In this paper, some endpoint estimates for the generalized multilinear fractional integrals I_(α,m) on the non-homogeneous metric spaces are established.
We prove that the fundamental semi-group eit(m 2I+|Δ|)1/2(m = 0) of the Klein-Gordon equation is bounded on the modulation space M ps,q(Rn) for all 0 < p,q ∞ and s ∈ R.Similarly,we prove that the wave semi-group eit|Δ|1/2 is bounded on the Hardy type modulation spaces μsp,q(Rn) for all 0 < p,q ∞,and s ∈ R.All the bounds have an asymptotic factor tn|1/p 1/2| as t goes to the infinity.These results extend some known results for the case of p 1.Also,some applications for the Cauchy problems related to the semi-group eit(m2I+|Δ|)1/2 are obtained.Finally we discuss the optimum of the factor tn|1/p 1/2| and raise some unsolved problems.
In the present paper, we prove the existence of global solutions for the Navier-Stokes equations in R^n when the initial velocity belongs to the weighted weak Lorentz space A^n,∞ （u） with a sufficiently small norm under certain restriction on the weight u. At the same time, self-similar solutions are induced if the initial velocity is, besides, a homogeneous function of degree -1. Also the uniqueness is discussed.
In this paper,we study certain Hausdorf operators in the high-dimensional product spaces.We obtain their power weighted boundedness from Lpto Lqand characterize the necessary and sufcient conditions for their boundedness on the power weighted Lpspaces.Moreover,in the case p=q,we obtain the sharp bound constants.
