This paper proves that the parametric Marcinkiewicz integral μΩρ(f)(x)=(∫∞0∣1/tρ∫∣x-y≤tΩ(x-y)∣/x-y∣n-ρf(y)dy∣2 dt/t)1/2 is an operator of type(H p,∞,Lp,∞)(0 < p< 1),with Ω being the homogeneous zero on R n and satisfying L1-Dini condition.

For the commutators of Multilinear Marcinkiewicz integral μΩ,A(f)(x),the endpoint estimates are obtained when function b(x)∈BMO,Ω satisfies the Dini type conditions.

For the commutators μ_(Ω,b)(f) of Marcinkiewicz integrals with Ω satisfying the Dini type condition,it is shown that the following inequality holdsu(｛x∈R~n:μ_(Ω,b)(f)(x)>λ｝)≤CCt~p∫_(R~n)|f|~pvdx,where(u,v) satisfy(1|Q|∫_Qu~rdx)~(1/rp)‖v~(-1/p)‖_(c,Q)≤C<∞.

Let {p_j}~m_(j=1) be positive continuous real-valued function on R~d,and {logp_j}~m_(j=1) satisfy the Dini condition.

Boundary regularity results for viscosity solutions of Poisson s equation and fully nonlinear uniformly elliptic equations under the Dini condition are obtained by use of Caffarelli s perturbation method, which give a simple proof of Theorem 4.

Estimation and application of kernel of singular integral operators with Dini-type condition;

It is proved that the Marcinkiewicz integralμΩis an operator of type (Hp,∞, Lp,∞) for 0 < p < I, hereΩis homogeneous of degree zero on Rn and satisfies a kind of Dini-type condition.