On the Riesz potential and its commutators on generalized Orlicz-Morrey spaces

We consider generalized Orlicz-Morrey spaces $M_{\Phi,\varphi}(\Rn)$ including their weak versions $WM_{\Phi,\varphi}(\Rn)$. In these spaces we prove the boundedness of the Riesz potential from $M_{\Phi,\varphi_1}(\Rn)$ to $M_{\Psi,\varphi_2}(\Rn)$ and from $M_{\Phi,\varphi_1}(\Rn)$ to $WM_{\Psi,\varphi_2}(\Rn)$. As applications of those results, the boundedness of the commutators of the Riesz potential on generalized Orlicz-Morrey space is also obtained. In all the cases the conditions for the boundedness are given either in terms of Zygmund-type integral inequalities on $(\varphi_{1},\varphi_{2})$, which do not assume any assumption on monotonicity of $\varphi_{1}(x,r)$, $\varphi_{2}(x,r)$ in r.


Introduction
The theory of boundedness of classical operators of the real analysis, such as the maximal operator, fractional maximal operator, Riesz potential and the singular integral operators etc, have been extensively investigated in various function spaces. Results on weak and strong type inequalities for operators of this kind in Lebesgue spaces are classical and can be found for example in [2,53,55]. These boundedness extended to several function spaces which are generalizations of L pspaces, for example, Orlicz spaces, Morrey spaces, Lorentz spaces, Herz spaces, etc.
Orlicz spaces, introduced in [45,46], are generalizations of Lebesgue spaces L p . They are useful tools in harmonic analysis and its applications. For example, the Hardy-Littlewood maximal operator is bounded on L p for 1 < p < ∞, but not on L 1 . Using Orlicz spaces, we can investigate the boundedness of the maximal operator near p = 1 more precisely (see [4,22,23]).
It is well known that the Riesz potential I α of order α (0 < α < n) plays an important role in harmonic analysis, PDE and potential theory (see [53]). Recall that I α is defined by The classical result by Hardy-Littlewood-Sobolev states that if 1 < p < q < ∞, then the operator I α is bounded from L p (R n ) to L q (R n ) if and only if α = n 1 p − 1 q and for p = 1 < q < ∞, the operator I α is bounded from L 1 (R n ) to W L q (R n ) if and only if α = n 1 − 1 q . For boundedness of I α on Morrey spaces M p,λ (R n ), see Peetre (Spanne) [47], Adams [1].
The boundedness of I α from Orlicz space L Φ (R n ) to L Ψ (R n ) was studied by O'Neil [44] and Torchinsky [54] under some restrictions involving the growths and certain monotonicity properties of Φ and Ψ. Moreover Cianchi [4] gave a necessary and sufficient condition for the boundedness of I α from L Φ (R n ) to L Ψ (R n ) and from L Φ (R n ) to weak Orlicz space W L Ψ (R n ), which contain results above.
In [8] the authors were study the boundedness of the maximal operator M and the Calderón-Zygmund operator T from one generalized Orlicz-Morrey space M Φ,ϕ 1 (R n ) to M Φ,ϕ 2 (R n ) and from M Φ,ϕ 1 (R n ) to the weak space W M Φ,ϕ 2 (R n ).
The main purpose of this paper is to find sufficient conditions on general Young functions Φ, Ψ and functions ϕ 1 , ϕ 2 which ensure the boundedness of the Riesz potential I α from one generalized Orlicz-Morrey spaces M Φ,ϕ 1 (R n ) to another M Ψ,ϕ 2 (R n ), from M Φ,ϕ 1 (R n ) to weak generalized Orlicz-Morrey spaces W M Ψ,ϕ 2 (R n ) and the boundedness of the commutator of the Riesz potential In the next section we recall the definitions of Orlicz and Morrey spaces and give the definition of Orlicz-Morrey and generalized Orlicz-Morrey spaces in Section 3. In Section 4 and Section 5 the results on the boundedness of the Riesz potential and its commutator on generalized Orlicz-Morrey spaces is obtained.
By A B we mean that A ≤ CB with some positive constant C independent of appropriate quantities. If A B and B A, we write A ≈ B and say that A and B are equivalent.

Some preliminaries on Orlicz and Morrey spaces
In the study of local properties of solutions to of partial differential equations, together with weighted Lebesgue spaces, Morrey spaces M p,λ (R n ) play an important role, see [12]. Introduced by C. Morrey [38] in 1938, they are defined by the norm f M p,λ := sup x, r>0 where 0 ≤ λ ≤ n, 1 ≤ p < ∞. Here and everywhere in the sequel B(x, r) stands for the ball in R n of radius r centered at x. Let |B(x, r)| be the Lebesgue measure of the ball B(x, r) and |B(x, r)| = v n r n , where v n is the volume of the unit ball in R n . Note that M p,0 = L p (R n ) and M p,n = L ∞ (R n ). If λ < 0 or λ > n, then M p,λ = Θ, where Θ is the set of all functions equivalent to 0 on R n .
We also denote by W M p,λ ≡ W M p,λ (R n ) the weak Morrey space of all func- where W L p (B(x, r)) denotes the weak L p -space. We refer in particular to [27] for the classical Morrey spaces. We recall the definition of Young functions. From the convexity and Φ(0) = 0 it follows that any Young function is increasing. If there exists s ∈ (0, +∞) such that Φ(s) = +∞, then Φ(r) = +∞ for r ≥ s.
Let Y be the set of all Young functions Φ such that is called Orlicz space. If Φ(r) = r p , 1 ≤ p < ∞, then L Φ (R n ) = L p (R n ). If Φ(r) = 0, (0 ≤ r ≤ 1) and Φ(r) = ∞, (r > 1), then L Φ (R n ) = L ∞ (R n ). The space L loc Φ (R n ) endowed with the natural topology is defined as the set of all functions f such that f χ B ∈ L Φ (R n ) for all balls B ⊂ R n . We refer to the books [24,25,48] for the theory of Orlicz Spaces.
We note that For a measurable set Ω ⊂ R n , a measurable function f and t > 0, let In the case Ω = R n , we shortly denote it by m(f, t).
For a Young function Φ and 0 ≤ s ≤ +∞, let If Φ ∈ Y, then Φ −1 is the usual inverse function of Φ. We note that for some k > 1. The function Φ(r) = r satisfies the ∆ 2 -condition but does not satisfy the ∇ 2 -condition. If 1 < p < ∞, then Φ(r) = r p satisfies both the conditions. The function Φ(r) = e r − r − 1 satisfies the ∇ 2 -condition but does not satisfy the ∆ 2 -condition. For a Young function Φ, the complementary function Φ(r) is defined by The complementary function Φ is also a Young function and Φ = Φ. If Φ(r) = r, then Φ(r) = 0 for 0 ≤ r ≤ 1 and Φ(r) = +∞ for r > 1. If 1 < p < ∞, Note that Young functions satisfy the properties The following analogue of the Hölder inequality is known, see [56].
For a Young function Φ and its complementary function Φ, the following inequality is valid The following lemma is valid.
Lemma 2.5. [2,31] Let Φ be a Young function and B a set in R n with finite Lebesgue measure. Then In the next sections where we prove our main estimates, we use the following lemma, which follows from Theorem 2.4, Lemma 2.5 and (2.2). Lemma 2.6. For a Young function Φ and B = B(x, r), the following inequality is valid

Orlicz-Morrey and Generalized Orlicz-Morrey Spaces
We also denote by According to this definition, we recover the spaces M Φ,λ and W M Φ,λ under According to this definition, we recover the generalized Morrey spaces M p,ϕ and weak generalized Morrey spaces W M p,ϕ under the choice Φ(r) = r p , 1 ≤ p < ∞: There are different kinds of Orlicz-Morrey spaces in the literature. We want to make some comment about these spaces. Let ϕ : (0, ∞) → (0, ∞) be a function and Φ : (0, ∞) → (0, ∞) a Young function.
1. For a cube Q, define (ϕ, Φ)-average over Q by and define its Φ-average over Q by The function space L ϕ,Φ is defined to be the Orlicz-Morrey space of the first kind as the set of all measurable functions f for which the norm f L ϕ,Φ is finite. 3 The function space M ϕ,Φ is defined to be the Orlicz-Morrey space of the second kind as the set of all measurable functions f for which the norm f M ϕ,Φ is finite.
In this section sufficient conditions on the pairs (ϕ 1 , ϕ 2 ) and (Φ, Ψ) for the boundedness of I α from one generalized Orlicz-Morrey spaces M Φ,ϕ 1 (R n ) to another M Ψ,ϕ 2 (R n ) and from M Φ,ϕ 1 (R n ) to the weak space W M Ψ,ϕ 2 (R n ) have been obtained.
Necessary and sufficient conditions on (Φ, Ψ) for the boundedness of I α from [4,Theorem 2]. In the statement of the theorem, Ψ p is the Young function associated with the Young function Ψ and p ∈ (1, ∞] whose Young conjugate is given by and p ′ , the Holder conjugate of p, equals either p/(p − 1) or 1, according to whether p < ∞ or p = ∞ and Φ p denotes the Young function defined by  We will use the following statement on the boundedness of the weighted Hardy operator where w is a weight.

Theorem 4.2.
Let v 1 , v 2 and w be weights on (0, ∞) and v 1 (t) be bounded outside a neighborhood of the origin. The inequality holds for some C > 0 for all non-negative and non-decreasing g on (0, ∞) if and only if Moreover, the value C = B is the best constant for (4.5).
The following lemma is valid.
Lemma 4.4. Let 0 < α < n, Φ and Ψ Young functions, f ∈ L loc Φ (R n ) and B = B(x 0 , r). If (Φ, Ψ) satisfy the conditions (4.4), then and if (Φ, Ψ) satisfy the conditions (4.3), then Proof. Suppose that the conditions (4.4) satisfied. For arbitrary x 0 ∈ R n , set B = B(x 0 , r) for the ball centered at x 0 and of radius r, 2B = B(x 0 , 2r). We represent f as and have Since f 1 ∈ L Φ (R n ), I α f 1 ∈ L Ψ (R n ) and from the boundedness of I α from L Φ (R n ) to L Ψ (R n ) (see Theorem 4.1) it follows that: We get By Lemma 2.6 and Lemma 4.3 for p = n/α we get Moreover, is valid. Thus On the other hand, using the property of Young function as it mentioned in (2.2) and we get (4.13) Thus Suppose that the conditions (4.4) satisfied. From the boundedness of I α from L Φ (R n ) to W L Ψ (R n ) (see Theorem 4.1) and (4.13) it follows that: (4.14) Then by (4.12) and (4.14) we get the inequality (4.10).
where C does not depend on x and r. Then for the conditions (4.4), I α is bounded from M Φ,ϕ 1 (R n ) to M Ψ,ϕ 2 (R n ) and for the conditions (4.3), I α is bounded from M Φ,ϕ 1 (R n ) to W M Ψ,ϕ 2 (R n ).
where C does not depend on x and r. Then I α is bounded from M p,ϕ 1 to M q,ϕ 2 for p > 1 and from M 1,ϕ 1 to W M q,ϕ 2 for p = 1.
from Theorem 4.5 we get the following Spanne type theorem for the boundedness of the Riesz potential on Orlicz-Morrey spaces.
We recall the definition of the space of BMO(R n ).
Modulo constants, the space BMO(R n ) is a Banach space with respect to the norm · * .
Lemma 5.5. [28] Let Φ be a Young function which is lower type p 0 and upper type p 1 with 1 ≤ p 0 ≤ p 1 < ∞. Let C be a positive constant. Then there exists a positive constant C such that for any ball B of R n and µ ∈ (0, ∞) In the following lemma we provide a generalization of the property (5.1) from L p -norms to Orlicz norms.
Lemma 5.6. Let f ∈ BMO(R n ) and Φ be a Young function. Let Φ is lower type p 0 and upper type p 1 with 1 ≤ p 0 ≤ p 1 < ∞, then Proof. By Hölder's inequality, we have ,r)) .
Now we show that sup x∈R n ,r>0 Without loss of generality, we may assume that f * = 1; otherwise, we replace f by f / f * . By the fact that Φ is lower type p 0 and upper type p 1 and (2.1) it follows that By Lemma 5.5 we get the desired result.
Remark 5.7. Note that statements of type of Lemma 5.6 are known in a more general case of rearrangement invariant spaces and also variable exponent Lebesgue spaces L p(·) , see [26] and [21], but we gave a short proof of Lemma 5.6 for completeness of presentation.
Remark 5.10. Remark 5.9 and Remark 5.4 show us that a Young function Φ is lower type p 0 and upper type p 1 with 1 The characterization of (L p , L q ) boundedness of the commutator [b, I α ] between M b and I α was given by Chanillo [3].
Theorem 5.12. [9] Let 0 < α < n and b ∈ BMO(R n ). Let Φ be a Young function and Ψ defined, via its inverse, by setting, for all t ∈ (0, ∞), We will use the following statement on the boundedness of the weighted Hardy operator where w is a weight.
The following lemma is valid.
Proof. For arbitrary x 0 ∈ R n , set B = B(x 0 , r) for the ball centered at x 0 and of radius r.
From the boundedness of [b, .
Let us estimate J 1 .