Riesz potential and its commutators on Orlicz spaces

In the present paper, we shall give necessary and sufficient conditions for the strong and weak boundedness of the Riesz potential operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$I_{\alpha}$\end{document}Iα on Orlicz spaces. Cianchi (J. Lond. Math. Soc. 60(1):247-286, 2011) found necessary and sufficient conditions on general Young functions Φ and Ψ ensuring that this operator is of weak or strong type from \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L^{\Phi}$\end{document}LΦ into \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L^{\Psi}$\end{document}LΨ. Our characterizations for the boundedness of the above-mentioned operator are different from the ones in (Cianchi in J. Lond. Math. Soc. 60(1):247-286, 2011). As an application of these results, we consider the boundedness of the commutators of Riesz potential operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$[b,I_{\alpha }]$\end{document}[b,Iα] on Orlicz spaces when b belongs to the BMO and Lipschitz spaces, respectively.


Introduction
Norm inequalities for several classical operators of harmonic analysis have been widely studied in the context of Orlicz spaces. It is well known that many of such operators fail to have continuity properties when they act between certain Lebesgue spaces and, in some situations, the Orlicz spaces appear as adequate substitutes.
The Hardy-Littlewood maximal operator M and the Riesz potential operator I α ( < α < n) are defined by Here and everywhere in the sequel B(x, r) is the ball in R n of radius r centered at x and |B(x, r)| = v n r n is its Lebesgue measure, where v n is the volume of the unit ball in R n .
The commutators generated by a suitable function b and the operator I α are formally defined by In [], Cianchi found necessary and sufficient conditions on general Young functions and ensuring that the operator I α is of weak or strong type from L into L . Another boundedness statement with only sufficient conditions for the operator I α on Orlicz spaces was given by Nakai []. Note that in [] a more general case of generalized fractional integrals was studied. Commutators of classical operators of harmonic analysis play an important role in various topics of analysis and PDE; see for instance [, ], and the references therein. The main purpose of this paper is to give characterizations for the strong and weak boundedness of the Riesz potential on Orlicz spaces. Our characterizations for the boundedness of the operator I α are different from the ones in []. As an application of these results, we consider the boundedness of the commutators of Riesz potential operator on Orlicz spaces when b belongs to the BMO and Lipschitz spaces, respectively.
We use the notation A B, which means 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.

Preliminaries; on Young functions and Orlicz spaces
We recall the definition of Young functions. From the convexity and () =  it follows that any Young function is increasing. The set of Young functions such that  < (r) < ∞ for  < r < ∞ is denoted by Y. If ∈ Y, then is absolutely continuous on every closed interval in [, ∞) and bijective from [, ∞) to itself.
For a Young function and  ≤ s ≤ ∞, let If ∈ Y, then - is the usual inverse function of . It is well known that where (r) is defined by A Young function is said to satisfy the  -condition, denoted also as ∈  , if (r) ≤ C (r), r > , for some C > . If ∈  , then ∈ Y. A Young function is said to satisfy the ∇ condition, denoted also by ∈ ∇  , if Definition . (Orlicz space) For a Young function , the set For a measurable set ⊂ R n , a measurable function f and t > , let m( , f , t) = |{x ∈ : |f (x)| > t}|. In the case = R n , we for brevity denote it by m(f , t).

Definition . The weak Orlicz space
We note that The following analogue of the Hölder inequality is well known (see, for example, []).
Theorem . Let ⊂ R n be a measurable set and functions f and g measurable on . For a Young function and its complementary function , the following inequality is valid: By elementary calculations we have the following property.
Lemma . Let be a Young function and B be a set in R n with finite Lebesgue measure. Then .
By Theorem ., Lemma . and (.) we get the following estimate.
Lemma . For a Young function and B = B(x, r), the following inequality is valid: In the next section, where we prove our main estimates, we use the following theorem.

Theorem . ([]) Let be a Young function.
(i) The operator M is bounded from L (R n ) to WL (R n ), and the inequality The operator M is bounded on L (R n ), and the inequality holds with constant C  independent of f if and only if ∈ ∇  .

Riesz potential in Orlicz spaces
In this section we find necessary and sufficient conditions for the strong/weak boundedness of the Riesz potential operator on Orlicz spaces. We recall that, for functions and from [, ∞) into [, ∞], the function is said to dominate globally if there exists a positive constant c such that (s) ≤ (cs) for all s ≥ .
In the theorem below we also use the notation where  < P ≤ ∞ and P (s) is the Young conjugate function to P (s), and where B - P (s) and A - P (s) are inverses to respectively. These functions P (s) and P (s) are used below with P = n α . In [], Cianchi found the necessary and sufficient conditions for the boundedness of I α on Orlicz spaces.
Let and Young functions and let n/α and n/α be the Young functions defined as in (.) and (.), respectively. Then dominates n/α globally and n/α dominates globally.
For proving our main results, we need the following estimate.
The following theorem gives necessary and sufficient conditions for the boundedness of the operator I α from L (R n ) to WL (R n ) and from L (R n ) to L (R n ).
() The condition for all r > , where C >  does not depend on r, is necessary for the boundedness of I α from L (R n ) to WL (R n ) and from L (R n ) to L (R n ).

() If the regularity condition
holds for all r > , where C >  does not depend on r, then the condition (.) is necessary and sufficient for the boundedness of and have For Consequently we have Thus, by (.) we obtain Therefore, we get Let C  be as in (.). Then by Theorem ., i.e.
() We shall now prove the second part. Let B  = B(x  , r  ) and x ∈ B  . By Lemma ., we have r α  ≤ CI α χ B  (x). Therefore, by Lemma ., we have and .
Since this is true for every r  > , we are done.
() The third statement of the theorem follows from the first and second parts of the theorem.
From Theorems . and . we have the following corollary. The following result is due to Nakai [].

Theorem . ([])
Let  < α < n and , ∈ Y. Assume that the conditions (.) and (.) hold. Then the operator I α is bounded from L (R n ) to WL (R n ). Moreover, if ∈ ∇  , then I α is bounded from L (R n ) to L (R n ).
Remark . Note that in Theorem . Nakai found the sufficient conditions which ensures the boundedness of the operator I α from L (R n ) to L (R n ), including weak version. Theorem . improves Theorem . by adding the necessity. Theorems . and . are different characterizations for the boundedness of the operator I α from L (R n ) to L (R n ), including a weak version.

Maximal commutator in Orlicz spaces
In this section we investigate the boundedness of the maximal commutator M b in Orlicz 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 · * . Before proving our theorems, we need the following lemmas and theorem.
where C is independent of b, x, r, and t.

Lemma . ([])
Let f ∈ BMO(R n ) and be a Young function with ∈  , then Then the operator M b is bounded on L (R n ), and the inequality holds with constant C  independent of f .
The following theorem is valid.
Theorem . Let b ∈ BMO(R n ) and be a Young function. Then the condition ∈ ∇  is necessary for the boundedness of M b on L (R n ).
Proof Assume that (.) holds. For the particular symbol b(x) = log |x| ∈ BMO(R n ) and where r = (a  uv) -/n , B r = B(, r), a r = |B r |, u >  and v > . By Lemma . and (.), we have On the other hand, if x / ∈ B r then B r ⊂ B(x, |x|) since for y ∈ B r we have |x -y| ≤ |x| + |y| ≤ |x| + r ≤ |x|.
Also for each y ∈ B r , we have Following the ideas of [], for g = - (u)χ B s with s = (a  u) -/n we obtain R n g(x) dx ≤ u|B s | = us n |B  | = .
Since the Luxemburg-Nakano norm is equivalent to the Orlicz norm for every t > , and so satisfies the  condition.
By Theorems . and . we have the following result.
Corollary . Let b ∈ BMO(R n ) and ∈ Y. Then the condition ∈ ∇  is necessary and sufficient for the boundedness of M b on L (R n ).

Theorem . b ∈ L  loc (R n ) and be a Young function. The condition b ∈ BMO(R n ) is necessary for the boundedness of M b on L
Thus b ∈ BMO(R n ).
By Theorems . and . we have the following result.
Corollary . Let be a Young function with ∈ ∇  . Then the condition b ∈ BMO(R n ) is necessary and sufficient for the boundedness of M b on L (R n ).

Commutators of Riesz potential in Orlicz spaces
In this section we find necessary and sufficient conditions for the boundedness of the commutators of Riesz potential on Orlicz spaces with the help of the previous section. Lemma . If  < α < n and f , b ∈ L  loc (R n ), then for all x ∈ R n and r >  we get The following theorem gives necessary and sufficient conditions for the boundedness of the operator |b, I α | from L (R n ) to L (R n ).
() If ∈ ∇  and ∈  , then the condition holds for all r > , where C >  does not depend on r, then the condition (.) is necessary and sufficient for the boundedness of |b, I α | from L (R n ) to L (R n ).
Choose r >  so that - (r -n ) = M b f (x) Therefore, we get Let C  be as in (.). Consequently by Theorem . we have In order to estimate J  , by (.), Lemma . and condition (.), we also get Consequently, we have Combining (.) and (.), we get By taking the supremum over B in (.), we get since the constants in (.) do not depend on x  and r.
() We shall now prove the second part. Let B  = B(x  , r  ) and x ∈ B  . By Lemma ., we have r α  |b(x)b B  | ≤ C|b, I α |χ B  (x). Therefore, by Lemmas . and . .
Since this is true for every r  > , we are done.
() The third statement of the theorem follows from the first and second parts of the theorem.
Remark . Theorems . and . give different sufficient conditions for the boundedness of the operator [b, I α ] from L (R n ) to L (R n ). But in Theorem . we also have necessary conditions for the boundedness of the operator |b, I α | from L (R n ) to L (R n ).
The following theorem is valid.
() The third statement of the theorem follows from the first and second parts of the theorem.

Characterization of Lipschitz spaces via commutators
In this section, as an application of Theorem . we consider the boundedness of [b, I α ] on Orlicz spaces when b belongs to the Lipschitz space, by which some new characterizations of the Lipschitz spaces are given. Such a characterization was given in [] as an application of the boundedness of M b on Lebesgue spaces. Definition . Let  < β < , we say a function b belongs to the Lipschitz space˙ β (R n ) if there exists a constant C such that, for all x, y ∈ R n , The smallest such constant C is called the˙ β (R n ) norm of b and is denoted by b ˙ β (R n ) .
To prove the theorems, we need auxiliary results. The first one is the following characterization of Lipschitz space, which is due to DeVore and Sharply [].
hold for all t > , where C >  does not depend on t, then the condition b ∈˙ β (R n ) is sufficient for the boundedness of [b, I α ] from L (R n ) to L (R n ).

() If the condition
holds for all t > , where C >  does not depend on t, then the condition b ∈˙ β (R n ) is necessary for the boundedness of |b, I α | from L (R n ) to L (R n ). . If ∈ ∇  , condition (.) holds and - (t) ≈ - (t)t -α+β n , then the condition b ∈˙ β (R n ) is necessary and sufficient for the boundedness of |b, I α | from L (R n ) to L (R n ).
Proof () The first statement of the theorem follows from Theorem . and Lemma ..
() We shall now prove the second part. Suppose that - (t) - (t)t -(α+β)/n and |b, I α | is bounded from L (R n ) to L (R n ). Choose any ball B in R n , by Lemmas . and Thus by Lemma . we get b ∈˙ β (R n ). () The third statement of the theorem follows from the first and second parts of the theorem.
The following theorem is valid.
Theorem . Let  < β < ,  < α < n,  < α + β < n, b ∈ L  loc (R n ), , ∈ Y. () If the conditions (.) and (.) are satisfied, then the condition b ∈˙ β (R n ) is sufficient for the boundedness of [b, I α ] from L (R n ) to WL (R n ). () If the condition (.) holds and t +ε (t) is almost decreasing for some ε > , then the condition b ∈˙ β (R n ) is necessary for the boundedness of |b, I α | from L (R n ) to WL (R n ). () If - (t) ≈ - (t)t -α+β n , condition (.) holds and t +ε (t) is almost decreasing for some ε > , then the condition b ∈˙ β (R n ) is necessary and sufficient for the boundedness of |b, I α | from L (R n ) to WL (R n ).
Proof () The first statement of the theorem follows from Theorem . and Lemma ..
() For any fixed ball B  such that x ∈ B  by Lemma . we have |B  | α/n |b(x)b B  | |b, I α |χ B  (x). Thus, together with the boundedness of |b, I α | from L (R n ) to WL (R n ) and Lemma ., .
Let t >  be a constant to be determined later, then where we use t +ε (t) being almost decreasing in the last step. Set t = C|B  | α+β n in the above estimate, we have Thus by Lemma . we get b ∈˙ β (R n ) since B  is an arbitrary ball in R n .
() The third statement of the theorem follows from the first and second parts of the theorem.