Operator Inequalities of Morrey Spaces Associated with Karamata Regular Variation

Karamata regular variation is a basic tool in stochastic process and the boundary blow-up problems for partial differential equations (PDEs). Morrey space is closely related to study of the regularity of solutions to elliptic PDEs. The aim of this paper is trying to bring together these two areas and this paper is intended as an attempt atmotivating some further research on these areas. A version of Morrey space associated with Karamata regular variation is introduced. As application, some estimates of operators, especially one-sided operators, on these spaces are considered.


Introduction
A positive measurable function  : R × R + → R + is called regularly varying at infinity with index , written as  ∈   , if, for each  > 0,  0 ∈ R and some  ∈ R, where ( 0 , ℎ) is an interval whose center at  0 and radius ℎ and ( 0 , ℎ) = ∫ ( 0 ,ℎ) ().In particular, when  = 0,  is called slowly varying at infinity. is the classical Karamata regular variation.Karamata regular variation theory was first introduced and established by Karamata in 1930.It is a basic tool in stochastic process [1,2] and has been applied to study the boundary behavior of solutions to boundary blow-up elliptic problems and singular nonlinear Dirichlet problems; for some of this work, see [3][4][5][6][7] and the references given there.
In [8], Nakai introduced a generalized weighted Morrey apace with the weight function  satisfying the following conditions: where any  ∈ R. Inspired by Nakai, a general case of ( 2) and (3) can be defined as where  ∈ [ 1 ,  2 ] with 0 <  1 <  2 and It is of interest to know that when ℎ → 0 or ℎ → ∞ in ( 6) and (7), the function  can be seen as Karamata regular variation at 0 and infinity, respectively.Let  satisfy ( 6) and (7).Then the Morrey space associated with Karamata regular variation (K-Morrey space) can be adopted from [8] as It is obvious that (R) is the classical Morrey space which was first introduced by Morrey [9] to investigate the local behavior of solutions to the second order elliptic PDEs.
In this paper, we shall consider some estimates for onesided operators on the K-Morrey space  ,  (R).Let us first recall some basic definitions of one-sided operators.The reasons to study one-sided operators involve the requirements of ergodic theory [11].The study of weighted theory for onesided operators was first introduced by Sawyer [12] and many authors thereafter ( [13][14][15][16][17][18][19]).The one-sided Hardy-Littlewood maximal operators [12] are defined by which arise in the ergodic maximal function.It is well known that  + and  − are bounded on   (R) spaces (1 <  < ∞) and bounded from  1 (R) spaces to weak  1 (R) spaces.Such operators are also bounded on K-Morrey spaces, which we now formulated as follows.
Theorem 1.Let  satisfy ( 6) and (7) Let  + be an one-sided integral operator with one-sided kernel  + (, ) supported in R − = (−∞, 0) and satisfy That is, Both the one-sided Calderón-Zygmund singular integral operator [13] and the one-sided oscillatory singular operator [17] are examples of operators  + .Our second result is as follows.
Theorem 2. Let  satisfy ( 6) and ( 7) with 0 <  ≤ 1.Then one has the following: (b) If  + is bounded from  1 (R) space to weak  1 (R) space, then there is a constant  > 0 such that for any  > 0 and for any      { ∈  : Remark 3. Theorem 2 provides a criterion for the boundedness of one-sided singular integral operators on K-Morrey spaces.
By the corresponding boundedness in [13,15,17], both the one-sided Calderón-Zygmund singular integral operator and the one-sided oscillatory singular operator satisfy Theorem 2.
In the fractional case, both the Rieman-Liouville fractional integral and the Weyl fractional integral are examples of one-sided fractional integrals.Without loss of generality, we take  +  as our model in the following analysis.The last goal of this section is to show that  +  is also bounded on K-Morrey space, which can be stated in the following theorem.
Theorem 4. Let 0 <  < 1, 0 <  ≤ 1, 1 ≤  < /, 1/ = 1/ − ,  satisfy (6), and (b) If  = 1, then there is a constant  > 0 such that for any  > 0 and for any Section 2 contains the proofs of Theorems 1-4.In Section 3, we extend the main results to -dimensional case, which cover the main results of [8].Throughout this paper,  is a constant which may change from line to line.

Preliminaries
In this section, some lemmas are described by some methods adopted from [20].
Lemma 5 (see [21]).Let  ≥ 0 be measurable functions.Then one has the following: (a) For every 1 <  < ∞, there is a constant  > 0 such that (b) There is a constant  > 0 such that The principal significance of Lemma 5 is that it allows one to obtain a version of one-sided Fefferman-Stein inequality.The following lemma will prove extremely useful in the proofs of the main results.Lemma 6.Let 0 <  ≤ 1, 0 <  ≤ 1,  satisfy (6), and Then for 1 ≤  < ∞, there is a constant  > 0 such that Proof.The proof of Lemma 6 has a root in [8, Lemma 1].We adopted its proof here for the one-sided case.Let   be the characteristic function of  = ( 0 , ℎ).Then  −   ≤ 1 for  ∈ 2.For  ∈ 2 +1 /2   and  ∈ , the following can be shown: This clearly forces We conclude from ( 6) that hence that and finally that Equations ( 27) and (28 On account of the estimates given above, Lemma 6 is proved. Lemma 7 (see [8]).Suppose that (ℎ) : R + → R + .If there is a constant  > 0 such that, for any ℎ > 0, ∫ ∞ ℎ (()/) ≤ (ℎ), then there are constants  > 0 and   > 0 such that, for any ℎ > 0, The remainder lemma of this section will be devoted to the boundedness of one-sided fractional integrals on Lebesgue spaces.Lemma 8 (see [14]).Let 0 <  < 1, 1 ≤  < /, 0 <  ≤ 1, and 1/ = 1/ − .Then there exists constant  > 0 such that Having disposed of the above lemmas, the estimates for the one-sided operators on K-Morrey spaces can be proved in this section.The method used here was partly adopted from [8].
Proof of Theorem 1.Let us first prove (a).Taking into account (20)  (37) Applying Lemma 7 to () = ( 0 , )/  , it is easy to check that Let  = ( − )/.Using Hölder's inequality, it may be concluded that Repeated application of Lemma 6 enables us to write For the term  2 , the fact that  −   () ≤ 1,  ∈ 2, and Since  + is bounded from  1 to  1,∞ , the following can be proved: Applying the same analysis as (a) and Lemma 6 with Therefore, We have thus completed the proof of Theorem 2.
Proof of Theorem 4.An argument similar to that of Theorem 2 can be used to prove Theorem 4. (a) For any , let Applying Lemma 8, we conclude that For  ∈  and  ∈ (2)  , the argument in (36) shows that Hence, Applying Lemma 7 to () = ( 0 , )/ − , the following is true: Let  = (−−)/.Hölder's inequality can be used to obtain The fact Applying the same analysis as that of (a) and Lemma 6 with  = 1,  = 1 −  = 1/ < 1, the following can be confirmed easily: This produces the following inequality: which is our desired result.

Boundedness of Operators on 𝑛-Dimensional 𝐾-Morrey Spaces
Since one-sided operators are defined on R, we built Theorems 1-4 in one dimension.The theorems in Section 2 gain interest if we realize that they are still hold for -dimension.
In fact, we can also define K-Morrey space on R  ( ≥ 2) with 0 <  ≤  and consider the boundedness of Hardy-Littlewood maximal operator, singular integral operator, and the Riesz potential on these spaces applying the method in [8] with only a slight modification.Let  : R  × R + → R + and Ω( 0 , ) be the cube whose center at  0 with edges has length  and is parallel to the coordinate axes.Then the definition of dimensional Morrey space associated with Karamata regular variation (K-Morrey space) can be defined by if  satisfies the following conditions: where  ∈ [  (74) When  = , it is of interest to know that Theorems 9-11 can be seen as an extension of that of [8] in the sense that these theorems agree with [8, Theorems 1-3], respectively.
With a slight modification in the proofs of [8, Theorems 1-3], Theorems 9-11 can be obtained easily; we omit its proof here for the similarity.
and Lemma 6 with  = 1, the following can be shown easily We conclude from the fact that  + is bounded on   (R  ) that The task is now to deal with the term  +  2 .The fact that  ∈  and  ∈ (2)  allows the user to estimate  −   as  −   () = sup The proof of Theorem 1 is completed.Proof of Theorem 2. (a) For any , let  =  2 + (2)  š  1 + 2 to produce  +  () ≤  +  1 () +  +  2 () .−   () .