In this paper, we introduce a new class of weights Ap （Rn） which retains many fine properties of the classical Muchenhoupt weights Ap （Rn）. While Ap （Rn） is too big a class to obtain the weighted norm inequalities for rough singular integrals and Marcinkiewicz integrals, our new class Ap （Rn） adapts well to these rough operators. As applications, we improve some known weighted estimates.
For a compact Riemannian manifold NRK without boundary, we establish the existence of strong solutions to the heat flow for harmonic maps from Rn to N, and the regularizing rate estimate of the strong solutions. Moreover, we obtain the analyticity in spatial variables of the solutions. The uniqueness of the mild solutions in C([0,T]; W1,n) is also considered in this paper.
Let T^2 be a flat two-dimensional torus with fundamental cell domain [-1/2,1/2]×[-1/2,1/2],h（x） a positive smooth function satisfying the symmetric property （8） on T^2.In this paper we give some sufficient condition under which the mean field equation △u = 16π - 16πe^u, has a smooth solution.
In this paper, the LP（Rn）-boundedness of the commutators generalized by BMO（Rn） function and the singular integral operator T with rough kernel Ω∈ L log＋ L（Sn-1）is proved by using the Bony＇s formula for the paraproduct of two functions.
