We prove that the fundamental semi-group e^it（m^2│△│）^1/2 （m≠ 0） of the Klein-Gordon equation is bounded on the modulation space M^8p,q（R^n） for all 0 〈 p, q ≤∞ and s ∈ R. Similarly, we prove that the wave semi-group e^it│△│^1/2 is bounded on the Hardy type modulation spaces μ^εp,q（R^n） for all 0 〈 p, q ≤ ∞, and s ∈R. All the bounds have an asymptotic factor t^n│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（m^2I＋│△│）1/2 are obtained. Finally we discuss the optimum of the factor t^n│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 Hausdorff operators in the high-dimensional product spaces. We obtain their power weighted boundedness from Lp to Lq and characterize the necessary and sufficient conditions for their boundedness on the power weighted Lp spaces. Moreover, in the case p = q, we obtain the sharp bound constants.
