A survey on oscillatory integral operators

This is a short survey on osicllatory integral operators. We summarize the main development and managing techniques of the field, and give some open problems and main references in the end.     

Estimates for oscillatory strongly singular integral operators

In this paper we consider oscillatory integral operators with strong singularities on . We obtain sharp decay estimates for L² operator norm. We also obtain H^p estimates for difference of oscillatory strongly singular integral operators and strongly singular integral operators, which gives L^p mapping properties of oscillatory strongly singular integral operators.     

Rough oscillatory singular integral operators of nonconvolution type

In this paper, we study a classes of oscillatory singular integral operators of nonconvolution type with phases more general than polynomials. We prove that such operators are bounded on Lp provided their kernels satisfy a very weak condition. In addition, we also study the related truncated oscillatory singular integral operators. Moreover, we present a class of unbounded oscillatory singular integral operators.     

Weighted estimate for the multilinear oscillatory singular integral operators

In this paper, the weighted L^p(R^n) boundedness for the multilinear oscillatory singular integral operators with polynomial phases is studied.     

Bounds for singular fractional integrals and related Fourier integral operators

We prove sharp L^p−L^q endpoint bounds for singular fractional integral operators and related Fourier integral operators, under the nonvanishing rotational curvature hypothesis.     

振荡积分算子及其交换子在加权Morrey空间上的有界性质

振荡积分算子的有界性质是调和分析研究的中心内容之一. 本文得到了由Ricci 和Stein 定义的一类振荡积分算子在加权Morrey 空间中的强型、弱型估计. 在此基础上, 得到了该类振荡积分算子与BMO 函数生成的交换子的强型估计, 还建立了分数次振荡积分算子的对应结果.     

Bounds for commutators of multilinear fractional integral operators with homogeneous kernels

We will show bounds for commutators of multilinear fractional integral operators with some homogeneous kernels.     

L~2 estimates for oscillatory singular integral operators with analytic phrases

L2 estimates are obtained for some oscillatory singular integral operators with analytic phrases by using the technology of almost orthogonality, oscillatory estimates and size estimates.     

Linear and bilinear estimates for oscillatory integral operators related to restriction to hypersurfaces

We obtain bilinear estimates for oscillatory integral operators which are variable coefficient generalizations of bilinear restriction estimates for hypersurfaces. As applications, we improve the known estimates for oscillatory integrals.     

Sharp Lp decay of oscillatory integral operators with certain homogeneous polynomial phases in several variables

In this paper, we obtain the L~p decay of oscillatory integral operators T_λ with certain homogeneous polynomial phase functions of degree d in(n + n)-dimensions; we require that d 2 n. If d/(d-n) p d/n,the decay is sharp and the decay rate is related to the Newton distance. For p = d/n or d/(d-n), we obtain the almost sharp decay, where "almost" means that the decay contains a log(λ) term. For otherwise, the L~p decay of T_λ is also obtained but not sharp. Finally, we provide a counterexample to show that d/(d-n) p d/n is not necessary to guarantee the sharp decay.     

Polysingular integral operators

Summary We study a class of linear singular integral operators of the Cauchy type on the n-torus; an application is given to a boundary value problem for functions of several complex variables.     

Strong integral operators

It is shown that a strong integral (strong B-integral) operator in L₂ is a Hilbert-Schmidt (nuclear) operator.Translated from Matematicheskie Zametki, Vol. 16, No. 6, pp. 907–912, December, 1974.     

Singular integral operators on complicated contours and pseudodifferential operators

We apply the pseudodifferential operator technique to the study of algebras of singular operators on complicated contours. This technique is used to construct a symbolic calculus for theC ^*-algebra generated by singular integral operators whose coefficients may have singularities of the second kind on complicated contours; the curves forming a node are not required to have a tangent at the node. Translated fromMatematicheskie Zametki, Vol. 58, No. 1, pp. 67–85, July, 1995. This work was partially supported by the Russian Foundation for Basic Research under grant No. 93-04-691.     

estimates for oscillatory integral operators

An endpoint boundedness result is established for a class of oscillatory integral operators.