Boundedness of fractional oscillatory integral operators and their commutators on generalized Morrey spaces
© Eroglu; licensee Springer. 2013
Received: 25 July 2012
Accepted: 17 March 2013
Published: 2 April 2013
In this paper, it is proved that both oscillatory integral operators and fractional oscillatory integral operators are bounded on generalized Morrey spaces . The corresponding commutators generated by BMO functions are also considered.
MSC:42B20, 42B25, 42B35.
1 Introduction and main results
The classical Morrey spaces, were introduced by Morrey  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.
Under this definition, becomes a Banach space; for , it coincides with and for with .
where denotes the weak -space.
Here and subsequently, C will denote a positive constant which may vary from line to line but will remain independent of the relevant quantities.
for all a, b with . Chiarenza and Frasca  showed the boundedness of on .
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 Calderón-Zygmund operators or fractional integrals. In , Chanillo and Christ established the weak type estimate of T.
where C does not depend on x and t. If K is a SCZK and the operator is of type , then for and any polynomial the operator S is bounded from to .
Moreover, for and K is a CZK operator, the operator T is bounded from to .
where C does not depend on x and t. Then for the operator is bounded from to and for the operator is bounded from to .
where C does not depend on x and t. If K is a SCZK and the operator is of type , then for any polynomial the operator is bounded from to .
where C does not depend on x and t. Then the operator is bounded from to .
2 Some known results in generalized Morrey spaces
The following statements were proved by Nakai .
where C does not depend on x and r. Then for the operators M and T are bounded in and for , M and T are bounded from to .
where C does not depend on x and r. Then for , the operators and are bounded from to and for , and are bounded from to .
where C does not depend on x and t. Then the operators M and T are bounded from to for and from to .
where C does not depend on x and r. Then the operators and are bounded from to for and from to for .
Theorem E Let and satisfy the condition (2.4). Then the operators M and T are bounded from to for and from to .
Theorem F Let , , and satisfy the condition (1.14). Then the operators and are bounded from to for and from to for .
Note that integral conditions of type (2.3) after the paper  of 1956 are often referred to as Bary-Stechkin or Zygmund-Bary-Stechkin conditions; see also . The classes of almost monotonic functions satisfying such integral conditions were later studied in a number of papers, see [19–21] 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 Matuszewska-Orlicz 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
Theorem G 
holds for any ball and for all .
holds for any ball and for all .
where constant is independent of f.
Then by (3.4) and (3.5), we get the inequality (3.1). □
Proof of Theorem 1.1
if . □
Proof of Theorem 1.2
4 Commutators of fractional oscillatory integral operators in the spaces
Let T be a Calderón-Zygmund singular integral operator and . A well known result of Coifman, Rochberg and Weiss  states that the commutator operator is bounded on for . The commutator of Calderón-Zygmund 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 .
If one regards two functions whose difference is a constant as one, then space is a Banach space with respect to norm .
where C is independent of f, x, r and t.
holds for any ball and for all .
where constant is independent of f.
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. □
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