Category Archives: 2021 Math

Helmholtz-Weyl decomposition

Today, one of my students, who is a cadet in Republic of Korea Air Force Academy, asked a proof of Helmholtz-Weyl decomposition for general vector field, which was considered in the electromagnetic class. In the mathematical fluid dynamics, we need to decompose a vector field into of the form     for some scalar function… Read More »

Stein’s spherical maximal function

1. Introduction Maximal function plays a central role in several places in analysis. As an example, we can prove the celebrated Lebesgue differentiation theorem by using Hardy-Littlewood maximal function. Another example for application of maximal function is the nontangential behavior of Poisson integral defined on the half-plane. From these examples, maximal function helps us to… Read More »

1.1. Newtonian Potentials

The goal of this note is to study solvability of second-order elliptic and parabolic equations in Hölder spaces. The prototype of elliptic equation is the Poisson equation     To understand the property of solution , one of the easiest ways is to use Newtonian potential. In Section 1.1, we derive Newtonian potential and study… Read More »