A note on fractional integral operators on Herz spaces with variable exponent

Indeed Gatto and Vagi’s result was proved in the setting of the spaces of homogeneous type. Also there are extensions like weighted function spaces theory, see []. Recently, Ramseyer et al. [] extended Gatto and Vagi’s result in the variable exponent function spaces case. For the sake of convenience, we briefly recall some basic elements of the Lebesgue spaces with variable exponent, while more results can be found in [, ] and the references therein. Let be a non-empty open set in Rn and p(·) : → [,∞) be a measurable function. The variable exponent Lebesgue space Lp(·)( ) is defined by


Introduction
Let R n be the n-dimensional Euclid space. For  < β < n, the fractional integral operator I β is defined by The famous Hardy-Littlewood-Sobolev theorem tells us that I β is a bounded operator from the usual Lebesgue spaces L p to L q with /q = /pβ/n, where  < p < n/β. Also I β is bounded from L n β into BMO. As for p > n/β, Gatto and Vagi [] proved thatĨ β is bounded from L p into Lipschitz spaces whose smoothness is controlled by p and α, whereĨ β is defined as |y| n-β f (y) dy. It is easy to check that L p(·) ( ) is a Banach space with the norm defined by We let and we denote by P( ) the set of measurable function p(·) on with value in [, ∞) such that  < p -( ) ≤ p(·) ≤ p + ( ) < ∞. For the sake of simplicity, we write L p(·) (R n ) as L p(·) and f L p(·) (R n ) as f p(·) , respectively. We say a function p(·) : R n − → R is locally log-Hölder continuous, if there exists a constant C such that for all x, y ∈ R n . If, for some p ∞ ∈ R and C > , we have for all x ∈ R n , then we say p(·) is log-Hölder continuous at infinity.
The notation P log (R n ) is used for all those exponents p(·) ∈ P(R n ) which are locally log-Hölder continuous and log-Hölder continuous at infinity with p ∞ := lim |x|→∞ p(x). Moreover, we can easily show that p(·) ∈ P log (R n ) implies p (·) ∈ P log (R n ) .
Ramseyer, Salinas and Viviani introduced the following function space, which can be viewed as the variable exponent counterpart of Lipschitz space defined by Peetre in [].
Definition  ([]) Let  < β < n and p(·) ∈ P(R n ) and denote the Lebesgue measure of B by |B|. We say that a locally integrable function f belongs to L β,p(·) (R n ) if there exists a constant C such that Ramseyer, Salinas and Viviani proved the following theorem.
() p(·) ∈ P β , i.e., there exists a positive constant C such that for any ball B, hold for every ball B, where x B denotes its center.
Corollary . in [] says that if p(·) ∈ P log (R n ) with  < β - < n p + , then p(·) satisfies (.). With the help of Theorem .,Ĩ β is bounded from L p(·) (R n ) to L β,p(·) (R n ). It is natural to ask what the target space is when L p(·) (R n ) is replaced by other more general spaces. The main result of this note is that the target space of mappingĨ β is just the variant Lipschitz space when L p(·) (R n ) is replaced by the so-called variable exponent Herz space.

Herz spaces and main results
Variable exponent Herz spaces were considered by many authors in recent years. Especially Herz spaces with two variable exponents and even with three variable exponents were produced by Almeida and Drihem [] and Samko [], respectively. For brevity, we only consider the Herz space with one variable exponent case, which was introduced by are classical Herz spaces. We can refer to [] for more properties of the classical one.
Our main result is to establish a result of mapping property ofĨ β onK α p(·),q (R n ). For this purpose we need to define a variant of the Lipschitz space.
Definition  Given -∞ < λ < +∞,  < β < n, and p(·) ∈ P(R n ). We say that a locally integrable function f belongs to L λ β,p(·) if there exists a constant C such that . Remark . It is easy to see that in Definition  the average m B f can be replaced by a constant in the following sense: Now we are in a position to state our results.
This shows that Theorem . is optimal.
We give some lemmas in Section  and then prove the above theorems in Section . C always means a positive constant independent of the main parameters and it may change from one occurrence to another. f ∼ g means C - g ≤ f ≤ Cg.

Technique lemmas
then for all f ∈ L p(·) ( ) and all g ∈ L p (·) ( ) we have Given a function f ∈ L  loc (R n ), the Hardy-Littlewood maximal operator M is defined by and we say B(R n ) is the set of p(·) ∈ P(R n ) satisfying the condition that M is bounded on L p(·) (R n ). Remark . According to Lemma ., the conclusion of Lemma . is correct when the condition p(·) ∈ B(R n ) is replaced by p(·) ∈ P log (R n ).
() If p(·) ∈ P β , then there exists a positive constant C such that for every ball B, where B is the ball having the same center as B but whose diameter is two times as large.
We point out that the two results collected in Lemma . are from []. The result () is Corollary . and () is Lemma . therein, respectively.

Lemma . Let p(·) ∈ P log (R n ), then there exists a constant C >  such that for all balls B and all measurable subsets S
Proof We proved the lemma in the following three cases: () |S| < |B| < ; () |S| <  < |B|; ()  ≤ |S| < |B|. Cases () and () are easy, we omit the details. Now for case (). By Lemma ., Indeed in the last inequality in the above equation, since |x Bx S | ≤ r  , we make use of the local-Hölder continuity of p(x), so The lemma is proved.

Proofs of theorems
Proof of Theorem . Fix a ball Q = B(x  , R). To prove Theorem ., we need to estimate Let k be the least integer such that Q ⊂ B(,  k ), hence |x  | + R ∼  k . We consider three cases: Case () or (). Note that |Q| ≥ C kn in both cases. We write |x-y| n-β dy. Then by Fubini's theorem, we have Using Lemma . and Lemma ., we derive the estimate Now we can distinguish three cases as follows, by Lemma .: () If j - ≤ k < , we get Here in the last inequality we using the following facts: If k ≥ , |x k | <  k , and |x j | <  j ≤  k , then the local-Hölder continuity of p(x) at the origin yields Since Thus by the condition αn + n p -< , it follows that Next we estimateĨ β f  . Let c  =Ĩ β f  (x  ). For any x ∈ Q, y ∈ A j and j ≥ k + , we have By Lemma . and Lemma ., we obtain Applying the arguments used in the corresponding step of the estimate ofĨ β f  , we arrive at the inequality Since αβ + n p + +  > , Combining (.)-(.), cases () and () are proved. Case (). We write |x-y| n-β dy. Then by Fubini's theorem and Lemma ., we obtain Note that B(x  , R) ⊂ k+ j=k- A j , so Next we estimateĨ β f  . Let c  =Ĩ β f  (x  ). By Lemma ., and then by the condition  < β < n p + +  with Lemma ., Finally we estimateĨ β f  . Let c  =Ĩ β f  (x  ). For any x ∈ Q, y ∈ R n \ B(,  k+ ), and j ≥ k + , we have |x  -y| ≥ |y| -|x  | >  k+ - k =  k . Then we write Applying Lemma . we obtain Since αβ + n p + +  > , by (.) and (.), Proof of Theorem . Let f i (x) =  -iα χ [ i , i +] (x) for i ≥ , then f i K α p(·),q (R  ) ∼  and This finishes the proof of Theorem ..