Boundedness of fractional oscillatory integral operators and their commutators on generalized Morrey spaces

Ahmet Eroglu

doi:10.1186/1687-2770-2013-70

Abstract

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.

Highlights

1 Introduction and main results The classical Morrey spaces, were introduced by Morrey [ ] in, 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
Mp,λ(Rn) becomes a Banach space; for λ =, it coincides with Lp(Rn) and for λ = with L∞(Rn)
Let ≤ p < ∞, and K is a standard Calderón-Zygmund kernel (SCZK) and the Calderón-Zygmund singular integral operator S is of type (L (Rn), L (Rn))

Summary

Rn and

Sf (x) = p.v. eiP(x,y)K (x, y)f (y) dy, Rn where P(x, y) is a real valued polynomial defined on Rn × Rn. In [ ], Lu and Zhang proved that S is bounded on Lp with < p < ∞. Sαf (x) = eiP(x,y)Kα(x, y)f (y) dy, Rn where P(x, y) is a real valued polynomial defined on Rn × Rn. Obviously, when α = , S = S and K = K. Satisfies the condition n q where C does not depend on x and t. Polynomial, and (φ , φ ) satisfies the condition where C does not depend on x and t. Theorem A Let ≤ p < ∞ and φ(x, r) satisfy the conditions c– φ(x, r) ≤ φ(x, t) ≤ cφ(x, r) whenever r ≤ t ≤ r, where c (≥ ) does not depend on t, r and x ∈ Rn and.

Theorem B

We get