Real Interpolation, Lorentz Spaces and the Navier-stokes Equation

In this note we give a brief sketch of real interpolations, the Lorentz spaces and their applications to the NavierStokes equations. Introduction. This note is a brief sketch of real interpolation and the Lorentz spaces, and its application to the Navier-Stokes equations. There are two methods of interpolation: complex method and real method. Both of them survive because each has its advantages to the other. Namely, the advantage of complex interpolation is the following: (i) Multilinear operators can be treated (ii) The operator may depend holomorphically with respect to the interpolation parameter. On the other hand, the advantage of real interpolation is the following: (i) The function spaces need not be normed. (ii) The operators need not be linear. (iii) One can “improve” the function space in the course of interpolation. Indeed, in the proof of the Mikhlin-Hörmander multiplier theorem we use properties (i) and (iii) of real interpolation, and in the proof of the boundedness of the Hardy-Littlewood maximal function, we use all of them. Section 1 to Section 5 are devoted to the sketch of real interpolation and the Lorentz spaces. Generally I followed [1], but I avoided to describe the whole theory and limited ourselves to the necessary part of the theory, and some proofs are modified accordingly. Beside, all of the aforementioned properties of real interpolation are employed in the application to the Navier-Stokes equations, and hence I emphasized such properties. Section 6 is devoted to the application of the theory above to the Navier-Stokes equations. There are two methods to the proof of the key inequality (6.7) in Lemma 6.2. One is the cut-off method developed by Prof. Shibata et al. This method can be applied to more general 1