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

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.


Introduction
This paper focuses on the Hardy's inequalities and the boundedness of the integral operators on Herz-Morrey spaces.
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 nd 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 de ne 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 L (R) = MK , , (R). On the other hand, our result shows that the Hardy operator is bounded on the Herz-Morrey MK α,λ , (R) when α < λ. The reader is referred to De nition 2.1 for the de nition of the Herz-Morrey space MK α,λ p,q (R n ). 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 de nition 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 R n while the remaining results are on Herz-Morrey spaces over R.

Herz-Morrey spaces and Boyd's indices
We give the de nition 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 De nition 2.1. Let α ∈ R, λ ≥ , < p ≤ ∞ and < q < ∞. The Herz-Morrey space MK α,λ p,q (R n ) consists of all Lebesgue measurable functions f satisfying When λ = , the Herz-Morrey space MK α,λ p,q (R n ) becomes the Herz spaceK α p,q (R n ) studied in [22]. In addition, when α = and p = q, the Herz-Morrey space MK α,λ p,q (R n ) reduces to the Lebesgue space L q . Furthermore, when α = , p = q and λ = nθ q with < θ < , the Herz-Morrey space MK α,λ p,q (R n ) is the central Morrey spaceḂ q,θ (R n ) [3-6, 23, 24]. Recall that the central Morrey spaceḂ q,θ (R n ) consists of all Lebesgue measurable functions f satisfying The reader is referred to [3][4][5] for the studies of central Morrey spaces. We use the de nition of central Morrey spaces from [5, De nition 2] while we use the notion for the central Morrey spaces from [3,4].
For β ∈ R, L ∈ Z, de ne f β,L (x) = |x| β χ L (x). For any j ∈ Z, we have where C is independent of j.
Therefore, (2.1) and (2.2) conclude that We establish the Minkowski inequality for the Herz-Morrey space MK α,λ p,q (R n ) in the following.
Proof. The Minkowski inequality for L q guarantees that By applying the Minkowski inequality for p , we obtain that for any Finally, by taking the supremum for k ∈ Z on both sides of the above inequality, we get As a special case of Theorem 2.1, we obtain which is the Minkowski inequality for the central Morrey spaceḂ q,θ (R n ). In addition, Theorem 2.1 gives the Minkowski inequality for the Herz space. That is, Next, we study dilation operators on Herz-Morrey spaces. For any s ∈ R\{ } and Lebesgue measurable function f , the dilation operator Ds is de ned as The following theorem gives us some estimates for the operator norms of Ds on MK α,λ p,q (R n ).
There is a C > such that for any s ∈ R\{ } Proof. It su ces to consider s > since we have for some C > independent of f and s.
where Ds is the operator norm of Ds : We also have the corresponding de nitions of the Boyd indices for central Morrey spacesḂ q,θ (R n ) and Herz spacesK α p,q (R n ). Theorem 2.2 yields the formula for the Boyd indices of Herz-Morrey spaces.
The above inequality and (2. 3) yield Consequently, by applying the logarithm and, then, dividing by log s on the above inequalities, we obtain Similarly, when < s < , we have The above inequalities ensure the existence of the limit lim s→ + log Ds log s and give We can also calculate the Boyd indices forK α p,q (R n ), namely, Moreover, for any s ∈ R\{ }, we also have Ds f Kα p,q (R n ) ≤ Cs n q +α f Kα p,q (R n ) .
The above inequalities give estimates for the dilation operators on central Morrey spaces and Herz spaces.

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 There exists a constant C > such that for any f ∈ MK α,λ p,q (R) Proof. By using the substitution u = s t , we nd that because K(·, ·) satis es (3.1). Theorem 2.1 guarantees that

Consequently, (2.4) yields
Finally, by letting M trending to in nity, we have For the estimates of the operator norms of integral operators on weighted Morrey spaces, see [26].
The boundedness of T on MK α,λ p,q (R) relies on the integral condition (3.2) where the Boyd's indices of MK α,λ p,q (R), − q − α + 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.
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 < θ < , ≤ q < ∞ and K : ( , ∞) × ( , ∞) → R be a Lebesgue measurable function. Suppose that K satis es (3.1) and There exists a constant C > such that for any f ∈Ḃ q,θ (R) Similar to the discussion on the lower estimate of the operator norm of T : MK α,λ p,q (R) → MK α,λ p,q (R), we also have the lower estimate of the operator norm of T :Ḃ q,θ (R) →Ḃ q,θ (R). We have when K is nonnegative.
In addition, we have the following result for Herz spaces. There exists a constant C > such that for any f ∈K α p,q (R)

. 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 α,λ p,q (R). They are applications of Theorem 3.1. We begin with the de nitions of the Hardy operators For the history, development and applications of the Hardy's inequality, the reader is referred to [32][33][34].
The following is the Hardy's inequality on Herz-Morrey spaces.
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 de ned as Theorem 3.6. Let α ∈ R, λ ≥ and ≤ p, q < ∞. If < q + α − λ < , then there is a constant C > such that for any f ∈ MK α,λ p,q (R) Hf MK α,λ p,q (R) ≤ C f MK α,λ p,q (R) .
In particular, we have the following results for central Morrey spaces and Herz spaces.
2. There is a constant C > such that for any f ∈Ḃ q,θ (R), 3. There is a constant C > such that for any f ∈Ḃ q,θ (R), We have the above results because < −θ q < is valid when < θ < and ≤ q < ∞.
1. If q + α < , then there is a constant C > such that for any f ∈K α p,q (R), we have 2. If < q + α, there is a constant C > such that for any f ∈K α p,q (R), 3. If < q + α < , then there is a constant C > such that for any f ∈K α p,q (R), In particular, when α < − q , we have Notice thatK α q,q (R) is the power weighted Lebesgue space L q (( , ∞), |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].

. Hadamard fractional integrals
The Hadamard fractional integrals are the fractional integrals corresponding to the Mellin transform see [43].
In [43], Butzer, Kilbas and Trujillo introduce and study the following generalizations of Hadamard fractional integrals. They are de ned by using the con uent hypergeometric function, which is also named as Kummer function. The con uent hypergeometric function Φ[a, c; z] is de ned for |z| < , c > and a ≠ −j, where (a) k , k ∈ N ∪ { }, is the Pochhammer symbol [44, Section 6.1] given by (a) = , (a) k = a(a + ) · · · (a + k − ), k ∈ N.
As Re(µ − σ) > q + α − λ − , by using the substitution y = − log u, we get For the studies of fractional Hadamard integrals on other function spaces such as amalgam spaces, BMO and modular spaces, see [27,30,46].