Estimates of Some Operators on One-Sided Weighted Morrey Spaces

and Applied Analysis 3 to study fractional integrals [6]. In addition to their contributions to harmonic analysis, fractional integrals also play an essential role in many fields. The Hardy-Littlewood-Sobolev inequality of fractional integral is still an indispensable tool in the establishment of time-space estimates for the heat semigroup of nonlinear evolution equations. Let 0 < α < 1; the one-sided fractional maximal operator and the one-sided fractional integrals were defined by M + α f (x) = sup h>0 1 h 1−α ∫ x+h x 󵄨 󵄨 󵄨 󵄨 f (y) 󵄨 󵄨 󵄨 󵄨 dy, M − α f (x) = sup h>0 1 h 1−α ∫ x x−h 󵄨 󵄨 󵄨 󵄨 f (y) 󵄨 󵄨 󵄨 󵄨 dy, I + α f (x) = ∫ ∞ x f (y)


Introduction
The reasons to study one-sided operators involve not only the generalization of the theory of two-sided operators but also the requirements of ergodic theory [1].The well-known Riemann-Liouville fractional integral can be viewed as the one-sided version of Riesz potential (the solution of Laplace equation) in harmonic analysis and PDE [2,3].The study of weighted theory for one-sided operators was first introduced by Sawyer [4] and many authors thereafter [5][6][7][8][9][10].Many of their results show that, for a class of smaller operators (one-sided operators) and a class of wider weights (one-sided weights), many of the famous findings of harmonic analysis still hold.
The study of one-sided spaces emerged naturally alongside the study of one-sided operators.In one previous study, the authors studied one-sided BMO spaces associated with one-sided sharp functions and their relationship to good weights for the one-sided Hardy-Littlewood maximal functions [9].Aimar and Crescimbeni [11] further investigated the structures of these one-sided regular functions and their basic properties.Other classical works regarding one-sided spaces have also been published [12][13][14].
A version of one-sided weighted Morrey spaces and Campanato spaces is introduced in this paper.The boundedness of some one-sided operators and its effects on these spaces are investigated.First recall some definitions of the classical Campanato spaces and Morrey spaces.
Let −1/ ≤  < 1 and 1 ≤  < ∞.Then the Campanato space C , (R) can be defined using the following norm where  I = (1/|I|) ∫ I (), I denotes an interval contained in R, and |I| is the Lebesgue measure of I.The excellent structures of Campanato spaces render them useful in the study of the regularity theory of PDEs.They allow the user to determine an integral characterization of the spaces of Hölder continuous functions.This allows generalization of the classical Sobolev embedding theorems; see [15][16][17], for example.It is also well known that C 1,1/−1 (R) is the dual space of Hardy space   (R) when 0 <  < 1 [18].There has been also a recent account of the theory on Campanato spaces [19][20][21].The original form of classical Morrey space was first introduced by Morrey Jr. [22] to investigate the local behavior of solutions to the second order elliptic PDEs, 2

Abstract and Applied Analysis
It is obvious that M ,−1/ (R) =   (R).Many properties of solutions to PDEs are concerned with the boundedness of some operators on Morrey type spaces.In fact, the better inclusion between the Morrey and the Hölder spaces permits obtaining higher regularity of the solutions to different elliptic and parabolic boundary problems.In recent years, there has been an explosion of interest in the study of the boundedness of operators on Morrey type spaces [23][24][25].
The study of weighted estimates and their effects on these spaces is important to harmonic analysis.Weighted inequalities arise naturally in Fourier analysis, but their use is best justified by the variety of applications in which they appear.For example, the theory of weights plays an important role in the study of boundary value problems inherent in Laplace's equations on Lipschitz domains.Many authors are interested in the study of the events that occur when the weight function belongs to one of the Muckenhoupt classes.Let 1 <  < ∞.The Muckenhoupt class   [26] consists of all positive locally integrable functions  for which sup where 1/+1/  = 1.We say that  ∈  1 if there is a constant  > 0 such that () ≤ ().Here  is the Hardy-Littlewood maximal operator The study of weights for one-sided operators is motivated by their natural emergence in harmonic analysis.For example, certain measures are required when the one-sided Hardy-Littlewood maximal operators [4] arising in the ergodic maximal function are treated.The classical Dunford-Schwartz ergodic theorem can be considered the first result regarding weights for these operators.In [4], Sawyer introduced the one-sided   classes  +  and  −  ; they are defined by the following conditions: with support in R − = (−∞, 0) or R + = (0, +∞).Equation ( 10) is also called the size condition for  and (11) is the continuous condition for .An example of such a kernel is where   denotes the characteristic function of a set . Aimar et al. [5] studied the one-sided Calderón-Zygmund singular integrals which were defined by where the kernels  are OCZKs.The one-sided   classes not only control the boundedness of one-sided Hardy-Littlewood maximal operators, but also serve as the right weight classes for one-sided singular integral operators.They also appear in PDEs [27].
Also, a result concerning the converse of Theorem 3 is given in [5].
In addition to singular integral operators, fractional integral operators also play an important role in harmonic analysis.The problem of fractional derivation was an early impetus Abstract and Applied Analysis 3 to study fractional integrals [6].In addition to their contributions to harmonic analysis, fractional integrals also play an essential role in many fields.The Hardy-Littlewood-Sobolev inequality of fractional integral is still an indispensable tool in the establishment of time-space estimates for the heat semigroup of nonlinear evolution equations.Let 0 <  < 1; the one-sided fractional maximal operator and the one-sided fractional integrals were defined by respectively. +  and  −  are also called the Rieman-Liouville and the Weyl fractional integral operators.The boundedness of  +  was determined by Andersen and Sawyer [6].
The one-sided Campanato space and one-sided Morrey space can now be introduced. where When  = 0 and  = 1, C + 1,0 () coincides with the dual space of the one-sided weighted Hardy space  1 + () [28], which consists of certain classes of one-sided weighted BMO functions, see also [11,13].

Main Results
In this section, the boundedness of the one-sided operators mentioned in Section 1 and its effects on one-sided Morrey spaces are described.The primary results are formulated as follows.
Theorem 7(a) is also true when C + , () is replaced by M + , () (see proof of Theorem 7).A corresponding substitution for Theorem 7(b) under certain assumption with respect to  is given in Section 3.
For the fractional case, the following is true.
(b) ∈  +  for 1 <  < ∞ if and only if there exists

and
According to the definitions of  +  and  + (,) , the following relationship between these two classes can be assessed easily.
If () ∈   ; then it is a doubling weight, that is, there exists  > 0 such that for all  ∈ R and ℎ > 0. However, one-sided   weights do not satisfy this property.But the weights  +  satisfy a onesided doubling condition.
Like the one-sided doubling condition, the following proposition also plays an important role in the present arguments.
Proposition 12. Let  > 0 and ,  ≥ 1.Then Proof.For the proof of (a), we first claim that In fact, we can apply Hölder's inequality with exponents  and   to get Applying (26) to the function  =   and putting  in the place of  in (26), we obtain  (() − ) ≤  +  ()    () ≤    () .
The proof of (b) is a byproduct of (a) and the fact that Proof of Theorem 7. The proof of (a) is given first.Because M + , () ⊆ C + , () when  < 0, it is sufficient to prove that there exists  > 0 such that Decompose  =  1 +  2 =  2 +  (2)  to obtain Using Theorem 1 and Lemma 11, the following is true: Hölder's inequality and Proposition 12(a) allow us to estimate Ĩ as M + , () ≤          M + , () . (32) Theorem 7(b) can now be proven.Decomposing  =  1 +  2 =  2 +  (2)  shows that The fact that, if  ∈  +  , then  + is bounded on   () allows the following to be shown: ≤          M + , () . (34) Here, Lemma 11 is used in the last inequality.
For the term , by (11) and Proposition 12(a), we can derive the following: On account of the estimates for  and  given above, the following can be proved: Proof of Theorem 8. We begin with the proof for  +  , which is similar to that of Theorem 7(a).It is sufficient to show that there exists constant  > 0 such that Decompose  =  1 +  2 =  2 +  (2)  to obtain By Theorem 4 and Lemma 11, By the same arguments as those of Ĩ, J can be estimated as M + , () The proof of (b) is a reprise of the argument given in the proof of Theorem 7(b).Set  =  1 +  2 =  2 +  (2)  to obtain Theorem 4 and Lemma 11 allow us to estimate  as we obtain by Hölder's inequality and Proposition 12(b) that We have thus proved Theorem 8.

Boundedness of One-Sided Sublinear Operators
The method used in the proof of Theorem 7(b) depends heavily on the convolution of the kernel function.However, there are some other one-sided operators with nonconvoluted kernels.Such operators appear in many places and PDEs.The one-sided oscillatory singular integral operators that were first introduced by the authors of this paper in a previous study are one such example [7].
A class of more general one-sided operators that do not necessarily have convolution kernels can now be studied.Let   = 2   and   =   \  −1 for  ∈ .In this section, a definition made in a previous study [30] can be adopted to introduce a one-sided sublinear operator satisfying the following size condition: where supp  ⊆   and 0 ≤  0 <  ≤  0 + 2 −1 ℎ with  ∈ .
For the fractional case, the corresponding size condition can be introduced: where supp  ⊆   and 0 ≤  0 <  ≤  0 + 2 Theorems 13 and 14 agree with Theorems 7(a) and 8(a) but are different from Theorems 7(b) and 8(b).The conditions of the kernel functions in Theorems 13 and 14 are weaker than those of Theorems 7(b) and 8(b), respectively, in that only the size conditions are used there.For this reason, Theorems 13 and 14 can be seen as an extension of Theorems 7 and 8, respectively.However, the present study was conducted under the assumptions that 1 <  < 1/(1 + ) and  <  < 1/(1 + ).These conditions are stronger than those of Theorems 7(b) and 8(b).
Proof of Theorem 13.It is sufficient to show that there exists  > 0 such that can be found easily.
In view of (45), the following is true:  (51) Proof of Theorem 14.An argument similar to that used in the proof of Theorem 13 can be used to produce where we have used Propositions 10 and 12.