Hardy’s inequalities and integral operators on Herz-Morrey spaces

Tat-Leung Yee
  • Department of Mathematics and Information Technology, The Education University of Hong Kong, 10 Lo Ping Road, Tai Po, Hong Kong, China
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and Kwok-Pun Ho
  • Corresponding author
  • Department of Mathematics and Information Technology, The Education University of Hong Kong, 10 Lo Ping Road, Tai Po, Hong Kong, China
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Abstract

We obtain some estimates for the operator norms of the dilation operators on Herz-Morrey spaces. These results give us the Hardy’s inequalities and the mapping properties of the integral operators on Herz-Morrey spaces. As applications of this general result, we have the boundedness of the Hadamard fractional integrals on Herz-Morrey spaces. We also obtain the Hilbert inequality on Herz-Morrey spaces.

1 Introduction

This paper focuses on the Hardy’s inequalities and the boundedness of the integral operators on Herz-Morrey spaces.

Herz-Morrey spaces are extensions of Herz spaces [1] and Morrey spaces [2]. They also include the central Morrey spaces [3, 4, 5, 6]. One of the pioneer studies on the Herz-Morrey spaces is from Lu and Xu [7] on the mapping properties of the singular integral operators on the Herz-Morrey spaces. Since then, the study of Herz-Morrey spaces inspires the introduction of a number of new function spaces including Herz-Morrey-Hardy spaces [8, 9], Herz-Morrey spaces with variable exponents [10, 11, 12, 13, 14, 15] and the Herz-Morrey-Besov spaces [16].

In this paper, we study the mapping properties of integral operators on Herz-Morrey spaces. In particular, we are interested in Hadamard fractional integrals, the Hardy operator and the Hilbert operator. We find that the mapping properties of these operators rely on the operator norms of dilation operators on Herz-Morrey spaces.

The use of the dilation operators to study the mapping properties of integral operators is well studied, especially for the rearrangement-invariant Banach function spaces [17]. The study in [17, Chapter 3, Section 5] relies on the notion of Boyd’s indices. The Boyd indices are also used in the study of the mapping properties of Fourier transform and the Hankel transform, see [18, 19, 20].

In this paper, we give some estimates for the operator norms of the dilation operators on Herz-Morrey spaces. With these estimates, we define and obtain the Boyd indices of Herz-Morrey spaces.

By using these indices, we establish the a general result on the mapping properties of integral operators on Herz-Morrey spaces. This general result yields the boundedness of Hadamard fractional integrals, the Hardy operator and the Hilbert operator on Herz-Morrey spaces. We are interested in Hadamard fractional integrals because they are fractional integrals for Mellin transform [21]. The reader is referred to [21] for the relation between Mellin transform and Hadamard fractional integrals.

It is well known that the Hardy operator is not bounded on L1(ℝ) = MK˙1,10,0(ℝ). On the other hand, our result shows that the Hardy operator is bounded on the Herz-Morrey MK˙1,1α,λ(ℝ) when α < λ. The reader is referred to Definition 2.1 for the definition of the Herz-Morrey space MK˙p,qα,λ(ℝn).

Since Herz spaces and central Morrey spaces are members of Herz-Morrey spaces, our results yield the mapping properties of Hadamard fractional integrals, the Hardy operator and the Hilbert operator on Herz spaces and central Morrey spaces.

This paper is organized as follows. Section 2 contains the definition of Herz-Morrey spaces. The Boyd’s indices of the Herz-Morrey spaces are obtained in this section. The main result for the Hardy’s inequalities and the boundedness of integral operators on Herz-Morrey spaces is established in Section 3. As applications for the general results on the boundedness of integral operators, we also obtain the boundedness of Hadamard fractional integrals and the Hilbert inequalities on Herz-Morrey spaces. Notice that in this paper, the results on dilation operators and Boyd’s indices are on Herz-Morrey spaces over ℝn while the remaining results are on Herz-Morrey spaces over ℝ.

2 Herz-Morrey spaces and Boyd’s indices

We give the definition of Herz-Morrey spaces in this section. We also obtain some estimates for the operator norms of the dilation operators on Herz-Morrey spaces. These estimates give the Boyd indices of the Herz-Morrey spaces.

Let Bk = {x ∈ ℝn : |x| ≤ 2k} and Rk = BkBk–1, k ∈ ℤ. Define χk = χRk.

Definition 2.1

Let α ∈ ℝ, λ ≥ 0, 0 < p ≤ ∞ and 0 < q < ∞. The Herz-Morrey space MK˙p,qα,λ(ℝn) consists of all Lebesgue measurable functions f satisfying

fMK˙p,qα,λ(Rn)=supkZ2kλj=k2jαpfχjLqp1p<.

When λ = 0, the Herz-Morrey space MK˙p,qα,λ(ℝn) becomes the Herz space K˙p,qα(ℝn) studied in [22]. In addition, when α = 0 and p = q, the Herz-Morrey space MK˙p,qα,λ(ℝn) reduces to the Lebesgue space Lq.

Furthermore, when α = 0, p = q and λ = nθq with 0 < θ < 1, the Herz-Morrey space MK˙p,qα,λ(ℝn) is the central Morrey space q,θ(ℝn) [3, 4, 5, 6, 23, 24]. Recall that the central Morrey space q,θ(ℝn) consists of all Lebesgue measurable functions f satisfying

fB˙q,θ(Rn)=supR>01|B(0,R)|θB(0,R)|f(y)|qdy1q<.

The reader is referred to [3, 4, 5] for the studies of central Morrey spaces. We use the definition of central Morrey spaces from [5, Definition 2] while we use the notion for the central Morrey spaces from [3, 4].

The study of Herz-Morrey spaces had been extended to Herz-Morrey-Hardy spaces [8, 9], Herz-Morrey-Besov spaces and the Herz-Morrey-Triebel-Lizorkin spaces [16]. Moreover, the Herz-Morrey spaces had been further generalized to the Herz-Morrey spaces built on Lebesgue spaces with variable exponents in [11, 13, 14, 16].

For β ∈ ℝ, L ∈ ℤ, define fβ,L(x) = |x|βχL(x). For any j ∈ ℤ, we have

fβ,LχjLq=2j12jrβqrn1dr1q=C2min(j,L)(β+nq),

where C is independent of j.

When α + β + nq > λ > α, we have

2kλj=k2jαpfβ,LχjLqp1p=C2kλj=k2jαp2pmin(j,L)(β+nq)1pC2kλ2kα2min(k,L)(β+nq)

for some C > 0 independent of k. For the case kL, we have

2kλj=k2jαpfβ,LχjLqp1pC2k(λα).

For the case k < L, we find that

2kλj=k2jαpfβ,LχjLqp1pC2k(λ+α+β+nq).

Therefore, (2.1) and (2.2) conclude that

fβ,LMK˙p,qα,λ(Rn)=supkZ2kλj=k2jαpfβ,LχjLqp1p<.

That is, if α + β + nq > λ > α, fβ,LMK˙p,qα,λ(ℝn).

We establish the Minkowski inequality for the Herz-Morrey space MK˙p,qα,λ(ℝn) in the following.

Theorem 2.1

Letα ∈ ℝ, λ ≥ 0 and 1 ≤ p, q < ∞. Letmbe the Lebesgue measure andμbe a signedσ-finite measure on ℝ. For anym × μmeasurable functionf(x, s) onn × ℝ, we have

Rf(,s)dμMK˙p,qα,λ(Rn)Rf(,s)MK˙p,qα,λ(Rn)d|μ|.

Proof

The Minkowski inequality for Lq guarantees that

Rf(,s)χj()dμLqRf(,s)χj()Lqd|μ|.

By applying the Minkowski inequality for p, we obtain that for any k ∈ ℤ

2kλj=k2jαpRf(,s)χj()dμLqp1p2kλj=kR2jαf(,s)χj()Lqd|μ|p1pR2kλj=k2jαpf(,s)Lqp1pd|μ|Rf(,s)MK˙p,qα,λ(Rn)d|μ|.

Finally, by taking the supremum for k ∈ ℤ on both sides of the above inequality, we get

Rf(,s)dμMK˙p,qα,λ(Rn)=supkZ2kλj=k2jαpRf(,s)χj()dμLqp1pRf(,s)MK˙p,qα,λ(Rn)d|μ|.

As a special case of Theorem 2.1, we obtain

Rf(,s)dμB˙q,θ(Rn)Rf(,s)B˙q,θ(Rn)d|μ|

which is the Minkowski inequality for the central Morrey space q,θ(ℝn).

In addition, Theorem 2.1 gives the Minkowski inequality for the Herz space. That is,

Rf(,s)dμK˙p,qα(Rn)Rf(,s)K˙p,qα(Rn)d|μ|.

Next, we study dilation operators on Herz-Morrey spaces. For any s ∈ ℝ ∖ {0} and Lebesgue measurable function f, the dilation operator Ds is defined as

(Dsf)(x)=f(x/s),xRn.

The following theorem gives us some estimates for the operator norms of Ds on MK˙p,qα,λ(ℝn).

Theorem 2.2

Letα ∈ ℝ, λ ≥ 0, 0 < p ≤ ∞ and 0 < q < ∞. There is aC > 0 such that for anys ∈ ℝ ∖ {0}

DsfMK˙p,qα,λ(Rn)Csnq+αλfMK˙p,qα,λ(Rn).

Proof

It suffices to consider s > 0 since

f()MK˙p,qα,λ(Rn)=f()MK˙p,qα,λ(Rn).

For any s > 0, there is a unique J ∈ ℤ such that 2J1s < 2J+1.

As D1/sχjχj+J–1 + χj+J + χj+J+1, j ∈ ℤ and

(Dsf)χjLq=snqf(D1/sχj)Lq,

we have

(Dsf)χjLqCsnqi=11fχj+J+iLq,jZ

for some C > 0 because ∥⋅∥Lq is a norm when 1 ≤ q < ∞ and ∥⋅∥Lq is a quasi-norm when 0 < q < 1.

Consequently,

2kλj=k2jαp(Dsf)χjLqp1pCsnqi=112kλj=k2jαpfχj+J+iLqp1pCsnq2Jλi=112(k+J+i)λ2(J+i)αj=k+J+i2jαpfχjLqp1pCsnq+αλfMK˙p,qα,λ(Rn)

for some C > 0 independent of f and s. □

We modify the definition of Boyd’s indices for rearrangement-invariant Banach function spaces from [17, Chapter 3, Definition 5.12] to define the Boyd indices for Herz-Morrey spaces.

Definition 2.2

Let α ∈ ℝ, λ ≥ 0, 0 < p ≤ ∞ and 0 < q < ∞. Define

α_MK˙p,qα,λ(Rn)=limslogDslogs,α¯MK˙p,qα,λ(Rn)=lims0+logDslogs,

where ∥Ds∥ is the operator norm of Ds : MK˙p,qα,λ(ℝn) → MK˙p,qα,λ(ℝn).

We also have the corresponding definitions of the Boyd indices for central Morrey spaces q,θ(ℝn) and Herz spaces K˙p,qα(ℝn).

Theorem 2.2 yields the formula for the Boyd indices of Herz-Morrey spaces.

Theorem 2.3

Letα ∈ ℝ, λ ≥ 0, 0 < p ≤ ∞ and 0 < q < ∞. We have

α_MK˙p,qα,λ(Rn)=α¯MK˙p,qα,λ(Rn)=nq+αλ.

Proof

Since D1/sDsf = f, ∀s > 0, (2.3) gives

D1/sDsfMK˙p,qα,λ(Rn)Csnqα+λDsfMK˙p,qα,λ(Rn).

That is,

C1snq+αλfMK˙p,qα,λ(Rn)DsfMK˙p,qα,λ(Rn).

The above inequality and (2.3) yield

C1snq+1pλDsCsnq+αλ.

Consequently, by applying the logarithm and, then, dividing by log s on the above inequalities, we obtain

logClogs+nq+αλlogDslogslogClogs+nq+αλ

when s > 1. By taking lims on the above inequalities, we find that the limit limslogDslogs exists and

α_MK˙p,qα,λ(Rn)=nq+αλ.

Similarly, when 0 < s < 1, we have

logClogs+nq+αλlogDslogslogClogs+nq+αλ.

The above inequalities ensure the existence of the limit lims0+logDslogs and give

α¯MK˙p,qα,λ(Rn)=nq+αλ.

Since q,θ(ℝn) = MK˙q,q0,nθq, we have

α_B˙q,θ(Rn)=α¯B˙q,θ(Rn)=n(1θ)q.

We can also calculate the Boyd indices for K˙p,qα(ℝn), namely,

α_K˙p,qα(Rn)=α¯K˙p,qα(Rn)=nq+α.

Moreover, for any s ∈ ℝ ∖ {0}, we also have

DsfB˙q,θ(Rn)Csn(1θ)qfB˙q,θ(Rn),DsfK˙p,qα(Rn)Csnq+αfK˙p,qα(Rn).

The above inequalities give estimates for the dilation operators on central Morrey spaces and Herz spaces.

3 Integral operators

In this section, we establish the main result of this paper, a general principle on the boundedness of integral operators and Hardy’s inequalities on Herz-Morrey spaces on ℝ. As applications of this principle, we get the boundedness of the Hadamard fractional integrals on Herz-Morrey spaces. We also obtain the Hilbert inequalities on Herz-Morrey spaces.

We consider the integral operator

Tf(t)=0K(s,t)f(s)ds,t0

and Tf(t) = 0, t < 0 where f is a Lebesgue measurable function on ℝ and K is a Lebesgue measurable function on (0, ∞) × (0, ∞). The mapping property of this operator on Lebesgue space is named as the Hardy-Littlewood-Pólya inequalities [25, Chapter IX].

The following theorem gives the boundedness of T on Herz-Morrey spaces.

Theorem 3.1

Letα ∈ ℝ, λ ≥ 0, 1 ≤ p, q < ∞ and K : (0, ∞) × (0, ∞) → ℝ be a Lebesgue measurable function. Suppose that

K(λs,λt)=λ1K(s,t),

0|K(x,1)|x1qα+λdx<.

There exists a constantC > 0 such that for anyfMK˙p,qα,λ(ℝ)

TfMK˙p,qα,λ(R)CfMK˙p,qα,λ(R).

Proof

By using the substitution u = st, we find that

|Tf(t)|0|K(ut,t)||(D1uf)(t)|tdu=0|K(u,1)||(D1uf)(t)|du

because K(⋅, ⋅) satisfies (3.1).

Theorem 2.1 guarantees that

TfMK˙p,qα,λ(R)0|K(u,1)||(D1uf)()|duMK˙p,qα,λ(R)0|K(u,1)|(D1uf)()MK˙p,qα,λ(R)du.

Theorem 2.3 yields

TfMK˙p,qα,λ(R)CfMK˙p,qα,λ(R)0|K(u,1)|u1qα+λdu

for some C > 0 because K(⋅, ⋅) fulfills (3.2). □

We give an estimate for the lower bound of the operator norm T : MK˙p,qα,λ(ℝ) → MK˙p,qα,λ(ℝ) when α + 1q > λ > α and K is a nonnegative Lebesgue measurable function satisfying (3.1), (3.2) and

0K(u,1)u1qα+λdu>0.

Let β ∈ (λα1q, 0). As α + β + 1q > λ > α, fβ,0MK˙p,qα,λ(ℝ). For any M ∈ ℕ, by using the substitution u = st, we find that

Tfβ,0(t)=0K(u,1)(D1ufβ,0)(t)du02MK(u,1)(ut)βχ(0,1)(ut)dutβχ(0,2M)(t)02MK(u,1)uβdu=2MβD2M(fβ,0(t))02MK(u,1)uβdu.

Consequently, (2.4) yields

Tfβ,0MK˙p,qα,λ(R)2MβD2Mfβ,0MK˙p,qα,λ(R)02MK(u,1)uβduC2M(1q+αλ+β)fMK˙p,qα,λ(Rn)02MK(u,1)uβdu

for some C > 0 independent of M and β. For any β ∈ (λα1q, 0), we have

TMK˙p,qα,λ(R)MK˙p,qα,λ(R)=supfMK˙p,qα,λ(R)1TfMK˙p,qα,λ(R)fMK˙p,qα,λ(R)C2M(1q+αλ+β)02MK(u,1)uβdu

for some C > 0 independent of M and β.

By applying the limit limβλα1q on both sides of the above inequalities, the dominated convergence theorem yields

TMK˙p,qα,λ(R)MK˙p,qα,λ(R)Climβλα1q02MK(u,1)uβdu=C02MK(u,1)u1qα+λdu

for some C > 0 independent of M because uβ2M(β+1q+αλ)u1qα+λ when u ∈ (0, 2M).

Finally, by letting M trending to infinity, we have

TMK˙p,qα,λ(R)MK˙p,qα,λ(R)C0K(u,1)u1qα+λdu.

For the estimates of the operator norms of integral operators on weighted Morrey spaces, see [26].

The boundedness of T on MK˙p,qα,λ(ℝ) relies on the integral condition (3.2) where the Boyd’s indices of MK˙p,qα,λ(ℝ), – 1qα + λ, involve in (3.2). This is the main reason for the introduction of the Boyd’s indices for Herz-Morrey spaces in the previous section.

For the boundedness of the above integral operator on Morrey spaces, block spaces, amalgam spaces, function space of bounded mean oscillation BMO, Campanato spaces and ball Banach function spaces, see [27, 28, 29, 30, 31], respectively.

As a consequence of Theorem 3.1, we have the following boundedness result for the integral operator T on central Morrey spaces.

Corollary 3.2

Let 0 < θ < 1, 1 ≤ q < ∞ and K : (0, ∞) × (0, ∞) → ℝ be a Lebesgue measurable function. Suppose thatKsatisfies (3.1) and

0|K(x,1)|x1θqdx<.

There exists a constantC > 0 such that for anyfq,θ(ℝ)

TfB˙q,θ(R)CfB˙q,θ(R).

Similar to the discussion on the lower estimate of the operator norm of T : MK˙p,qα,λ(ℝ) → MK˙p,qα,λ(ℝ), we also have the lower estimate of the operator norm of T : q,θ(ℝ) → q,θ(ℝ). We have

TB˙q,θ(R)B˙q,θ(R)C0K(x,1)x1θqdx

when K is nonnegative.

In addition, we have the following result for Herz spaces.

Corollary 3.3

Letα ∈ ℝ, 1 ≤ p, q < ∞ and K : (0, ∞) × (0, ∞) → ℝ be a Lebesgue measurable function. Suppose thatKsatisfies (3.1) and

0|K(x,1)|x1qαdx<.

There exists a constantC > 0 such that for anyfK˙p,qα(ℝ)

TfK˙p,qα(R)CfK˙p,qα(R).

3.1 Hardy’s inequality and Hilbert’s inequality

In this section, we present another main result of this paper, the Hardy’s inequalities on Herz-Morrey spaces. We also study the Hilbert inequality on the Herz-Morrey space MK˙p,qα,λ(ℝ). They are applications of Theorem 3.1.

We begin with the definitions of the Hardy operators

Hf(t)=1t0tf(s)ds,H~f(t)=tf(s)sds.

For the history, development and applications of the Hardy’s inequality, the reader is referred to [32, 33, 34].

For the Hardy’s inequalities on non-Lebesgue space such as the Morrey spaces, the block spaces, the amalgam spaces, Hardy type spaces and rearrangement-invariant Banach function spaces, see [28, 29, 30, 35, 36, 37, 38, 39, 40].

The following is the Hardy’s inequality on Herz-Morrey spaces.

Theorem 3.4

Letα ∈ ℝ, λ ≥ 0 and 1 ≤ p, q < ∞. If1q + αλ < 1, then there is a constantC > 0 such that for anyfMK˙p,qα,λ(ℝ)

HfMK˙p,qα,λ(R)CfMK˙p,qα,λ(R).

Proof

Let K(s, t) = t–1χE(s, t) where E = {(s, t) : s < t}. We find that for any λ > 0, K(λs, λt) = λ–1K(s, t). Moreover, K satisfies

0|K(x,1)|x1qα+λdx=01x1qα+λdx=x1qα+λ+11qα+λ+1|01<

because 1q + αλ < 1. Theorem 3.1 gives the Hardy’s inequality on MK˙p,qα,λ(ℝ). □

We also have the corresponding result for the operator .

Theorem 3.5

Letα ∈ ℝ, λ ≥ 0 and 1 ≤ p, q < ∞. If 0 < 1q + αλ, there is a constantC > 0 such that for anyfMK˙p,qα,λ(ℝ)

H~fMK˙p,qα,λ(R)CfMK˙p,qα,λ(R).

Since the proof of the preceding theorem is similar to the proof of Theorem 3.4, for simplicity, we leave the details to the reader.

Next, we establish the Hilbert’s inequality on the Herz-Morrey spaces. For any Lebesgue measurable function f, the Hilbert operator is defined as

Hf(t)=0f(s)s+tds.

Theorem 3.6

Letα ∈ ℝ, λ ≥ 0 and 1 ≤ p, q < ∞. If 0 < 1q + αλ < 1, then there is a constantC > 0 such that for anyfMK˙p,qα,λ(ℝ)

HfMK˙p,qα,λ(R)CfMK˙p,qα,λ(R).

Proof

We have

Hf(t)=0K(s,t)f(s)ds

where K(s, t) = 1s+t. K obviously satisfies (3.1). Since 0 < 1q + αλ < 1, we obtain

0|K(x,1)|x1qα+λdx=0(x+1)1x1qα+λdx01x1qα+λdx+1x1qα+λ1dx<.

Consequently, the boundedness of 𝓗 on MK˙p,qα,λ(ℝ) is assured by Theorem 3.1. □

In particular, we have the following results for central Morrey spaces and Herz spaces.

Corollary 3.7

Let 0 < θ < 1 and 1 ≤ q < ∞.

  1. There is a constantC > 0 such that for anyfq,θ(ℝ), we have
    HfB˙q,θ(R)CfB˙q,θ(R).
  2. There is a constantC > 0 such that for anyfq,θ(ℝ),
    H~fB˙q,θ(R)CfB˙q,θ(R).
  3. There is a constantC > 0 such that for anyfq,θ(ℝ),
    HfB˙q,θ(R)CfB˙q,θ(R).

We have the above results because 0 < 1θq < 1 is valid when 0 < θ < 1 and 1 ≤ q < ∞.

Corollary 3.8

Letα ∈ ℝ and 1 ≤ p, q < ∞.

  1. If1q + α < 1, then there is a constantC > 0 such that for anyfK˙p,qα(ℝ), we have
    HfK˙p,qα(R)CfK˙p,qα(R).
  2. If 0 < 1q + α, there is a constantC > 0 such that for anyfK˙p,qα(ℝ),
    H~fK˙p,qα(R)CfK˙p,qα(R).
  3. If 0 < 1q + α < 1, then there is a constantC > 0 such that for anyfK˙p,qα(ℝ),
    HfK˙p,qα(R)CfK˙p,qα(R).

In particular, when α < 1 – 1q, we have

HfK˙q,qα(R)CfK˙q,qα(R).

Notice that K˙q,qα(ℝ) is the power weighted Lebesgue space Lq((0, ∞), |x|αq) [22, Remark 1.1.3]. This result recovers the well known results for the Hardy’s inequality on power weighted Lebesgue spaces, see [40, 41, 42].

Furthermore, Corollary 3.8 also gives the Hilbert inequality on power weighted Lebesgue spaces Lq((0, ∞), |x|αq) when 0 < 1q + α < 1.

3.2 Hadamard fractional integrals

The Hadamard fractional integrals are the fractional integrals corresponding to the Mellin transform

Mf(s)=0us1f(u)du,s=c+it,c,tR,

see [43].

In [43], Butzer, Kilbas and Trujillo introduce and study the following generalizations of Hadamard fractional integrals. They are defined by using the confluent hypergeometric function, which is also named as Kummer function. The confluent hypergeometric function Φ[a, c; z] is defined for |z| < 1, c > 0 and a ≠ –j, j ∈ ℕ ∪ {0} by

Φ[a,c;z]=k=0(a)k(c)kzkk!,

where (a)k, k ∈ ℕ ∪ {0}, is the Pochhammer symbol [44, Section 6.1] given by

(a)0=1,(a)k=a(a+1)(a+k1),kN.

For β > 0, γ ∈ ℝ and μ, σ ∈ ℂ, the generalized Hadamard fractional integrals J0+,μ;γ,σβf,J,μ;γ,σβf,I0+,μ;γ,σβf and I,μ;γ,σβf are defined as

J0+,μ;γ,σβf(x)=1Γ(β)0xtxμlogxtβ1Φγ,β;σlogxtf(t)dtt,J,μ;γ,σβf(x)=1Γ(β)xxtμlogtxβ1Φγ,β;σlogtxf(t)dtt,I0+,μ;γ,σβf(x)=1Γ(β)0xtxμlogxtβ1Φγ,β;σlogxtf(t)dtx,I,μ;γ,σβf(x)=1Γ(β)xxtμlogtxβ1Φγ,β;σlogtxf(t)dtx,

and

(J0+,μ;γ,σβf)(x)=(J,μ;γ,σβf)(x)=(I0+,μ;γ,σβf)(x)=(I,μ;γ,σβf)(x)=0,x0,

where Γ(β) is the Gamma function.

Note that Φ[a, c; 0] = 1, when σ = 0, the above Hadamard fractional integral J0+,μ;γ,0β becomes the Hadamard fractional integral J0+β. Additionally, J,μ;γ,σβ,I0+,μ;γ,σβ and I,μ;γ,σβ are the Hadamard type fractional integrals introduced and studied in [21]. For the studies of these integrals and their applications on fractional calculus, see [21, 43, 45].

In order to obtain the mapping of the generalized Hadamard fractional integrals, we need to use the following asymptotic behaviours for Φ[a, c; x]

Φ[a,c;x]=Γ(c)Γ(a)exxac1+O1xasx.

Moreover, the limit

limk(a)k+1(c)k+1(k+1)!(a)k(c)kk!=limka+kc+k1k+1=0

assures that

Φ[a,c;x]=1+O(x)asx0+.

We are now ready to establish the boundedness of the generalized Hadamard fractional integrals on Herz-Morrey spaces.

Theorem 3.9

Let 1 ≤ p, q < ∞, β > 0, λ ≥ 0, α ∈ ℝ, γ ∈ ℝ andμ, σ ∈ ℂ.

  1. If Re(μσ) > 1q + αλ, then there exists a constantC > 0 such that for anyfMK˙p,qα,λ(ℝ), we have
    J0+,μ;γ,σβfMK˙p,qα,λ(R)CfMK˙p,qα,λ(R).
  2. If Re(μσ) > – 1qα + λ, then there exists a constantC > 0 such that for anyfMK˙p,qα,λ(ℝ), we have
    J,μ;γ,σβfMK˙p,qα,λ(R)CfMK˙p,qα,λ(R).
  3. If Re(μσ) > 1q + αλ – 1, then there exists a constantC > 0 such that for anyfMK˙p,qα,λ(ℝ), we have
    I0+,μ;γ,σβfMK˙p,qα,λ(R)CfMK˙p,qα,λ(R).
  4. If Re(μσ) > 1 – 1qα + λ, then there exists a constantC > 0 such that for anyfMK˙p,qα,λ(ℝ), we have
    I,μ;γ,σβfMK˙p,qα,λ(R)CfMK˙p,qα,λ(R).

Proof

Let E = {(u, x) ∈ (0, ∞) × (0, ∞) : u < x} and F = {(u, x) ∈ (0, ∞) × (0, ∞) : x < u}. We first consider integral J0+,μ;γ,σβ. We have

J0+,μ;γ,σβf(x)=0K1(u,x)f(u)du,

where

K1(u,x)=1Γ(β)uxμlogxuβ1Φγ,β;σlogxu1uχE(u,x).

For any λ > 0, we find that

K1(λu,λx)=1Γ(β)λuλxμlogλxλuβ1Φγ,β;σlogλxλu1λuχE(λu,λx)=λ1K1(u,x)

since χE(λu, λx) = χE(u, x). Therefore, (3.1) is fulfilled.

Since

K1(u,1)=1Γ(β)uμ1(logu)β1Φγ,β;σlog1uχ{u:0<u<1},

(3.3) and (3.4) give

K1(u,1)=1Γ(γ)uμσ1(logu)β1+γβσγβ1+O1σlog1u

as u → 0+ and

K1(u,1)=1Γ(β)uμσ1(logu)β11+Oσlog1u

as u → 1.

By using the substitution y = –log u, we have

0|K1(u,1)|u1qα+λdu=01|K1(u,1)|u1qα+λdu=0|K1(ey,1)|ey(1qα+λ+1)dyC(02ey(Re(μσ)1qα+λ)yβ1dy+2ey(Re(μσ)1qα+λ)y1+γdy).

Since Re(μσ) > 1q + αλ, we have a constant C > 0 such that

02ey(Re(μσ)1qα+λ)yβ1dyC02yβ1dy<C.

Furthermore, we also have an ϵ > 0 such that Re(μσ) > 1q + αλ + ϵ and

2ey(Re(μσ)1qα+λ)y1+γdy<2ey(Re(μσ)1qα+λϵ)dy<C.

Consequently, (3.2) is fulfilled and Theorem 3.1 guarantees (3.5).

Next, we consider J,μ;γ,σβ. We have

J,μ;γ,σβf(x)=0K2(u,x)f(u)du,

where

K2(u,x)=1Γ(β)xuμloguxβ1Φγ,β;σlogux1uχF(u,x).

Obviously, for any λ > 0, K2(λu, λx) = λ–1K2(u, x) and

K2(u,1)=1Γ(β)uμ(logu)β1Φγ,β;σloguu1χ{u:1<u}.

By using the substitution y = log u, we find that

0|K2(u,1)|u1qα+λdu=1|K2(u,1)|u1qα+λdu=0|K2(ey,1)|ey(1qα+λ+1)dy.

Consequently, (3.3) and (3.4) yield

0|K2(u,1)|u1qα+λduC02yβ1dy+2ey(Re(μσ)1qα+λ)y1+γdy<

because Re(μσ) > – 1qα + λ. Therefore, Theorem 3.1 yields (3.6).

We consider the operator I0+,μ;γ,σβ. We have

I0+,μ;γ,σβf(x)=0K3(u,x)f(u)du,

where

K3(u,x)=1Γ(β)uxμlogxuβ1Φγ,β;σlogxu1xχE(u,x).

The function K3 fulfills (3.1) and

K3(u,1)=1Γ(β)uμ(logu)β1Φγ,β;σlog1uχ{u:0<u<1}.

As Re(μσ) > 1q + αλ – 1, by using the substitution y = –log u, we get

0|K3(u,1)|u1qα+λdu=01|K3(u,1)|u1qα+λdu=0|K3(ey,1)|ey(1qα+λ+1)dyC02yβ1dy+2ey(Re(μσ)1qα+λ+1)y1+γdy<

because Re(μσ) > 1q + αλ – 1. Theorem 3.1 gives the boundedness of I0+,μ;γ,σβ.

Finally, we consider I,μ;γ,σβ. We have

I,μ;γ,σβf(x)=0K4(u,x)f(u)du,

where

K4(u,x)=1Γ(β)xuμloguxβ1Φγ,β;σlogux1xχF(u,x).

We see that K4 satisfies (3.1) and

K4(u,1)=1Γ(β)uμ(logu)β1Φγ,β;σloguχ{u:1<u}.

Since Re(μσ) > 1 – 1qα + λ, by using the substitution y = log u, we obtain

0|K4(u,1)|u1qα+λdu=1|K4(u,1)|u1qα+λdu=0|K4(ey,1)|ey(1qα+λ+1)dyC02yβ1dy+2ey(Re(μσ)1qα+λ+1)y1+γdy<

Theorem 3.1 guarantees the validity of (3.8). □

Since the central Morrey space q,θ(ℝ) and the Herz space K˙p,qα(ℝ) are members of Her-Morrey spaces, as special cases of Theorem 3.9, we have the mapping properties of the Hadamard fractional integrals on central Morrey spaces and Herz spaces.

Corollary 3.10

Let 1 ≤ q < ∞, β > 0, 0 < θ < 1, γ ∈ ℝ andμ, σ ∈ ℂ.

  1. If Re(μσ) > 1θq, then there exists a constantC > 0 such that for anyfq,θ(ℝ), we have
    J0+,μ;γ,σβfB˙q,θ(R)CfB˙q,θ(R).
  2. If Re(μσ) > – 1θq, then there exists a constantC > 0 such that for anyfq,θ(ℝ), we have
    J,μ;γ,σβfB˙q,θ(R)CfB˙q,θ(R).
  3. If Re(μσ) > 1θq – 1, then there exists a constantC > 0 such that for anyfq,θ(ℝ), we have
    I0+,μ;γ,σβfB˙q,θ(R)CfB˙q,θ(R).
  4. If Re(μσ) > 1 – 1θq, then there exists a constantC > 0 such that for anyfq,θ(ℝ), we have
    I,μ;γ,σβfB˙q,θ(R)CfB˙q,θ(R).

Corollary 3.11

Let 1 ≤ p, q < ∞, β > 0, α ∈ ℝ, γ ∈ ℝ andμ, σ ∈ ℂ.

  1. If Re(μσ) > 1q + α, then there exists a constantC > 0 such that for anyfK˙p,qα(ℝ), we have
    J0+,μ;γ,σβfK˙p,qα(R)CfK˙p,qα(R).
  2. If Re(μσ) > – 1qα, then there exists a constantC > 0 such that for anyfK˙p,qα(ℝ), we have
    J,μ;γ,σβfK˙p,qα(R)CfK˙p,qα(R).
  3. If Re(μσ) > 1q + α – 1, then there exists a constantC > 0 such that for anyfK˙p,qα(ℝ), we have
    I0+,μ;γ,σβfK˙p,qα(R)CfK˙p,qα(R).
  4. If Re(μσ) > 1 – 1qα, then there exists a constantC > 0 such that for anyfK˙p,qα(ℝ), we have
    I,μ;γ,σβfK˙p,qα(R)CfK˙p,qα(R).

For the studies of fractional Hadamard integrals on other function spaces such as amalgam spaces, BMO and modular spaces, see [27, 30, 46].

Acknowledgments

The authors would like to thank the referees for careful reading of the paper and valuable suggestions.

References

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    C.S. Herz, Lipschitz spaces and Bernstein’s theorem on absolutely convergent Fourier transforms, J. Math. Mech. 18 (1968/69), 283–323.

  • [2]

    C. Morrey, On the solutions of quasi-linear elliptic partial differential equations, Trans. Amer. Math. Soc. 43 (1938), 126–166.

  • [3]

    J. Alvarez, M. Guzmán-Partida, and J. Lakey, Spaces of bounded λ-central mean oscillation, Morrey spaces, and λ-central Carleson measures, Collect. Math. 51 (2000), 1–47.

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    Z. Fu, Y. Lin, and S.-Z. Lu, λ-central BMO estimates for commutators of singular integral operators with rough kernels, Acta Math. Sinica (Engl. Ser.) 24 (2008), 373–386.

  • [5]

    Y. Komori-Furuya and E. Sato, Fractional integral operators on central Morrey spaces, Math. Inequal. Appl. 20 (2017), 801–813.

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    V.S. Guliyev, S.G. Hasanov, and Y. Sawano, Decompositions of local Morrey-type spaces, Positivity 21 (2017), 1223–1252.

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    S. Lu and L. Xu, Boundedness of rough singular integral operators on the homogeneous Morrey-Herz spaces, Hokkaido Math. J. 34 (2005), 299–314.

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    J. Xu and X. Yang, Herz-Morrey-Hardy spaces with variable exponents and their applications, J. Funct. Spaces (2015), Article ID 160635, .

    • Crossref
    • Export Citation
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    J. Xu and X. Yang, The molecular decomposition of Herz-Morrey-Hardy spaces with variable exponents and its application, J. Math. Inequal. 10 (2016), 977–1008.

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    M. Izuki, Boundedness of vector-valued sublinear operators on Herz-Morrey spaces with variable exponent, Math. Sci. Res. J. 13 (2009), 243–253.

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    M. Izuki, Fractional integrals on Herz-Morrey spaces with variable exponent, Hiroshima Math. J. 40 (2010), 343–355.

  • [12]

    Y. Mizuta and T. Ohno, Sobolev’s theorem and duality for Herz-Morrey spaces of variable exponent, Ann. Acad. Sci. Fenn. Math. 39 (2014), 389–416.

  • [13]

    Y. Mizuta and T. Ohno, HerzMorrey spaces of variable exponent, Riesz potential operator and duality, Complex Var. Theory Appl. 60 (2015), 211–240.

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    Y. Mizuta, Duality of Herz-Morrey spaces of variable exponent, Filomat 30 (2016), 1891–1898.

  • [15]

    J. Wu, Boundedness of some sublinear operators on HerzMorrey spaces with variable exponent, Georgian Math. J. 21 (2014), 101–111.

  • [16]

    B. Dong and J. Xu, Herz-Morrey type Besov and Triebel-Lizorkin spaces with variable exponents, Banach J. Math. Anal. 9 (2015), 75–101.

  • [17]

    C. Bennett and R. Sharpley, Interpolations of Operators, Academic Press, 1988.

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    K.-P. Ho, Fourier integrals and Sobolev embedding on rearrangement invariant quasi-Banach function spaces, Ann. Acad. Sci. Fenn. Math. 41 (2016), 897–922.

  • [19]

    K.-P. Ho, Modular interpolation and modular estimate of Fourier transform and related operators, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 28 (2017), 349–368.

  • [20]

    K.-P. Ho, Fourier type transforms on rearrangement-invariant quasi-Banach function spaces, Glasgow Math. J. 61 (2019), 231–248.

  • [21]

    P. Butzer, A. Kilbas, and J. Trujillo, Fractional calculus in the Mellin setting and Hadamard-type fractional integrals, J. Math. Anal. Appl. 269 (2002), 1–27.

  • [22]

    S.-Z. Lu, D. Yang, and G. Hu, Herz type spaces and their applications, Science Press, Beijing, 2008.

  • [23]

    Y. Komori-Furuya, K. Matsuoka, E. Nakai, and Y. Sawano, Applications of Littlewood-Paley theory for σ-Morrey spaces to the boundedness of integral operators, J. Funct. Spaces Appl. (2013), Art. ID 859402.

  • [24]

    Y. Komori-Furuya, K. Matsuoka, E. Nakai, and Y. Sawano, Integral operators on Bσ-Morrey-Campanato spaces, Rev. Mat. Complut. 26 (2013), 1–32.

  • [25]

    G. Hardy, J. Littlewood, and G. Pólya, Inequalities, Cambridge University Press, 1934.

  • [26]

    T. Batbold and Y. Sawano, Sharp bounds for m-linear Hilbert-type operators on the weighted Morrey spaces, Math. Inequal. Appl. 20 (2017), 263–283.

  • [27]

    K.-P. Ho, Integral operators on BMO and Campanato spaces, Indag. Mat. 30 (2019), 1023–1035.

  • [28]

    K.-P. Ho, Hardy’s inequality and Hausdorff operators on rearrangement-invariant Morrey spaces, Publ. Math. Debrecen 88 (2016), 201–215.

  • [29]

    K.-P. Ho, Hardy-Littlewood-Pólya inequalities and Hausdorff operators on block spaces, Math. Inequal. Appl. 19 (2016), 697–707.

  • [30]

    K.-P. Ho, Dilation operators and Integral operators on amalgam (Lp, lq), Ricerche Mat. 68 (2019), 661–677.

  • [31]

    K.-P. Ho, Erdélyi-Kober fractional integral operators on ball Banach function spaces, Rend. Semin. Mat. Univ. Padova, 2019.

  • [32]

    A. Kufner, L. Maligranda, and L.-E. Persson, The prehistory of the Hardy inequality, Amer. Math. Monthly 113 (2006), 715–732.

  • [33]

    A. Kufner, L.-E. Persson, and N. Samko, Weighted inequalities of Hardy type, World Scientific Publishing Company, 2017.

  • [34]

    B. Opic and A. Kufner, Hardy-type inequalities, Pitman Reserach Notes in Math. Series 219, Longman Sci. and Tech, Harlow, 1990.

  • [35]

    K.-P. Ho, Hardy’s inequality on Hardy spaces, Proc. Japan Acad. Ser. A Math. Sci. 92 (2016), 125–130.

  • [36]

    K.-P. Ho, Hardy’s inequality on Hardy-Morrey spaces with variable exponents, Mediterr. J. Math. 14 (2017), 79–98.

  • [37]

    K.-P. Ho, Atomic decompositions and Hardy’s inequality on weak Hardy-Morrey spaces, Sci. China Math. 60 (2017), 449–468.

  • [38]

    K.-P. Ho, Discrete Hardy’s inequality with 0 < p ≤ 1, J. King Saud Univ. Sci. 30 (2018), 489–492.

  • [39]

    L. Maligranda, Generalized Hardy inequalities in rearrangement invariant spaces, J. Math. Pures Appl. 59 (1980), 405–415.

  • [40]

    E. Sawyer, Weighted Lebesgue and Lorentz norm inequalities for the Hardy operators, Trans. Amer. Math. Soc. 281 (1984), 329–337.

  • [41]

    K. Andersen and B. Muckenhoupt, Weighted weak type Hardy inequalities with applications in Hilbert transforms and maximal functions, Studia Math. 72 (1982), 9–26.

  • [42]

    B. Muckenhoupt, Hardy’s inequality with weights, Studia Math. 34 (1972), 31–38.

  • [43]

    P. Butzer, A. Kilbas, and J. Trujillo, Compositions of Hadamard-type fractional integration operators and the semigroup property, J. Math. Anal. Appl. 269 (2002), 387–400.

  • [44]

    A. Erdelyi, W. Magnus, F. Oberhettinger, and F. Tricomi, Higher Transcendental Functions, Vol 1, McGraw-Hill, New York, 1953.

  • [45]

    S. Samko, A. Kilbas, and O. Marichev, Fractional Integrals and Derivatives, Theory and Applications, Gordan and Breach, New York, 1993.

  • [46]

    K.-P. Ho, Modular Hadamard, Riemann-Liouville and Weyl fractional integrals, (preprint).

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  • [1]

    C.S. Herz, Lipschitz spaces and Bernstein’s theorem on absolutely convergent Fourier transforms, J. Math. Mech. 18 (1968/69), 283–323.

  • [2]

    C. Morrey, On the solutions of quasi-linear elliptic partial differential equations, Trans. Amer. Math. Soc. 43 (1938), 126–166.

  • [3]

    J. Alvarez, M. Guzmán-Partida, and J. Lakey, Spaces of bounded λ-central mean oscillation, Morrey spaces, and λ-central Carleson measures, Collect. Math. 51 (2000), 1–47.

  • [4]

    Z. Fu, Y. Lin, and S.-Z. Lu, λ-central BMO estimates for commutators of singular integral operators with rough kernels, Acta Math. Sinica (Engl. Ser.) 24 (2008), 373–386.

  • [5]

    Y. Komori-Furuya and E. Sato, Fractional integral operators on central Morrey spaces, Math. Inequal. Appl. 20 (2017), 801–813.

  • [6]

    V.S. Guliyev, S.G. Hasanov, and Y. Sawano, Decompositions of local Morrey-type spaces, Positivity 21 (2017), 1223–1252.

  • [7]

    S. Lu and L. Xu, Boundedness of rough singular integral operators on the homogeneous Morrey-Herz spaces, Hokkaido Math. J. 34 (2005), 299–314.

  • [8]

    J. Xu and X. Yang, Herz-Morrey-Hardy spaces with variable exponents and their applications, J. Funct. Spaces (2015), Article ID 160635, .

    • Crossref
    • Export Citation
  • [9]

    J. Xu and X. Yang, The molecular decomposition of Herz-Morrey-Hardy spaces with variable exponents and its application, J. Math. Inequal. 10 (2016), 977–1008.

  • [10]

    M. Izuki, Boundedness of vector-valued sublinear operators on Herz-Morrey spaces with variable exponent, Math. Sci. Res. J. 13 (2009), 243–253.

  • [11]

    M. Izuki, Fractional integrals on Herz-Morrey spaces with variable exponent, Hiroshima Math. J. 40 (2010), 343–355.

  • [12]

    Y. Mizuta and T. Ohno, Sobolev’s theorem and duality for Herz-Morrey spaces of variable exponent, Ann. Acad. Sci. Fenn. Math. 39 (2014), 389–416.

  • [13]

    Y. Mizuta and T. Ohno, HerzMorrey spaces of variable exponent, Riesz potential operator and duality, Complex Var. Theory Appl. 60 (2015), 211–240.

  • [14]

    Y. Mizuta, Duality of Herz-Morrey spaces of variable exponent, Filomat 30 (2016), 1891–1898.

  • [15]

    J. Wu, Boundedness of some sublinear operators on HerzMorrey spaces with variable exponent, Georgian Math. J. 21 (2014), 101–111.

  • [16]

    B. Dong and J. Xu, Herz-Morrey type Besov and Triebel-Lizorkin spaces with variable exponents, Banach J. Math. Anal. 9 (2015), 75–101.

  • [17]

    C. Bennett and R. Sharpley, Interpolations of Operators, Academic Press, 1988.

  • [18]

    K.-P. Ho, Fourier integrals and Sobolev embedding on rearrangement invariant quasi-Banach function spaces, Ann. Acad. Sci. Fenn. Math. 41 (2016), 897–922.

  • [19]

    K.-P. Ho, Modular interpolation and modular estimate of Fourier transform and related operators, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 28 (2017), 349–368.

  • [20]

    K.-P. Ho, Fourier type transforms on rearrangement-invariant quasi-Banach function spaces, Glasgow Math. J. 61 (2019), 231–248.

  • [21]

    P. Butzer, A. Kilbas, and J. Trujillo, Fractional calculus in the Mellin setting and Hadamard-type fractional integrals, J. Math. Anal. Appl. 269 (2002), 1–27.

  • [22]

    S.-Z. Lu, D. Yang, and G. Hu, Herz type spaces and their applications, Science Press, Beijing, 2008.

  • [23]

    Y. Komori-Furuya, K. Matsuoka, E. Nakai, and Y. Sawano, Applications of Littlewood-Paley theory for σ-Morrey spaces to the boundedness of integral operators, J. Funct. Spaces Appl. (2013), Art. ID 859402.

  • [24]

    Y. Komori-Furuya, K. Matsuoka, E. Nakai, and Y. Sawano, Integral operators on Bσ-Morrey-Campanato spaces, Rev. Mat. Complut. 26 (2013), 1–32.

  • [25]

    G. Hardy, J. Littlewood, and G. Pólya, Inequalities, Cambridge University Press, 1934.

  • [26]

    T. Batbold and Y. Sawano, Sharp bounds for m-linear Hilbert-type operators on the weighted Morrey spaces, Math. Inequal. Appl. 20 (2017), 263–283.

  • [27]

    K.-P. Ho, Integral operators on BMO and Campanato spaces, Indag. Mat. 30 (2019), 1023–1035.

  • [28]

    K.-P. Ho, Hardy’s inequality and Hausdorff operators on rearrangement-invariant Morrey spaces, Publ. Math. Debrecen 88 (2016), 201–215.

  • [29]

    K.-P. Ho, Hardy-Littlewood-Pólya inequalities and Hausdorff operators on block spaces, Math. Inequal. Appl. 19 (2016), 697–707.

  • [30]

    K.-P. Ho, Dilation operators and Integral operators on amalgam (Lp, lq), Ricerche Mat. 68 (2019), 661–677.

  • [31]

    K.-P. Ho, Erdélyi-Kober fractional integral operators on ball Banach function spaces, Rend. Semin. Mat. Univ. Padova, 2019.

  • [32]

    A. Kufner, L. Maligranda, and L.-E. Persson, The prehistory of the Hardy inequality, Amer. Math. Monthly 113 (2006), 715–732.

  • [33]

    A. Kufner, L.-E. Persson, and N. Samko, Weighted inequalities of Hardy type, World Scientific Publishing Company, 2017.

  • [34]

    B. Opic and A. Kufner, Hardy-type inequalities, Pitman Reserach Notes in Math. Series 219, Longman Sci. and Tech, Harlow, 1990.

  • [35]

    K.-P. Ho, Hardy’s inequality on Hardy spaces, Proc. Japan Acad. Ser. A Math. Sci. 92 (2016), 125–130.

  • [36]

    K.-P. Ho, Hardy’s inequality on Hardy-Morrey spaces with variable exponents, Mediterr. J. Math. 14 (2017), 79–98.

  • [37]

    K.-P. Ho, Atomic decompositions and Hardy’s inequality on weak Hardy-Morrey spaces, Sci. China Math. 60 (2017), 449–468.

  • [38]

    K.-P. Ho, Discrete Hardy’s inequality with 0 < p ≤ 1, J. King Saud Univ. Sci. 30 (2018), 489–492.

  • [39]

    L. Maligranda, Generalized Hardy inequalities in rearrangement invariant spaces, J. Math. Pures Appl. 59 (1980), 405–415.

  • [40]

    E. Sawyer, Weighted Lebesgue and Lorentz norm inequalities for the Hardy operators, Trans. Amer. Math. Soc. 281 (1984), 329–337.

  • [41]

    K. Andersen and B. Muckenhoupt, Weighted weak type Hardy inequalities with applications in Hilbert transforms and maximal functions, Studia Math. 72 (1982), 9–26.

  • [42]

    B. Muckenhoupt, Hardy’s inequality with weights, Studia Math. 34 (1972), 31–38.

  • [43]

    P. Butzer, A. Kilbas, and J. Trujillo, Compositions of Hadamard-type fractional integration operators and the semigroup property, J. Math. Anal. Appl. 269 (2002), 387–400.

  • [44]

    A. Erdelyi, W. Magnus, F. Oberhettinger, and F. Tricomi, Higher Transcendental Functions, Vol 1, McGraw-Hill, New York, 1953.

  • [45]

    S. Samko, A. Kilbas, and O. Marichev, Fractional Integrals and Derivatives, Theory and Applications, Gordan and Breach, New York, 1993.

  • [46]

    K.-P. Ho, Modular Hadamard, Riemann-Liouville and Weyl fractional integrals, (preprint).

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