥rLr (Qn ) = ω
p
=
=
−n−α−
|x|p
r
γ2
ω(x)dx
| x| p ≥1 ∞
p
−k n+α+
k=0
Sk
∞
−k n+α+
r γ2
p
r
ω(x)dx
γ2
· pkn+kα
1
=
ω(y)dy S0
k=0
1−p
ω(y)dy.
·
r
−
γ2
S0
Thus fγ ∈ Lrω Qnp for each γ . Then we have
− n+α r −
Uψ,s fγ (x) = |x|p
1
− n+α r −
γ2
S (x)
|s(t )|p
1
γ2
ψ(t )dt , β
where S (x) = {t ∈ Z⋆p : |s(t ) · x|p ≥ 1}. For almost everywhere t ∈ Z⋆p , since |s(t )|p ≥ |t |p , we have that |t |p ≥ implies that t ∈ S (x). Consequently, we have
r Lrω Qnp
∥Uψ,s fγ ∥ ( ) ≥
−n−α−
|x|p
r
γ2
S (x)
|x|p ≥1
−n−α−
|x|p
≥ |x|p ≥pγ r Lrω Qnp
− n+α r −
= ∥fγ ∥ ( ) · p
−r /γ
|s(t )|p
ψ(t )dt
r
γ2
r
1
γ2
ω(x)dx
ω(x)dx − n+α r −
1≥|t |p ≥p−γ /β
|s(t )|p
− n+α r −
· 1≥|t |p ≥p−γ /β
1 1/β
|x|p
|s(t )|p
ψ(t )dt
r
1
γ2
r
1
γ2
ψ(t )dt
,
which implies that − n+α r −
1≥|t |p ≥p−γ /β
|s(t )|p
1
γ2
ψ(t )dt ≤ p1/γ ·
∥Uψ,s fγ ∥Lrω (Qnp ) ∥fγ ∥Lrω (Qnp )
≤ C · p1/γ .
Letting γ to infinity, then, by Fatou’s lemma, we obtain
− n+α r
Z⋆p
|s(t )|p
ψ(t )dt ≤ C < +∞.
As we have shown, Uψ,s is bounded on Lrω Qnp , and hence
− n+α r
Z⋆p
|s(t )|p
ψ(t )dt ≤ ∥Uψ,s ∥Lrω (Qnp )→Lrω (Qnp ) .
From (14) and (16), we have
∥Uψ,s ∥Lrω (Qnp )→Lrω (Qnp ) =
− n+α r
Z⋆p
|s(t )|p
ψ(t )dt .
(16)
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H.D. Hung / J. Math. Anal. Appl. 409 (2014) 868–879
5. Proof of Corollary 3.2
First, we observe that inequality (8) is trivial if
< ∞ and
r j∈Z xj
be assumed that
y k < ∞.
∞
k=0
j∈Z
xrj = ∞ or
∞
k=0
yk = ∞. Thus, without loss of generality, it may
For each t ∈ Z⋆p and x ∈ Qnp , we put s(t ) = t β , ψ(t ) = φ(|t |p ), and ω(x) = |x|αp . We also choose f (x) = g (|x|p ). Here, α > −n, and functions φ and g will be chosen later. We have Uψ,s f (x) =
=
g |t |βp |x|p φ |t |p dt
Z⋆p
g pβ k |x|p φ pk pkn
1−
k≤0
=
−β k
g p
|x|p φ
1
p
pk
k≥0
−kn
1
pn
1−
1
pn
.
Thus,
∥Uψ,s f ∥rLr (Qn ) = ω
p
=
Qnp
Uψ,s f (x)r ω(x)dx
pα j
Uψ,s f (x)r ω(x)dx Sγ
j∈Z
r ∞ j−β k 1 1 jn 1 −kn αj 1− n . 1− n p p g p = φ k p k=0 p p p j∈Z r ∞ 1 1 1 −kn g pj−β k φ p(n+α)j 1 − = p 1 − . k=0 pk pn pn j∈Z Similar calculations give us
∥f ∥rLr (Qn ) = ω
p
Qnp
g |x|p
r
|x|αp dx =
g p−j r j∈Z
1−
p(n+α)j
1 pn
,
and
− n+α r
Z⋆p
|s(t )|p
ψ(t )dt =
k≤0
−β n+α r
|t |p
φ pk dt
Sk
φ p−k 1 1− n . = k(n−β n+α p r ) k≥0 p For each j ∈ Z and each k = 0, −1, −2, . . ., we put g pj =
x−j p
n+α r j
and φ pk = y−k · p−kn+kβ
n+α r
.
It follows that
1/r 1/r ∥f ∥Lrω (Qnp ) = 1 − n · xrj < +∞, p j∈Z 1 − n+α |s(t )|p r ψ(t )dt = 1 − n · yk < +∞,
1
p
Z⋆p
k≥0
and
r ∞ 1 1 1 r −kn (n+α)j j−β k ∥Uψ,s f ∥Lr (Qn ) = p g p φ k p 1− n 1− n ω p k=0 p p p j∈Z r 1+r ∞ 1 n+α = p(α+n)j xβ k−j yk p− r j 1 − n k=0 p j∈Z
(17)
H.D. Hung / J. Math. Anal. Appl. 409 (2014) 868–879
=
1−
=
1 +r ∞ · x β k −j y k n
1 p
j∈Z
875
r
k=0
r 1 +r ∞ 1 1− n · x β k +j y k . p
j∈Z
k=0
Applying Theorem 3.1, Uψ,s is bounded on Lrω Qnp , and
∥Uψ,s f ∥Lrω (Qnp ) ≤ ∥f ∥Lrω (Qnp ) ·
− n+α r
Z⋆p
|s(t )|p
ψ(t )dt .
(18)
Combining the above equalities together with (18), we arrive immediately at (8). 6. Proof of Theorem 3.3 Lemma 6.1. If ω belongs to W =
α>−n
Wα , then log |x|p ∈ BMOω Qnp .
Note that, in Euclid’s case, ω needs to have the doubling property, but we can remove this condition in the p-adic case. Proof. Suppose that ω ∈ Wα for some α > −n. We just need to show there exists a positive constant C such that, for all x0 ∈ Qnp and γ ∈ Z, we can find a constant cγ ,x0 such that Ax0 ,γ =
1
ω Bγ (x0 )
Bγ (x0 )
log |x|p − cγ ,x ω(x)dx ≤ C . 0
By using the homogeneity of ω, if we use the change of variable z = p−γ x and replace x0 by p−γ x0 , we may assume that γ = 0. Thus, it suffices to check the alternative assertion that there is a constant C > 0 such that, for any x0 ∈ Qnp ,
1
ω (B0 (x0 ))
B0 (x0 )
log |x|p ω(x)dx ≤ C ,
for |x0 |p ≤ 1,
and
1
ω (B0 (x0 ))
B0 (x0 )
log |x|p − log |x0 |p ω(x)dx ≤ C ,
for |x0 |p ≥ p.
If |x0 | ≥ p then for each x ∈ B0 (x0 ), |x|p = |x − x0 + x0 |p = |x0 |p . Thus
1
ω (B0 (x0 ))
B0 (x0 )
log |x|p − log |x0 |p ω(x)dx = 0.
If |x0 |p ≤ 1, then B0 (x0 ) = B0 . Hence
1
ω (B0 (x0 ))
B0 (x0 )
log |x|p ω(x)dx =
1
ω (B0 )
log |x|p ω(x)dx B0
(pn+α − 1) log p γ = ω (S0 ) pn+α ω (S0 ) p(n+α)γ γ ≥0 =: C < ∞. (i) If Uψ,s is bounded on BMOω Qnp , then Uψ,s f0 ∈ BMOω Qnp , where f0 (x) = 1 for all x ∈ Qnp . This implies that Uψ,s f0 (x)
has a finite value at almost everywhere x ∈ Qnp . Hence some α > −n. Let f be in BMOω
Uψ,s f
B,ω
= =
1
ω(B) Z⋆p
Qnp
Z⋆p
ψ(t )dt < ∞. Now, suppose that (9) holds and that ω ∈ Wα for
, and let B be any ball of Qnp . By using Fubini’s theorem, we have
B
Z⋆p
1
ω(B)
f (s(t ) · x) ψ(t )dt f (y) · |s(t )|
−n−α
s(t )·B
ω(x)dx
ω(y)dy ψ(t )dt =
Z⋆p
fs(t )·B,ω ψ(t )dt .
876
H.D. Hung / J. Math. Anal. Appl. 409 (2014) 868–879
Thus 1
ω(B)
Uψ,s f (x) − Uψ,s f B,ω ω(x)dx ≤ B
=
1
ω(B)
Z⋆p
1
ω(B)
Z⋆p
=
B
=
B
ω(B)
Z⋆p
f (s(t ) · x) − fs(t )·B,ω ω(x)dx ψ(t )dt f (y) − fs(t )·B,ω ω(y)|s(t )|−n−α dy ψ(t )dt
1
f (s(t ) · x) − fs(t )·B,ω ψ(t )dt ω(x)dx
s(t )·B
1
f (y) − fs(t )·B,ω ω(y)dy ψ(t )dt
ω(s(t ) · B) s(t )·B ψ(t )dt . ≤ ∥f ∥BMOω (Qnp ) · Z⋆p
Z⋆p
Hence, ∥Uψ,s f ∥BMOω (Qnp ) ≤ ∥f ∥BMOω (Qnp ) ·
∥Uψ,s ∥BMOω (Qnp )→BMOω (Qnp ) ≤
Z⋆p
ψ(t )dt, so Uψ,s is bounded on BMOω Qnp , and in this case
ψ(t )dt .
Z⋆p
(ii) First, we assume that n = 1 and both integrals
∥Uψ,s ∥BMOω (Qp )→BMOω (Qp ) ≥ Uψ,s f1 (x) = f1 (x)
Z⋆p
Z⋆p
(19)
Z⋆p
log |s(t )|p ψ(t )dt and
ψ(t )dt. Let f1 (x) = log |x|p . Then f1 ∈ BMOω
ψ(t )dt +
Z⋆p
ψ(t )dt are finite. We need to show that
Z⋆p n Qp and
∥f1 ∥BMOω (Qp ) > 0. Also, we get
log |s(t )|p ψ(t )dt .
(20)
Since the second integral of (20) has to be finite, by taking the BMO norm on both sides of (20), we have
∥f1 ∥BMOω (Qp )
Z⋆p
ψ(t )dt = ∥f1 (·)
= ∥f1 (·)
Z⋆p
Z⋆p
ψ(t )dt ∥BMOω (Qp ) ψ(t )dt +
Z⋆p
log |s(t )|p ψ(t )dt ∥BMOω (Qp )
= ∥Uψ,s f1 ∥BMOω (Qp )
≤ ∥Uψ,s ∥BMOω (Qp )→BMOω (Qp ) ∥f1 ∥BMOω (Qp ) .
From this, we get
Z⋆p
ψ(t )dt ≤ ∥Uψ,s ∥BMOω (Qp )→BMOω (Qp ) .
Now let us assume that n > 1 and
Z⋆p
(21)
ψ(t )dt < ∞. For each x = (x1 , . . . , xn ) ∈ Qnp , let
|x1 |p , if x ̸= 0 f2 (x) = |x| 0, p if x = 0. Since f2 is not constant and f2 ∈ L∞ Qnp , we have f2 ∈ BMOω Qnp and ∥f2 ∥BMOω (Qnp ) > 0. From f2 (tx) = f2 (x) for any t ∈ Q⋆p , we obtain
Uψ,s f2 (x) = f2 (x)
Z⋆p
ψ(t )dt .
Taking the BMO norm on both sides, we get
∥f2 ∥
( )
BMOω Qnp
Z⋆p
ψ(t )dt = ∥Uψ,s f2 ∥BMOω (Qnp ) ≤ ∥Uψ,s ∥BMOω (Qnp )→BMOω (Qnp ) ∥f2 ∥BMOω (Qnp ) .
So
Z⋆p
ψ(t )dt ≤ ∥Uψ,s ∥BMOω (Qnp )→BMOω (Qnp ) . This completes the proof.
H.D. Hung / J. Math. Anal. Appl. 409 (2014) 868–879
877
7. Proof of Theorem 3.5 First of all, since ψ(t ) = φ |t |p , we have
Z⋆p
ψ(t )dt =
Z⋆p
φ |t |p dt = γ ≤0
=
1
1−
∞ φ
p
1 pγ
= 1.
pγ
γ =0
φ (pγ ) dt Sγ
So, by Theorem 3.3, Uψ f ∈ BMO(Qp ) for all f ∈ BMO(Qp ). Moreover, from the definition of Sp , we have
ψ(t )dt =
1 α =1− , φ |t |p dt = φ (1) · 1 −
S0
p
S0
p
where
α = sup γ ≥0
φ
φ
1 pγ +1
1 pγ
= sup t ∈Z⋆p
ψ(pt ) . ψ(t )
Assume that f (x) = g |x|p , where g is decreasing. Then it is obvious that Uψ f (x) ≥ f (x) for all x. By definition,
Uψ f p−1 x =
f tp−1 x ψ(t )dt
Z⋆p
= ≤ ≤
f tp−1 x ψ(t )dt + f p−1 x
(B−1 )⋆
1 p
α p
Z⋆p
ψ(t )dt S0
f (tx) ψ(pt )dt +
Uψ f (x) +
1−
α
1−
p
α p
f (x)
f (x) .
Thus Uψ f p−1 x ≤
α p
Uψ f (x) +
1−
α
p
f (x) ≤ Uψ f (x).
(22)
Put Vf (x) = αp Uψ f (x) + 1 − αp f (x). Once observing the elementary fact that, if a ≤ c ≤ b, then |c | ≤ max{|a|, |b|}, with the help of (22), we obtain
Vf (x) − Uψ f Bγ (x) ≤ max Uψ f (x) − Uψ f Bγ (x) , Uψ f p−1 x − Uψ f Bγ (x) . The second one is estimated as
Uψ f p−1 x − Uψ f Bγ (x) ≤ Uψ f ≤ Uψ f ∥Vf ∥BMO(Qp ) ≤ 2 sup γ ∈Z
1 pγ
Bγ (x)
+ U f − U f ψ p−1 Bγ (x) ψ Bγ (x) p−1 Bγ (x) −1 p x − Uψ f p−1 B (x) + 2∥Uψ f ∥BMO(Qp ) . γ p −1 x − U ψ f
Uψ f p−1 x − Uψ f Bγ (x) dx ≤ 6∥Uψ f ∥BMO(Qp ) .
On the other hand,
α α ∥Vf ∥BMO(Qp ) ≥ 1 − ∥f ∥BMO(Qp ) − ∥Uψ f ∥BMO(Qp ) . p
p
It follows that
∥Uψ f ∥BMO(Qp ) ≥
p−α 6p + α
which completes the proof.
∥f ∥BMO(Qp ) =
(p − 1)φ(1) 7p − (p − 1)φ(1)
∥f ∥BMO(Qp ) ,
878
H.D. Hung / J. Math. Anal. Appl. 409 (2014) 868–879
8. Proof of Theorem 3.6 β
b r n (a) Suppose that |s(t )|p ≥ |t |p for almost everywhere t ∈ Z⋆p for some β > 0, and that Uψ, s is bounded on Lω Qp for
any b ∈ BMOω Qp . From Lemma 6.1, we get b(x) = log |x|p ∈ BMOω Qp . For any γ ∈ Z+ , let fγ be as in (15). Then
n
− n+α r −
b Uψ, s fγ (x) = −|x|p
1
− n+α r −
γ2
S (x)
|s(t )|p
n
1
γ2
log |s(t )|p · ψ(t )dt ,
where S (x) = {t ∈ Z⋆p | so that |s(t ) · x|p > 1}. We have b
r 1 − n+α 1 r − γ2 |x|p ω(x) · log ψ(t )dt dx |s(t )|p S (x) |s(t )|p Qnp r 1 −n−α− r2 − n+α r − γ2 γ ω(x) · log |s(t )|p ψ(t )dt dx |x|p |s(t )|p 1≥|t |p ≥min{1/|x|1/β ,1} Qnp r 1 − n+α+ r2 − n+α r − γ2 γ · log |s(t )|p · ψ(t )dt ω(x)dx · |x|p |s(t )|p 1≥|t |p >p−γ /β |x|p >pγ r 1 − n+α+ r2 − n+α r r − γ2 γ |s(t )|p |y|p ω(y)dy p γ · · log |s(t )|p · ψ(t )dt −γ /β 1≥|t |p >p |y|>1 r 1 1 − n+α r − γ2 ∥fγ ∥rLr (Qn ) · p γ · log |s(t )|p · ψ(t )dt . |s(t )|p ω p −γ /β 1≥|t |p >p
r Lrω Qnp
∥Uψ,s fγ ∥ ( ) = ≥
≥
=
=
−n−α−
r
γ2
n > 0, so ω (Qp ) 1 − n+α 1 r − γ2 b pγ |s(t )|p · log |s(t )|p · ψ(t )dt ≤ ∥Uψ, s ∥Lrω (Qnp )→Lrω (Qnp ) . 1≥|t |p >p−γ /β
As in the proof of Theorem 3.1, ∥fγ ∥rLr
Letting γ to infinity, then
− n+α b r |s(t )|p · log |s(t )|p · ψ(t )dt ≤ ∥Uψ, s ∥Lrω (Qnp )→Lrω (Qnp ) < +∞. Z⋆p (b) Now we set b(x) = f (x) = χB0 (x), the characteristic function of the ball B0 = {x ∈ Qnp : |x|p ≤ 1}. Note that f ∈ Lrω Qnp , since α > −n and b ∈ BMOω (Qnp ). We have
b Uψ, s f (x) =
Z⋆p
(b(x) − b (s(t ) · x)) f (s(t ) · x) ψ(t )dt
= (b(x) − 1) ·
S (x)
ψ(t )dt ,
where S (x) = {t ∈ Z⋆p : |s(t ) · x|p < 1}. Since |s(t )|p ≤ |t |θp for almost everywhere t in Z⋆p , we can find a set E of measure zero such that S (x) ⊃ {t ∈ Z⋆p : |t |p <
1 1/θ
} \ E. Thus r b r ∥Uψ, ω(x) · ψ(t )dt dx s f ∥Lrω (Qnp ) ≥ θ − 1 1<|x|p ≤p 0<|t |p ≤p r ≥ ω(x)dx · ψ(t )dt 1<|t |p ≤p−1 1<|x|p ≤pθ r (n+α)θ r = ∥f ∥Lr (Qn ) · p −1 · ψ(t )dt . ω p 0<|t |p ≤p−1 |x|p
Therefore, we get
0<|t |p ≤p−1
ψ(t )dt ≤
C
p(n+α)θ
b where C = ∥Uψ, s ∥Lrω (Qnp )→Lrω (Qnp ) .
−1
1/r < ∞,
H.D. Hung / J. Math. Anal. Appl. 409 (2014) 868–879
879
Acknowledgments I would like to thank the referee for his or her helpful suggestions to improve this paper. I am also very grateful to Professor Nguyen Minh Chuong for his constant help and useful suggestions. This work is supported by a grant from the Viet Nam NAFOSTED (National Foundation for Science and Technology Development). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25]
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