We investigate the regularity properties of discrete multisublinear fractional maximal operators,both in the centered and uncentered versions.We prove that these operators are bounded and continuous from l^1（Z^d）×l^1（Z^d）×…×l^1（Z^d）to BV（Z^d）,where BV（Z^d）is the set of functions of bounded variation defined on Zd.Moreover,two pointwise estimates for the partial derivatives of discrete multisublinear fractional maximal functions are also given.As applications,we present the regularity properties for discrete fractional maximal operator,which are new even in the linear case.
The Lp bounds for the parametric Marcinkiewicz integrals associated to compound mapq pings, which contain many classical surfaces as model examples, are given, where the kernels of our operators are rather rough on the unit sphere as well as in the radial direction. These results substan- tially improve and extend some known results.
In this paper, the authors establish the LV-mapping properties for a class of singular integrals along surfaces in Rn of the form {Ф（lul）u＇ ： u ε ]t^n} as well as the related maimal operators provided that the function Ф satisfies certain oscillatory integral estimates of Van der Corput type, and the integral kernels are given by the radial function h E ε△γ（R＋） for γ 〉 1 and the sphere function ΩεFβ（S^n-1） for someβ 〉 0 which is distinct from HI（Sn-1）.
