If you are involved in the world of mathematics and science at all then you have more than likely heard about the Langlands Program. Langlands aims to connect algebra and representation theory to harmonic analysis and classical analysis. If this means nothing to you then don't worry. I was in the same boat. Today I want to look at the harmonic part of harmonic analysis. The neat part about harmonic functions is that we need only a basic calculus level of knowledge to define, and understand the basics of harmonic functions. I will however, be a little verbose at times.

We first need to define the Laplacian operator. The Laplacian acts on a $C^2$ (twice differentiable) function by summing all of the homogeneous second derivatives, explicitly, we have $$ \Delta f = \sum_{j=1}^n \left( \frac{\partial f}{\partial x_j}\right)^2 $$ That notation is a mess but it really is just the sum of homogeneous second derivatives of $f$. Further, we say a function $f$ is harmonic if $\Delta f = 0$. For a concrete example, lets take $f(x,y)=x^3-3xy^2$. Now lets compute the derivatives $$ \begin{align*} f_x &= 3x^2 - 3y^2\\ f_{xx} &= 6x\\ f_y &= -6xy\\ f_{yy} &= 6x \end{align*} $$ Now we just sum $f_{xx}+f_{yy} = 6x - 6x = 0$. So $f$ is in fact harmonic. This is pretty straightforward, but its the properties of harmonic functions that we are most interested in. I could just give a laundry list of things but a few properties I would like to give some motivation behind.

A huge amount of the convenient properties of harmonic functions were discovered or motivated by the study of PDE's. So I want to start with one of the most basic properties of harmonic functions, one that is used extensively in basic PDE thoery. We call this first property the Mean Value Property.

Def: A harmonic function $f \in C^2(\mathbb{R})$ satisfies the mean value property: $$ f(x) = \int_{\partial B(x,r)} f(y) dS(y) = \int_{B(x,r)} f(y)dy $$ where $B(x,r)$ is the ball centered at $x$ with radius $r$.

The integral here is technically the average value integral and the $dS(y)$ indicates a surface measure. In not so many words, the average value of a harmonic function around the boundary of the ball $B(x,r)$ is the same as the average value everywhere inside the ball. The proof of this statement isn't too bad and just requires some vector calculus. The results of this may not be obvious but it is a very neat property that seems almost too convenient. These kinds of theorems are one of my favorite parts about math.