In this paper we shall give a weighted version of Igari's estimate on the Kakeya maximal operator with a special base.

In this paper, one uses a discrete representation to characterize two singular integral kernel spaces with two others simple Banach, and to improve a Meyer's result: David and Journé's $T(1)$ theorem under a weaker kernel condition.

We construct the eigenspinors of the Dirac Operator $D_3$ on $S^3$ from a representation theoretical point of view and give a representation of $Spin(4)$ on them explicitly. These eigenspinors are extended to zero mode spinors of the Dirac operator $D_{4}^{\pm}$ on upper or lower hemisphere of $S^4$.

We determine homotopically energy-minimizing harmonic maps of tori into the 3-dimensional real projective space $\mathbf{RP}^3$ of constant sectional curvature 1.