Abstract. In this note we consider the parametric Marcinkiewicz integrals with mixed homogeneity along polynomials compound curves. Under the rather weakened size conditions on the integral kernels both on the unit sphere and in the radial direction, the L p bounds of such operators are given by an extrapolation argument. Some previous results are greatly extended and improved.

This paper is concerned with the singular integral operators along polynomial curves. The boundedness for such operators on Triebel-Lizorkin spaces and Besov spaces is established, provided the kernels satisfy rather weak size conditions both on the unit sphere and in the radial direction. Moreover, the corresponding results for the singular integrals associated to the compound curves formed by polynomial with certain smooth functions are also given. MSC: 42B20; 42B25

In this note we establish the L p boundedness of Marcinkiewicz integrals with mixed homogeneity along compound surfaces, which improve and extend some previous results. The main ingredient is to present a systematic treatment with several singular integral operators. MSC: 42B20; 42B15; 42B25

This paper is concerned with the norm estimates for the multilinear singular integral operators and their commutators formed by BMO functions on the weighted amalgam spacesLvw→q,Lpαℝn. Some criterions of boundedness for such operators inLvw→q,Lpαℝnare given. As applications, the norm inequalities for the multilinear Calderón-Zygmund operators and multilinear singular integrals with nonsmooth kernels as well as the corresponding commutators onLvw→q,Lpαℝnare obtained.

In this paper, we give a criterion of boundedness for the operators of convolution type on Triebel-Lizorkin spaces. As applications, under rather weak size conditions, the bounds for the singular integrals along certain compound surfaces on the above function spaces are obtained. Some previous results are improved and extended.