Integral operators on BMO and Campanato spaces

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ScienceDirect Indagationes Mathematicae 30 (2019) 1023–1035 www.elsevier.com/locate/indag

Integral operators on BMO and Campanato spaces Kwok-Pun Ho Department of Mathematics and Information Technology, The Education University of Hong Kong, 10 Lo Ping Road, Tai Po, Hong Kong, China Received 13 December 2018; received in revised form 25 April 2019; accepted 10 May 2019 Communicated by M. Veraar

Abstract This paper establishes the mapping properties of integral operators on space of bounded mean oscillation and Campanato spaces. In particular, we have the Hardy’s inequality and the boundedness of the Hadamard fractional integrals on space of bounded mean oscillation and Campanato spaces. c 2019 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. All rights reserved. ⃝

1. Introduction This paper gives the mapping properties of some integral operators on space of bounded mean oscillation B M O [16], Campanato spaces [3] and Lipschitz spaces. The space of bounded mean oscillation B M O and Lipschitz spaces are important function spaces in analysis. They play several roles on the studies of harmonic analysis. For instance, the T (1) theorem for the Calder´on–Zygmund operators relied on a condition involves space of bounded mean oscillation [6]. Furthermore, the dual spaces of Hardy spaces are Lipschitz spaces. Campanato introduced the Campanato spaces [3,23]. They are generalizations of B M O and Lipschitz spaces. The reader is referred to [5,7,19,21,26,27] for the recent developments of Campanato spaces. The main motivation of this paper is the study of Hardy’s inequalities on Campanato spaces. We find that the validity of Hardy’s inequalities relies on the dilation properties of Campanato spaces. During the investigation of the dilation properties of Campanato spaces, we find that these properties can be also used to study a general kind of integral operator including Hardy–Littlewood average and Hadamard fractional integrals. Therefore, we first obtain a E-mail address: [email protected]. https://doi.org/10.1016/j.indag.2019.05.007 c 2019 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. All rights reserved. 0019-3577/⃝

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general principle for the boundedness of integral operators on Campanato spaces. Then, we apply this general principle to establish the Hardy’s inequalities and the boundedness of Hadamard fractional integrals on Campanato spaces. This paper is organized as follows. The general principle for the boundedness of integral operators on Campanato spaces is presented in Section 2. The Hardy’ inequalities, the mapping properties of the Hardy–Littlewood average and Hadamard fractional integrals are given in Section 3. 2. Campanato spaces and integral operators Let R+ = (0, ∞) and I denote the family of open connected intervals in R+ . For any I ∈ I, the Lebesgue measure of I is denoted by |I |. For any r ∈ N ∪ {0}, let Pr be the collection of all polynomials of degree r . Definition 2.1. Let α ≥ 0 and 1 ≤ p < ∞ and r ∈ N ∪ {0}. The Campanato space Lαp,r (R+ ) consists of all locally integrable function f satisfying 1 ∥χ I ( f − PIr f )∥ L p < ∞ ∥ f ∥Lαp,r (R+ ) = sup 1 α+ p I ∈I |I | r where PI ( f ) is the unique polynomial of degree r such that ∫ ( f (t) − PIr f (t))Q(t)dt = 0, ∀Q ∈ Pr . I

Note that the Campanato space Lαp,r (R+ ) is a quotient space modulo the space of polynomials of degree at most r . For simplicity, we also denote these quotient spaces by Lαp,r (R+ ). For the other equivalent definitions of Campanato space, see [21]. An important member of the Campanato space is the space of bounded mean oscillation B M O(R+ ). Recall that B M O(R+ ) consists of all locally integrable function f satisfying ∫ 1 ∥ f ∥ B M O(R+ ) = sup | f (t) − f I |dt < ∞ I ∈I |I | I ∫ where f I = |I1| I f (y)dy. Therefore, L01,0 = B M O(R+ ). Another important members of Campanato spaces are Lipschitz spaces. We recall the definition of Lipschitz spaces from [9, Chapter III, Definition 5.15]. Definition 2.2. Let θ > 0. (1) When θ ̸∈ N. Suppose θ = [θ ] + θ ′ with 0 < θ ′ < 1. The Lipschitz space Λθ (R+ ) consists of those functions f ∈ C [θ ] (R+ ) satisfying ∥ f ∥Λθ (R+ ) = sup x̸= y x,y>0

|D [θ ] f (x) − D [θ ] f (y)| |x − y|θ

′

< ∞.

(2) When θ ∈ N. The Lipschitz space Λθ (R+ ) consists of those functions f ∈ C θ −1 (R+ ) satisfying ∥ f ∥Λθ (R+ ) =

sup h̸=0,x>0 x+h,x+2h>0

|D θ−1 f (x + 2h) − 2D θ −1 f (x + h) + D θ −1 f (x)| < ∞. |h|

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It is easy to see that eθ (x) = x θ belongs to Λθ (R+ ) with ∥eθ ∥Λθ (R+ ) > 0. One of the remarkable features of Lipschitz spaces is that they are dual spaces of Hardy spaces [9, Chapter III, Theorem 5.30]. According to [9, Chapter III, Theorem 5.30], we have Λθ (R+ ) = Lαp,r (R+ ) when 1 ≤ p < ∞, α = θ and r = [θ ]. We now study the dilation property of Campanato spaces. For any s > 0 and Lebesgue measurable function f , define Ds f (t) = f (t/s). For any I = (a, b) ∈ I, write Is = (a/s, b/s). Obviously, |Is | = |I |/s. Proposition 1. Let α ≥ 0 and 1 ≤ p < ∞ and r ∈ N ∪ {0}. For any f ∈ Lαp,r (R+ ), we have PIr (Ds f ) = Ds (PIrs f ).

(2.1)

Proof. Let I = (a, b) ∈ I and s > 0. For any Q ∈ Pr , the substitution t = su yields ∫ ∫ b r ((Ds f )(t) − (Ds PIs f )(t))Q(t)dt = ( f (t/s) − PIrs f (t/s))Q(t)dt I a ∫ b/s = ( f (u) − PIrs f (u))Q(su)sdu a/s ∫ = s ( f (u) − PIrs f (u))Q(su)du = 0 Is

since Q(s·) ∈ Pr . According to the definition of PIr , we establish (2.1).

■

Proposition 2. Let α ≥ 0 and 1 ≤ p < ∞ and r ∈ N ∪ {0}. For any f ∈ Lαp,r (R+ ), we have ∥Ds f ∥Lαp,r (R+ ) = s −α ∥ f ∥Lαp,r (R+ ) .

(2.2)

Proof. Let I ∈ I and s > 0. By using the substitution t = su and (2.1), we obtain 1 1 ∥χ I (Ds f − PIr Ds f )∥ L p = ∥χ I (Ds f − Ds (PIrs f ))∥ L p 1 α+ p α+ 1p |I | |I | 1 1 = s p ∥χ Is ( f − PIrs f )∥ L p 1 |I |α+ p 1 = s −α ∥χ Is ( f − PIrs f )∥ L p 1 |Is |α+ p ≤ s −α ∥ f ∥Lαp,r (R+ ) . By taking supremum over I ∈ I, we get ∥Ds f ∥Lαp,r (R+ ) ≤ s −α ∥ f ∥Lαp,r (R+ ) .

(2.3)

Since D1/s Ds f = f , we have ∥ f ∥Lαp,r (R+ ) = ∥D1/s Ds f ∥Lαp,r (R+ ) ≤ s α ∥Ds f ∥Lαp,r (R+ ) . Therefore, (2.3) and (2.4) yield (2.2).

■

(2.4)

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Particularly, as L01,0 = B M O(R+ ), for any s > 0 and f ∈ B M O(R+ ), we have ∥Ds f ∥ B M O(R+ ) = ∥ f ∥ B M O(R+ ) . In addition, for any s > 0 and f ∈ Λθ (R+ ), ∥Ds f ∥Λθ (R+ ) = s −θ ∥ f ∥Λθ (R+ ) . We now introduce the integral operator for our studies. Let K : (0, ∞) × (0, ∞) → R be a Lebesgue measurable function. We consider the integral operator ∫ ∞ K (s, t) f (s)ds, t ≥ 0. T f (t) = 0

Theorem 3. Let α ≥ 0 and 1 ≤ p < ∞ and r ∈ N ∪ {0}. Let K : (0, ∞) × (0, ∞) → R be a Lebesgue measurable function. If K (λs, λt) = λ−1 K (s, t), λ > 0 ∫ ∞ |K (u, 1)|u α du < ∞,

(2.5) (2.6)

0

and T f , f ∈ Lαp,r (R+ ), is well defined, then ) (∫ ∞ ∥T f ∥Lαp,r (R+ ) ≤ |K (u, 1)|u α du ∥ f ∥Lαp,r (R+ ) .

(2.7)

0

Proof. By using the substitution u = st , we find that ∫ ∞ ∫ ∞ K (u, 1)(D 1 f )(t)du T f (t) = K (ut, t)(D 1 f )(t)tdu = u

0

0

u

(2.8)

since K (·, ·) satisfies (2.5). For any I ∈ I and Q ∈ Pr by using (2.1), we find that ⏐ ( ) ∫ ∫ ∞⏐ ⏐ ⏐ ⏐ K (u, 1) (D 1 f )(t) − (P r D 1 f )(t) Q(t)⏐ dudt I ⏐ ⏐ u u I 0 ⏐ ⏐ ∫ ∫ ∞ ⏐ ⏐ ≤ |K (u, 1)| ⏐⏐(D 1 f )(t) − (D 1 PIr1 f )(t)⏐⏐ |Q(t)|dudt. I

0

u

u

u

The H¨older inequality and (2.6) guarantee that ⏐ ) ( ∫ ∫ ∞⏐ ⏐ ⏐ ⏐ K (u, 1) (D 1 f )(t) − (P r D 1 f )(t) Q(t)⏐ dudt I ⏐ ⏐ u u I 0 ∫ ∞   ≤ |K (u, 1)|χ I (D 1 f − D 1 PIr1 f ) L p ∥χ I Q∥ L p′ du u u u 0 ∫ ∞  1 1 α+ 1p = ∥χ I Q∥ L p′ |I | |K (u, 1)| u − p χ I 1 ( f − PIr1 f ) L p du 1 α+ u u 0 |I | p ∫ ∞   1 1 = ∥χ I Q∥ L p′ |I |α+ p |K (u, 1)| u α χ I 1 ( f − PIr1 f ) L p du 1 u u 0 |I 1 |α+ p u (∫ ∞ ) 1 ≤ ∥χ I Q∥ L p′ |I |α+ p |K (u, 1)|u α du ∥ f ∥Lαp,r (R+ ) < ∞. 0

Therefore, K (u, 1)((D 1 f )(t) − (PIr D 1 f )(t))Q(t) is absolutely integrable over the domain u u I × (0, ∞).

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Next, we verify that for any I ∈ I, we have ∫ ∞ PIr (T f )(t) = K (u, 1)(PIr D 1 f )(t)du.

(2.9)

u

0

1027

For any Q ∈ Pr , (2.8) shows that ) ∫ ( ∫ ∞ r T f (t) − K (u, 1)(PI D 1 f )(t)du Q(t)dt u I 0 ( ) ∫ ∫ ∞ = K (u, 1) (D 1 f )(t) − (PIr D 1 f )(t) Q(t)dudt. u

0

I

u

(PIr D 1 u

f )(t))Q(t) is absolutely integrable, we are allowed to use Since K (u, 1)((D 1 f )(t) − u Fubini’s theorem. As a result of the definition of PIr , we obtain ) ∫ ( ∫ ∞ K (u, 1)(PIr D 1 f )(t)du Q(t)dt T f (t) − u I 0 ((∫ ) ) ∫ ∞ r (D 1 f )(t) − (PI D 1 f )(t) Q(t)dt du = 0 K (u, 1) = 0

u

I

u

because Q ∈ Pr . Thus, (2.9) is valid. Consequently, (2.1), (2.8) and (2.9) yield ∫ ∞ ( ) T f (t) − PIr (T f )(t) = K (u, 1) (D 1 f )(t) − (PIr D 1 f )(t) du u u ( ) ∫0 ∞ r K (u, 1) (D 1 f )(t) − (D 1 PI 1 f )(t) du. = u

0

u

u

The Minkowski’s inequality gives 1 |I |

α+ 1p

∥χ I (T f − PIr (T f ))∥ L p ∫

1

≤

α+ 1p

|I | ∫ ∞

0

∞

  |K (u, 1)|χ I (D 1 f − D 1 PIr1 f ) L p du

|K (u, 1)|

= 0

∫

∞

|K (u, 1)|

=

u

1 1

|I |α+ p 1

u

− 1p

u

u

1

u

  u α χ I 1 ( f − PIr1 f ) L p du

u |I 1 |α+ p u (∫ ∞ ) ≤ |K (u, 1)|u α du ∥ f ∥Lαp,r (R+ ) .

0

u

  χ I ( f − P r f ) p du I1 1 L

u

0

By taking supremum over I ∈ I on both sides of the above inequalities, we obtain (2.7).

■

As special cases of the preceding theorem, we have the boundedness of the integral operator T on B M O(R+ ) and Λθ (R+ ). Corollary 4. Let K : (0, ∞) × (0, ∞) → R be a Lebesgue measurable function. If K satisfies (2.5) and ∫ ∞ |K (u, 1)|du < ∞, 0

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then for any f ∈ B M O(R+ ), (∫ ∞ ) ∥T f ∥ B M O(R+ ) ≤ |K (u, 1)|du ∥ f ∥ B M O(R+ ) . 0

Corollary 5. Let θ > 0 and K : (0, ∞) × (0, ∞) → R be a Lebesgue measurable function. If K satisfies (2.5) and ∫ ∞ |K (u, 1)|u θ du < ∞, 0

and T f , f ∈ Λθ (R+ ), is well defined, then (∫ ∞ ) θ ∥T f ∥Λθ (R+ ) ≤ |K (u, 1)|u du ∥ f ∥Λθ (R+ ) . 0

3. Applications This section gives the applications of Theorem 3. We establish the mapping properties of the Hardy–Littlewood average and Hadamard fractional integrals on B M O and Campanato spaces. 3.1. Hardy–Littlewood average and Hardy’s inequalities We first recall the definition of Hardy–Littlewood average from [25]. Let ψ : [0, 1] → [0, ∞) be a Lebesgue measurable function. The Hardy–Littlewood average Uψ is defined by ∫ 1 f (t x)ψ(t)dt. Uψ f (x) = 0

For the mapping properties of the Hardy–Littlewood average on B M O and L p , see [4,25]. In particular, when ψ(t) ≡ 1, Uψ reduces to the classical Hardy operator defined as ∫ 1 t f (s)ds. H f (t) = t 0 For the history, development and applications of the Hardy inequality, the reader is referred to [17,18,20]. We now establish the boundedness of the Hardy–Littlewood average on Campanato spaces. Theorem 6. Let α ≥ 0 and 1 ≤ p < ∞ and r ∈ N ∪ {0}. If ψ : [0, 1] → [0, ∞) satisfies ∫ 1 ψ(y)dy < ∞, (3.10) 0

then for any f ∈ Lαp,r (R+ ) (∫ ∥Uψ f ∥Lαp,r (R+ ) ≤

1

) ψ(y)y α dy ∥ f ∥Lαp,r (R+ ) .

0

Proof. We find that ∫ ∞ K (s, t) f (s)ds Uψ f (t) = 0

where K (s, t) = 1t ψ(s/t)χ{(s,t):s≤t} . It obviously satisfies (2.5). Moreover, (3.10) also guarantees that (2.6) is fulfilled. Therefore, Theorem 3 yields the boundedness of Uψ . ■

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We can also compute the operator norm of Uψ : Λθ (R+ ) → Λθ (R+ ). Corollary 7. Let θ > [θ ] and ψ satisfy (3.10). We have ∫ 1 ∥Uψ ∥Λθ (R+ )→Λθ (R+ ) = ψ(y)y θ dy. 0

Proof. Since Lθp,[θ ] (R+ ) = Λθ (R+ ), Theorem 6 shows that ∥Uψ ∥Λθ (R+ )→Λθ (R+ ) ≤ Next, as eθ (t) = t θ belongs to Λθ (R+ ) with ∥eθ ∥Λθ (R+ ) > 0 and (∫ 1 ) Uψ eθ (t) = ψ(y)y θ dy eθ (t).

∫1 0

ψ(y)y θ dy.

0

∫1 We have ∥Uψ ∥Λθ (R+ )→Λθ (R+ ) = 0 ψ(y)y θ dy. ■ ∫1 α 1 Since 0 y dy = α+1 , we have the following Hardy’s inequalities on Campanato spaces. Theorem 8. Let α ≥ 0 and 1 ≤ p < ∞ and r ∈ N ∪ {0}. For any f ∈ Lαp,r (R+ ) 1 ∥ f ∥Lαp,r (R+ ) . α+1 In particular, we have the Hardy’s inequality on B M O(R+ ) ∥H f ∥Lαp,r (R+ ) ≤

∥H f ∥ B M O(R+ ) ≤ ∥ f ∥ B M O(R+ ) ,

∀ f ∈ B M O(R+ ).

Moreover, Corollary 7 assures that 1 ∥H ∥Λθ (R+ )→Λθ (R+ ) = . θ +1 The classical Hardy inequality is given in [10]. The reader is referred to [4,24,25] and [9,11–15] for the Hardy inequalities on B M O and Hardy type spaces such as Hardy spaces, Hardy–Morrey spaces with variable exponents and discrete Hardy spaces, respectively. 3.2. Hadamard fractional integrals The Hadamard fractional integrals are the fractional integrals associated with the Mellin transform ∫ ∞ M f (s) = u s−1 f (u)du, s = c + it, c, t ∈ R, 0

see [1]. Butzer, Kilbas and Trujillo introduce the generalizations of Hadamard fractional integrals β β β β J0+,µ;γ ,σ f , J−,µ;γ ,σ f , I0+,µ;γ ,σ f and I−,µ;γ ,σ f in [1]. The preceding integrals in [1] 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] =

∞ ∑ (a)k z k (c)k k! k=0

where (a)k , k ∈ N ∪ {0} is the Pochhammer symbol [8, Section 6.1] given by (a)0 = 1,

(a)k = a(a + 1) · · · (a + k − 1),

k ∈ N.

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For β > 0, γ ∈ R and µ, σ ∈ C, the generalized Hadamard fractional integrals J0+,µ;γ ,σ f , β β β J−,µ;γ ,σ f , I0+,µ;γ ,σ f and I−,µ;γ ,σ f are defined as ∫ x ( )µ ( x )β−1 [ t x] dt 1 β log Φ γ , β; σ log f (t) , J0+,µ;γ ,σ f (x) = Γ (β) 0 x t t t )β−1 [ ] ∫ ∞ ( )µ ( t dt x 1 t β J−,µ;γ ,σ f (x) = log f (t) , Φ γ , β; σ log Γ (β) x t x x t ∫ x ( )µ ( ) [ ] β−1 1 x x dt t β I0+,µ;γ ,σ f (x) = log Φ γ , β; σ log f (t) , Γ (β) 0 x t t x )β−1 [ ] ∫ ∞ ( )µ ( 1 x t t dt β I−,µ;γ ,σ f (x) = f (t) , log Φ γ , β; σ log Γ (β) x t x x x β

β

β

β

and (J0+,µ;γ ,σ f )(x) = (J−,µ;γ ,σ f )(x) = (I0+,µ;γ ,σ f )(x) = (I−,µ;γ ,σ f )(x) = 0, x ≤ 0, where Γ (β) is the Gamma function. β Note that Φ[a, c; 0] = 1, when σ = µ = 0, the above Hadamard fractional integral J0+,0;γ ,0 β β β β becomes the Hadamard fractional integral J0+ . Additionally, J−,µ;γ ,σ , I0+,µ;γ ,σ and I−,µ;γ ,σ are the Hadamard type fractional integrals introduced and studied in [2]. For the studies of these integral such as the mapping properties on Lebesgue spaces, the group properties and their applications on fractional calculus, see [1,2,22]. In order to present the mapping of the Hadamard fractional integrals, we need the following asymptotic behaviours for Φ[a, c; x] ( ( )) 1 Γ (c) x a−c e x 1+O as x → ∞. (3.11) Φ[a, c; x] = Γ (a) x The limit (a)k+1 (c)k+1 (k+1)! lim (a)k k→∞ (c)k k!

a+k 1 =0 k→∞ c + k k + 1

= lim

assures that Φ[a, c; x] = 1 + O(x)

as

x → 0+ .

(3.12)

The following is the mapping properties of the Hadamard fractional integrals on Campanato spaces Lαp,r (R+ ). Theorem 9. Let α ≥ 0, 1 ≤ p < ∞ and r ∈ N ∪ {0}. Let β > 0, γ ∈ R and µ, σ ∈ C. (1) If Re(µ − σ ) > 0, then there exists a constant C > 0 such that for any f ∈ Lαp,r (R+ ), we have β

∥J0+,µ;γ ,σ f ∥Lαp,r (R+ ) ≤ C∥ f ∥Lαp,r (R+ ) .

(3.13)

(2) If Re(µ − σ ) > max(α, r ), then there exists a constant C > 0 such that for any f ∈ Lαp,r (R+ ), we have β

∥J−,µ;γ ,σ f ∥Lαp,r (R+ ) ≤ C∥ f ∥Lαp,r (R+ ) .

(3.14)

(3) If Re(µ − σ ) > −1, then there exists a constant C > 0 such that for any f ∈ Lαp,r (R+ ), we have β

∥I0+,µ;γ ,σ f ∥Lαp,r (R+ ) ≤ C∥ f ∥Lαp,r (R+ ) .

(3.15)

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(4) If Re(µ − σ ) > 1 + max(α, r ), then there exists a constant C > 0 such that for any f ∈ Lαp,r (R+ ), we have β

∥I−,µ;γ ,σ f ∥Lαp,r (R+ ) ≤ C∥ f ∥Lαp,r (R+ ) .

(3.16)

Proof. Let E = {(u, x) ∈ (0, ∞)×(0, ∞) : u < x} and F = {(u, x) ∈ (0, ∞)×(0, ∞) : x < u}. β For the integral J0+,µ;γ ,σ , we have ∫ ∞ β J0+,µ;γ ,σ f (x) = K 1 (u, x) f (u)du 0

where x] 1 x )β−1 [ 1 ( u )µ ( Φ γ , β; σ log log χ E (u, x). Γ (β) x u u u For any λ > 0, as χ E (λu, λx) = χ E (u, x). [ ( )µ ( ) ] 1 λx 1 λu λx β−1 K 1 (λu, λx) = Φ γ , β; σ log χ E (λu, λx) log Γ (β) λx λu λu λu = λ−1 K 1 (u, x). K 1 (u, x) =

Hence, (2.5) is fulfilled. Since [ ] 1 µ−1 1 β−1 K 1 (u, 1) = u (− log u) Φ γ , β; σ log χ{u:0
)) ( ( 1 µ−1 1 u (− log u)β−1 1 + O σ log Γ (β) u

as u → 1− . By using the substitution y = − log u, we obtain ∫ ∞ ∫ 1 α |K 1 (u, 1)|u du = |K 1 (u, 1)|u α du 0 0 ∫ ∞ = |K 1 (e−y , 1)|e−y(α+1) dy 0 (∫ 2 ) ∫ ∞ ≤C e−y(Re(µ)+α) y β−1 dy + e−y(Re(µ−σ )+α) y γ −1 dy . 0

2

There is a constant C > 0 such that ∫ 2 ∫ 2 e−y(Re(µ)+α) y β−1 dy ≤ C y β−1 dy < C. 0

0

Next, we have a ϵ > 0 such that Re(µ − σ ) > −α + ϵ and ∫ ∞ ∫ ∞ e−y(Re(µ−σ )+α) y γ −1 dy < e−y(Re(µ−σ )+α−ϵ) dy < C. 2

2

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The above inequalities assure that (2.6) is fulfilled. Therefore, Theorem 3 yields (3.13). β Next, for the integral operator J−,µ;γ ,σ , we have ∫ ∞ β J−,µ;γ ,σ f (x) = K 2 (u, x) f (u)du 0

where K 2 (u, x) =

1 ( x )µ ( u )β−1 [ u] 1 log Φ γ , β; σ log χ F (u, x). Γ (β) u x x u

It is easy to see that for any λ > 0, K 2 (λu, λx) = λ−1 K 2 (u, x) and K 2 (u, 1) = [ ] (log u)β−1 Φ γ , β; σ log u u −1 χ{u:1
1

0

In view of (3.11), (3.12) and Re(µ − σ ) > α, we have ∫ ∞ |K 2 (u, 1)|u α du 0 (∫ 2 ) ∫ ∞ β−1 y(−Re(µ−σ )+α) γ −1 ≤C y dy + e y dy < ∞. 0

2

Therefore, Theorem 3 yields (3.14). β For the operator I0+,µ;γ ,σ , we have ∫ ∞ β I0+,µ;γ ,σ f (x) = K 3 (u, x) f (u)du 0

where K 3 (u, x) =

x] 1 x )β−1 [ 1 ( u )µ ( Φ γ , β; σ log χ E (u, x). log Γ (β) x u u x

The function K 3 satisfies (2.5) and K 3 (u, 1) =

[ ] 1 1 µ u (− log u)β−1 Φ γ , β; σ log χ{u:0
As Re(µ − σ ) > −α − 1, by using the substitution y = − log u, we get ∫ ∞ ∫ 1 |K 3 (u, 1)|u α du = |K 3 (u, 1)|u α du 0 0 ∫ ∞ −y −y(α+1) = |K 3 (e , 1)|e dy 0 (∫ 2 ) ∫ ∞ β−1 −y(Re(µ−σ )+α+1) γ −1 ≤C y dy + e y dy < ∞ 0

2 β

because Re(µ − σ ) > −α − 1. Theorem 3 guarantees the boundedness of I0+,µ;γ ,σ . β Finally, for the operator I−,µ;γ ,σ , we have ∫ ∞ β I−,µ;γ ,σ f (x) = K 4 (u, x) f (u)du 0

1 u −µ Γ (β)

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where 1 ( x )µ ( u )β−1 [ u] 1 log Φ γ , β; σ log χ F (u, x). Γ (β) u x x x We see that K 4 satisfies (2.5) and [ ] 1 −µ K 4 (u, 1) = u (log u)β−1 Φ γ , β; σ log u χ{u:1 1 + α, by using the substitution y = log u, we obtain ∫ ∞ ∫ ∞ |K 4 (u, 1)|u α du = |K 4 (u, 1)|u α du 0 1 ∫ ∞ |K 4 (e y , 1)|e y(α+1) dy = 0 (∫ 2 ) ∫ ∞ β−1 y(−Re(µ−σ )+α+1) γ −1 ≤C y dy + e y dy < ∞. K 4 (u, x) =

0

2

Then, Theorem 3 asserts the validity of (3.16).

■

Corollaries 10 and 11 give the mapping properties of Hadamard fractional integrals on B M O(R+ ) and Λθ (R+ ). Let β > 0, γ ∈ R and µ, σ ∈ C.

Corollary 10.

(1) If Re(µ − σ ) > 0, then there exists a constant C > 0 such that for any f ∈ B M O(R+ ), we have β

∥J0+,µ;γ ,σ f ∥ B M O(R+ ) ≤ C∥ f ∥ B M O(R+ ) . (2) If Re(µ − σ ) > 0, then there exists a constant C > 0 such that for any f ∈ B M O(R+ ), we have β

∥J−,µ;γ ,σ f ∥ B M O(R+ ) ≤ C∥ f ∥ B M O(R+ ) . (3) If Re(µ−σ ) > −1, then there exists a constant C > 0 such that for any f ∈ B M O(R+ ), we have β

∥I0+,µ;γ ,σ f ∥ B M O(R+ ) ≤ C∥ f ∥ B M O(R+ ) . (4) If Re(µ − σ ) > 1, then there exists a constant C > 0 such that for any f ∈ B M O(R+ ), we have β

∥I−,µ;γ ,σ f ∥ B M O(R+ ) ≤ C∥ f ∥ B M O(R+ ) . Corollary 11. Let θ > 0. Let β > 0, γ ∈ R and µ, σ ∈ C. (1) If Re(µ − σ ) > 0, then there exists a constant C > 0 such that for any f ∈ Λθ (R+ ), we have β

∥J0+,µ;γ ,σ f ∥Λθ (R+ ) ≤ C∥ f ∥Λθ (R+ ) . (2) If Re(µ − σ ) > θ, then there exists a constant C > 0 such that for any f ∈ Λθ (R+ ), we have β

∥J−,µ;γ ,σ f ∥Λθ (R+ ) ≤ C∥ f ∥Λθ (R+ ) .

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(3) If Re(µ − σ ) > −1, then there exists a constant C > 0 such that for any f ∈ Λθ (R+ ), we have β

∥I0+,µ;γ ,σ f ∥Λθ (R+ ) ≤ C∥ f ∥Λθ (R+ ) . (4) If Re(µ − σ ) > 1 + θ , then there exists a constant C > 0 such that for any f ∈ Λθ (R+ ), we have β

∥I−,µ;γ ,σ f ∥Λθ (R+ ) ≤ C∥ f ∥Λθ (R+ ) . Acknowledgement The author would like to thank the referee for careful reading of the paper and valuable suggestions for improving the presentation of this paper. References [1] P. Butzer, A. Kilbas, J. Trujillo, Compositions of Hadamard-type fractional integration operators and the semigroup property, J. Math. Anal. Appl. 269 (2002) 387–400. [2] P. Butzer, A. Kilbas, J. Trujillo, Fractional calculus in the Mellin setting and Hadamard-type fractional integrals, J. Math. Anal. Appl. 269 (2002) 1–27. [3] S. Campanato, Proprietá di hölderianitá di alcune classi di funzioni, Ann. Sc. Norm. Super. Pisa 17 (3) (1963) 175–188. [4] C. Carton-Lebrun, M. Fosset, Moyennes et quotients de Taylor dans B M O, Bull. Soc. Roy. Sci. Liége 53 (1984) 85–87. [5] A. Cianchi, L. Pick, Sobolev embeddings into spaces of Campanato, Morrey and Hölder type, J. Math. Anal. Appl. 282 (2003) 128–150. [6] G. David, .J.-L. Journé, A boundedness criterion for generalized Calderón–Zygmund operators, Ann. of Math. 120 (1984) 371–397. [7] Nakai E., The Campanato, Morrey and Hölder spaces on spaces of homogeneous type, Studia Math. 176 (2006) 1–19. [8] A. Erdelyi, W. Magnus, F. Oberhettinger, F. Tricomi, Higher Transcendental Functions, Vol. 1, McGraw-Hill, New York, 1953. [9] J. Garcia-Cuerva, J. Rubio De Francia, Weighted Norm Inequalities and Related Topics, North-Holland, 1985. [10] G. Hardy, J. Littlewood, G. Pólya, Inequalities, Cambridge University Press, 1934. [11] K.-P. Ho, Hardy’s inequality on Hardy spaces, Proc. Japan Acad. Ser. A Math. Sci. 92 (2016) 125–130. [12] K.-P. Ho, Atomic decompositions and Hardy’s inequality on weak Hardy–Morrey spaces, Sci. China Math. 60 (2017) 449–468. [13] K.-P. Ho, Hardy’s inequality on Hardy–Morrey spaces with variable exponents, Mediterr. J. Math. 14 (2017) 79–98. [14] K.-P. Ho, Discrete Hardy’s inequality with 0 < p ≤ 1, J. King Saud Univ. Sci. 30 (2018) 489–492. [15] K.-P. Ho, Hardy’s inequality on Hardy–Morrey spaces, Georgian Math. J. (2019) published online https://doi .org/10.1515/gmj-2017-0046. [16] F. John, L. Nirenberg, On functions of bounded mean oscillation, Comm. Pure Appl. Math. 14 (1961) 415–426. [17] A. Kufner, L. Maligranda, L.-E. Persson, The prehistory of the Hardy inequality, Amer. Math. Monthly 113 (2006) 715–732. [18] A. Kufner, L.-E. Persson, N. Samko, Weighted Inequalities of Hardy Type, World Scientific Publishing Company, 2017. [19] E. Nakai, Singular and fractional integral operators on Campanato spaces with variable growth conditions, Rev. Mat. Complut. 23 (2010) 355–381. [20] B. Opic, A. Kufner, Hardy-type Inequalities Pitman Reserach Notes in Math. Series, Vol. 219, Longman Sci. and Tech, Harlow, 1990. [21] H. Rafeiro, N. Samko, S. Samko, Morrey–Campanato spaces: an overview, in: Operator Theory: Advances and Applications, Vol. 228, 2013, pp. 293–323. [22] S. Samko, A. Kilbas, O. Marichev, Fractional Integrals and Derivatives, Theory and Applications, Gordan and Breach, New York, 1993.

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