Global solutions of Navier-Stokes equations with large $L^2$ norms in a new function space

First we prove certain pointwise bounds for the fundamental solutions of the perturbed linearized Navier-Stokes equation (Theorem 1.1). Next, utilizing a new framework with very little $L^p$ theory or Fourier analysis, we prove existence of global classical solutions for the full Navier-Stokes equation when the initial value has a small norm in a new function class of Kato type (Theorem 1.2). The smallness in this function class does not require smallness in $L^2$ norm. Furthermore we prove that a Leray-Hopf solution is regular if it lies in this class, which allows much more singular functions then before (Corollary 1). For instance this includes the well-known result in [25]. A further regularity condition (form boundedness) was given in Section 5. We also give a different proof about the $L^2$ decay of Leray-Hopf solutions and prove pointwise decay of solutions for the three-dimensional Navier-Stokes equations (Corollary 2, Theorem 1.2). Whether such a method exists was asked in a survey paper [2].