Abstract
Keywords:
generalized Morrey space; oscillatory integral; commutator; BMO spaces1 Introduction and main results
The classical Morrey spaces, were introduced by Morrey [1] in 1938, have been studied intensively by various authors and together with weighted Lebesgue spaces play an important role in the theory of partial differential equations; they appeared to be quite useful in the study of local behavior of the solutions of elliptic differential equations and describe local regularity more precisely than Lebesgue spaces.
Morrey spaces are defined as the set of all functions such that
Under this definition, becomes a Banach space; for , it coincides with and for with .
We also denote by the weak Morrey space of all functions for which
where denotes the weak space.
Definition 1 Let be a positive measurable function on and . We denote by the generalized Morrey space, the space of all functions with finite quasinorm
Also, by , we denote the weak generalized Morrey space of all functions for which
According to this definition, we recover the spaces and under the choice :
The theory of boundedness of classical operators of the real analysis, such as the maximal operator, fractional maximal operator, Riesz potential and the singular integral operators etc., from one weighted Lebesgue space to another one is well studied. Let . The fractional maximal operator and the Riesz potential are defined by
If , then is the HardyLittlewood maximal operator. In [2], Chiarenza and Frasca obtained the boundedness of M on . In [3], Adams established the boundedness of on .
Here and subsequently, C will denote a positive constant which may vary from line to line but will remain independent of the relevant quantities.
The CalderónZygmund singular integral operator is defined by
where K is a CalderónZygmund kernel (CZK). We say a kernel is a CZK if it satisfies
and
for all a, b with . Chiarenza and Frasca [2] showed the boundedness of on .
It is worth pointing out that the kernel in (1.1) is convolution kernel. However, there were many kinds of operators with nonconvolution kernels, such as Fourier transform and Radon transform [4], which both are versions of oscillatory integrals. The object we consider in this paper is a class of oscillatory integrals due to Ricci and Stein [5]
where is a real valued polynomial defined on , and K is a CZK.
It is well known that the oscillatory factor makes it impossible to establish the norm inequalities of (1.5) by the method as in the case of CalderónZygmund operators or fractional integrals. In [6], Chanillo and Christ established the weak type estimate of T.
A distribution kernel K is called a standard CalderónZygmund kernel (SCZK) if it satisfies the following hypotheses:
and
The corresponding CalderónZygmund integral operator and oscillatory integral operator S are defined by
and
where is a real valued polynomial defined on . In [7], Lu and Zhang proved that S is bounded on with . In [5], Ricci and Stein also introduced the standard fractional CalderónZygmund kernel (SFCZK) with , where the conditions (1.6) and (1.7) were replaced by
and
The corresponding fractional oscillatory integral operator is defined by (see [8])
where is also a real valued polynomial defined on . Obviously, when , and . Partly motivated by the idea from [9,10] and the results of [11], we now give the results of this paper in the following.
Theorem 1.1Let, andsatisfies the condition
whereCdoes not depend onxandt. IfKis a SCZK and the operatoris of type, then forand any polynomialthe operatorSis bounded fromto.
Moreover, forandKis a CZK operator, the operatorTis bounded fromto.
Theorem 1.2Let, , , is a polynomial, andsatisfies the condition
whereCdoes not depend onxandt. Then forthe operatoris bounded fromtoand forthe operatoris bounded fromto.
For a locally integrable function b, the commutator operator formed by S (or ) and b are defined by
and
Theorem 1.3Let, andsatisfies the condition
whereCdoes not depend onxandt. IfKis a SCZK and the operatoris of type, then for any polynomialthe operatoris bounded fromto.
Theorem 1.4Let, , , , is a polynomial, andsatisfies the condition
whereCdoes not depend onxandt. Then the operatoris bounded fromto.
2 Some known results in generalized Morrey spaces
In [9,10,12,13] and [14], there were obtained sufficient conditions on weights and for the boundedness of the singular operator T from to .
The following statements were proved by Nakai [14].
Theorem ALetandsatisfy the conditions
whenever, wherec (≥1) does not depend ont, randand
whereCdoes not depend onxandr. Then forthe operatorsMandTare bounded inand for, MandTare bounded fromto.
Theorem BLet, , andsatisfy the conditions (2.1) and
whereCdoes not depend onxandr. Then for, the operatorsandare bounded fromtoand for, andare bounded fromto.
The following statements, containing Nakai results obtained in [13,14] was proved by Guliyev in [9,10] (see also [15,16]).
Theorem CLetandsatisfy the condition
whereCdoes not depend onxandt. Then the operatorsMandTare bounded fromtoforand fromto.
Theorem DLet, , andsatisfy the condition
whereCdoes not depend onxandr. Then the operatorsandare bounded fromtoforand fromtofor.
The following statements, containing Guliyev results obtained in [9,10] was proved by Guliyev et al. in [11,12].
Theorem ELetandsatisfy the condition (2.4). Then the operatorsMandTare bounded fromtoforand fromto.
Theorem FLet, , andsatisfy the condition (1.14). Then the operatorsandare bounded fromtoforand fromtofor.
Note that integral conditions of type (2.3) after the paper [17] of 1956 are often referred to as BaryStechkin or ZygmundBaryStechkin conditions; see also [18]. The classes of almost monotonic functions satisfying such integral conditions were later studied in a number of papers, see [1921] and references therein, where the characterization of integral inequalities of such a kind was given in terms of certain lower and upper indices known as MatuszewskaOrlicz indices. Note that in the cited papers the integral inequalities were studied as . Such inequalities are also of interest when they allow to impose different conditions as and ; such a case was dealt with in [22,23].
3 The fractional oscillatory integral operators in the spaces
In this section, we are going to use the following statement on the boundedness of the Hardy operator:
Theorem G[24]
The inequality
holds for all nonnegative and nonincreasinggonif and only if
Lemma 3.1Let, andKis a SCZK and the CalderónZygmund singular integral operatoris of type. Then forand any polynomialthe inequality
holds for any balland for all.
holds for any balland for all.
Proof Let . For arbitrary , set for the ball centered at and radius r, . We represent f as
and have
It is known that (see [5], see also [7,25,26]), if K is a SCZK and the operator is of type , then for and any polynomial the operator S is bounded on . Since , and boundedness of S in (see [5]) it follows that
where constant is independent of f.
It is clear that , implies . We get
By Fubini’s theorem and applying Hölder inequality, we have
Moreover, for all the inequality
is valid. Thus,
On the other hand,
Hence,
Let . From the weak boundedness of T (see [6]) and (3.4), it follows that:
Then by (3.4) and (3.5), we get the inequality (3.1). □
Proof of Theorem 1.1
By Lemma 3.1 and Theorem G, we get
Proof of Theorem 1.2
The proof of Theorem 1.2 follows from Theorem F and the following observation:
□
4 Commutators of fractional oscillatory integral operators in the spaces
Let T be a CalderónZygmund singular integral operator and . A well known result of Coifman, Rochberg and Weiss [27] states that the commutator operator is bounded on for . The commutator of CalderónZygmund operators plays an important role in studying the regularity of solutions of elliptic partial differential equations of second order (see, for example, [2,28,29]).
First, we recall the definition of the space .
Definition 2 Suppose that , let
where
Define
If one regards two functions whose difference is a constant as one, then space is a Banach space with respect to norm .
Remark 1 (1) The JohnNirenberg inequality: there are constants , such that for all and
(2) The JohnNirenberg inequality implies that
(3) Let . Then there is a constant such that
where C is independent of f, x, r and t.
Lemma 4.1Let, , Kis a SCZK and the CalderónZygmund singular integral operatoris of type. Then forand any polynomialthe inequality
holds for any balland for all.
Proof Let . For arbitrary , set for the ball centered at and radius r, . We represent f as
and have
It is known that (see [5], see also [7,25,26]), if K is a SCZK and the operator is of type , then for and any polynomial the commutator operator is bounded on . Since , and boundedness of in (see [5]) it follows that
where constant is independent of f.
Then
Applying Hölder’s inequality and by (4.1), (4.2), we get
In order to estimate note that
By (4.1), we get
Thus, by (3.2)
Summing up and , for all we get
Finally,
and statement of Lemma 4.1 follows by (3.4). □
Proof of Theorem 1.3 The statement of Theorem 1.3 follows by Lemma 4.1 and Theorem G in the same manner as in the proof of Theorem G. □
Proof of Theorem 1.4 The proof of Theorem 1.4 follows from the Theorem 7.4 in [11] and the following observation:
□
Competing interests
The author declares that they have no competing interests.
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