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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 at motivating 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.
A positive measurable function is called regularly varying at infinity with index , written as , if, for each and some ,where is an interval whose center at and radius and . In particular, when , 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 [3–7] and the references given there.
If , thenholds uniformly for with . And for and , the following asymptotic behavior is true:Karamata regular variation at can also be defined by a positive measurable function with replaced by .
In , Nakai introduced a generalized weighted Morrey apace with the weight function satisfying the following conditions:where any . Inspired by Nakai, a general case of (2) and (3) can be defined aswhere with andIt is of interest to know that when or in (6) and (7), the function can be seen as Karamata regular variation at and infinity, respectively.
Let satisfy (6) and (7). Then the Morrey space associated with Karamata regular variation (K-Morrey space) can be adopted from  as It is obvious that is a Banach space with norm . If , then . If , then . And if with , then is the classical Morrey space which was first introduced by Morrey  to investigate the local behavior of solutions to the second order elliptic PDEs.
It is also worth pointing out that when , , and (6) and (7) are exactly the same as (4) and (5), is the generalized Morrey space introduced by Nakai  with . In his well-known paper, Nakai proved the interesting result that the Hardy-Littlewood maximal operator, singular integral operator, and the Riesz potential were bounded on certain space. As stated in , we can also prove that with and  is an example of functions satisfying (6) and (7).
In this paper, we shall consider some estimates for one-sided operators on the K-Morrey space . Let us first recall some basic definitions of one-sided operators. The reasons to study one-sided operators involve the requirements of ergodic theory . The study of weighted theory for one-sided operators was first introduced by Sawyer  and many authors thereafter ([13–19]). The one-sided Hardy-Littlewood maximal operators  are defined bywhich arise in the ergodic maximal function. It is well known that and are bounded on spaces () and bounded from spaces to weak spaces. Such operators are also bounded on K-Morrey spaces, which we now formulated as follows.
Let be an one-sided integral operator with one-sided kernel supported in and satisfy That is,Both the one-sided Calderón-Zygmund singular integral operator  and the one-sided oscillatory singular operator  are examples of operators . Our second result is as follows.
Theorem 2. Let satisfy (6) and (7) with . Then one has the following:
(a) If is bounded on with , then there is a constant such that (b) If is bounded from space to weak space, then there is a constant such that for any and for any
Remark 3. Theorem 2 provides a criterion for the boundedness of one-sided singular integral operators on K-Morrey spaces.
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 , , , , satisfy (6), and with . Then one has the following:
(a) If , then there is a constant such that (b) If , then there is a constant such that for any 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 . Throughout this paper, is a constant which may change from line to line.
In this section, some lemmas are described by some methods adopted from .
Lemma 5 (see ). Let be measurable functions. Then one has the following:
(a) For every , there is a constant such that(b) There is a constant 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 , , satisfy (6), and Then for , there is a constant such that
Proof. The proof of Lemma 6 has a root in [8, Lemma ]. We adopted its proof here for the one-sided case. Let be the characteristic function of . Then for . For and , the following can be shown: This clearly forces We conclude from (6) that hence thatand finally thatEquations (27) and (28) show that is comparable to , which derives that On account of the estimates given above, Lemma 6 is proved.
Lemma 7 (see ). Suppose that . If there is a constant such that, for any , , then there are constants and such that, for any ,
The remainder lemma of this section will be devoted to the boundedness of one-sided fractional integrals on Lebesgue spaces.
Lemma 8 (see ). Let , , , and . Then there exists constant 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 .
Proof of Theorem 1. Let us first prove (a). Taking into account (20) and Lemma 6 with , the following can be shown easilyApplying (21) and Lemma 6 with , we note that The proof of Theorem 1 is completed.
Proof of Theorem 2. (a) For any , let to produce We conclude from the fact that is bounded on that The task is now to deal with the term . The fact that and allows the user to estimate asThis clearly forces Applying Lemma 7 to , 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 , the fact that , and for shows that Hence, The proof of (a) is completed by showing that (b) For and , let to produce Since is bounded from to , 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 to produce Applying Lemma 8, we conclude that which implies For and , the argument in (36) shows that Hence, Applying Lemma 7 to , the following is true: Let . Hölder’s inequality can be used to obtain The fact and Lemma 6 with allow us to estimate as which produces (b) For and , let produce The fact that is bounded from to allows us to get Applying the same analysis as that of (a) and Lemma 6 with , , the following can be confirmed easily: This produces the following inequality: which is our desired result.
3. Boundedness of Operators on -Dimensional -Morrey Spaces
Since one-sided operators are defined on , 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 with and consider the boundedness of Hardy-Littlewood maximal operator, singular integral operator, and the Riesz potential on these spaces applying the method in  with only a slight modification. Let and be the cube whose center at 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 with and
Theorem 9. Let be the Hardy-Littlewood maximal operator Assume that satisfies (63) and (64) with . Then one has the following:
(a) For , there is a constant such that (b) There is a constant such that for any and for any cube
Theorem 10. Let be a singular integral operator with the kernel satisfying Assume that satisfies (63) and (64) with . Then one has the following:
(a) If is bounded on with , then there is a constant such that (b) If is bounded from space to weak space, then there is a constant such that, for any and for any cube , one has
The last result is about the Riesz potential
Theorem 11. Let , , , , satisfy (63), and with . Then one has the following:
(a) If , then there is a constant such that (b) If , then there is a constant such that for any and for any cube
Conflicts of Interest
The authors declare that they have no conflicts of interest.
All authors read and approved the final manuscript.
This work was partially supported by National Natural Science Foundation of China (Grant nos. 11671185, 11771195, and 11771196), the key Laboratory of Complex Systems and Intelligent Computing in University of Shandong (Linyi University), and the Applied Mathematics Enhancement Program of Linyi University.
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