Energy measures of harmonic functions on the Sierpiński Gasket

In the development of analysis on fractals, there are three basic concepts: energy, measure and Laplacian. For $SG$, which may be regarded as the “poster child” for the theory, Kigami has developed an elegant construction of a self-similar Laplacian based on a self-similar energy $\mathcal{E}$ and self-similar measure $\mu$ via an analog of integration by parts. This theory is briefly sketched below, and is described in complete detail in the book [8]. This standard Laplacian $\Delta$ is not the only Laplacian on $SG$ that has been widely studied. A competing energy Laplacian $\Delta_{\nu}$, based on the same energy but using the Kusuoka measure $\nu$ in place of $\mu$, has received a lot of attention in recent years([4, 5, 6, 10, 11]). It has two striking advantages over $\Delta$, namely the existence of a Leibniz-type product formula and the fact that the associated heat kernel satisfies Gaussian type estimates. The main drawback so far has been that the Kusuoka measure is not algorithmically transparent. The purpose of this paper is to eliminate this drawback. We will provide algorithms that describe the nature of this measure and more generally the family of energy measures $\nu_{h}$ based on harmonic functions $h$, and the Radon-Nikodym derivatives $\frac{d\nu_{h}}{d\nu}$. Some of the results of this paper, especially Theorem 5.5, may also be derived from [6].

The Sierpiński gasket, or simply SG, is the unique nonempty compact set satisfying

where $F_{i}=\frac{1}{2}(x+q_{i})$ and $\{q_{i}\}_{i=0}^{2}$ are vertices of an equilateral triangle in counter-clockwise direction. $\{q_{i}\}_{i=1}^{2}$ are called the boundary points of SG. If $w=(w_{1},\cdots,w_{n})$ is a finite word, we define the mapping $F_{w}=F_{w_{1}}\circ\cdots\circ F_{w_{n}}$ where $|w|$ is defined to be the length of the word. Define $V_{0}=\{q_{i}\}$, $V_{n}=\bigcup F_{i}\left(V_{n-1}\right)$ and $V_{*}=\bigcup_{n=1}^{\infty}V_{n}$. For the boundary of a cell $C=F_{w}SG$ , denoted by $\partial C$, we mean the three vertices $\{F_{w}(q_{i})\}_{i=0}^{2}$ of the cell.

For functions $u$, $v$ defined on SG, we define the energy on level $m$ by

and energy $\mathcal{E}(u,v)=\lim\limits_{m\to\infty}\mathcal{E}_{m}(u,v)$ provided that $\mathcal{E}(u,u)$ and $\mathcal{E}(v,v)$ are finite. For each cell $F_{w}SG$, we define the energy of $u$, $v$ on this cell by

Observe that the energy is non-negative if $u=v$, but in general it is not. In the case $u=v$, we write $\mathcal{E}(u)$ instead of $\mathcal{E}(u,u)$. We define a function $h$ to be harmonic if it minimizes the energy from level $m$ to level $m+1$, as defined in (1.1). A simple computation shows that this minimization occurs when $\mathcal{E}_{m}(h)=\mathcal{E}_{m+1}(h)$ at which to each cell $F_{w}SG$, $h(F_{w0}(q_{1}))=\frac{2}{5}h(F_{w}(q_{0}))+\frac{2}{5}h(F_{w}(q_{1}))+\frac{1}{5}h(F_{w}(q_{2}))$. Equivalently, a function $h$ on SG is harmonic if the sequence, taking $u=h=v$ in (1.1), is a constant sequence, i.e. constantly $\sum_{i<j}(h(q_{i})-h(q_{j}))^{2}$. Given three values on the boundary, in order to have a constant sequence in (1.1), all the other values are completely determined. Consequently, all the harmonic functions form a three-dimensional space and hence any values on the boundary of a cell can extend to a unique harmonic function on SG. A harmonic function $h$ is said to be symmetric(resp. skew-symmetric) if $h(q_{i})=h(q_{i+1})$(resp. $h(q_{i})=-h(q_{i+1})$) for some $i$. Harmonic functions characterized by $h_{i}(q_{j})=\delta_{ij}$ are three symmetric functions and the notation $h_{i}$ will be used throughout this paper. On the space $\mathcal{H}$ of harmonic functions modulo constants, $(\mathcal{H},\mathcal{E}$) is a two-dimensional inner product space. A harmonic function $h$ is said to be symmetric (resp. skew-symmetric) in $\mathcal{H}$ if there exists a constant $c$ and a symmetric (resp. skew-symmetric) harmonic function $f$ such that $h=c+f$. Explicitly, for a symmetric harmonic function $h$, we mean $h=ah_{i}+c$ for some $a,c\in\mathbb{R}$ and some $h_{i}$; and for a skew-symmetric harmonic function $h$, we mean $a(h_{i}-h_{j})+c$ for some $a>0$ and $i\not=j$. To each harmonic $h$, we write $h^{\perp}$ the harmonic function having the same energy as $h$ and orthogonal to $h$ unless we specify $h^{\perp}$ “orthonormal to $h$” in which case we mean having energy $1$ and orthogonal to $h$.

The standard measure $\mu$ is a self-similar measure characterized by $\mu(F_{w}F_{i}SG)=\frac{1}{3}\mu(F_{w}SG)$ for every cell $C$. The standard Laplacian $\Delta u$ of $u$ is a continuous function satisfying

for every $v$ vanishing at the boundary with $\mathcal{E}(v)<\infty$. Harmonic functions on SG are precisely those whose Laplacian is zero. However, for a function $u$ whenever $\Delta u$ is defined as a function, $\Delta(u^{2})$ is not defined as a function. We can view $\Delta u$ as a measure $m=\Delta ud\mu$ and $\Delta u$ exists as a function if and only if this measure is absolutely continuous with respect to the standard measure. The carré du champs formula given by, for all $f,u,v$ of finite energy, $\int_{SG}fd\nu_{u,v}=\frac{1}{2}\mathcal{E}(fu,v)+\frac{1}{2}\mathcal{E}(u,fv)-\frac{1}{2}\mathcal{E}(f,uv)$ shows that

for all $u,v$ with $\Delta u$ as a function. Thus if $\Delta u$ exists as a function and if we first view $\Delta(u^{2})$ as a measure,

then $\Delta(u^{2})$ would exist as a function if this measure were absolutely continuous with respect to the standard measure $\mu$, but this is almost never the case [2].

Suppose that $u$, $v$ are functions of finite energy. We can define a signed measure $\nu_{u,v}$ by, on each cell,

As usual, if it happens that $u=v$, we write $\nu_{u}$ instead. In this paper, we will restrict the attention to both $u$, $v$ being harmonic. The energy measures of harmonic functions form a three-dimensional space. The Kusuoka measure $\nu$ is defined to be $\nu_{0}+\nu_{1}+\nu_{2}$. An easy computation shows that for any harmonic function $h$, we have $\nu_{h}+\nu_{h^{\perp}}=c\nu$ where $c=\frac{1}{3}\mathcal{E}(h)$. Now, we define the energy Laplacian $\Delta_{\nu}u$ of $u$ by, for every finite energy $v$ vanishing on the boundary,

By Theorem 5.3.1 of [8], every energy measure of harmonic functions is absolutely continuous with respect to the Kusuoka measure $\nu$ and it makes sense to consider the Radon-Nikodym derivatives. For the energy measures of the symmetric harmonic functions, we will denote these by $\nu_{i}$ instead of $\nu_{h_{i}}$. $\{\nu_{i}\}$ forms a basis of the space of all energy measures of harmonic functions. Since $h_{0}+h_{1}+h_{2}=1$, we have $\nu_{h_{0},h_{j}}+\nu_{h_{1},h_{j}}+\nu_{h_{2},h_{j}}=\nu_{1,h_{j}}=0$ for $j=0,1,2$. Since $\nu_{f,g}=\nu_{g,f}$ for all $f$, $g$, this gives us three equations that we solve to obtain

We will keep using the notation $\nu_{f,g}$ for any energy measure of harmonic functions without mentioning $f$ and $g$.

By [1], we have that there exist matrices $E_{i}$ such that for every cell $C$,

The $E_{i}$’s are all diagonalizable, so it is easy to calculate $E_{i}^{m}$. Explicitly, after diagonalization, we have

It is easy to calculate that the limits of quotients which are used to compute derivatives are as follows:

where

For each infinite word $w$, let $w_{m}$ be the word consisting of the first $m$ letters of $w$; then $\lim\limits_{m\to\infty}E_{w_{m}}=0$. It follows that every energy measure is continuous.

It is natural to ask whether the energy Laplacian behaves in a similar manner to the standard Laplacian; in Section 2, we express the “self-similarity” of the energy Laplacian. In Section 3, we will study the decay rates of energy measure from a cell to its subcells. Through studying this, we give sharp bounds for the Radon-Nikodym derivatives for each $\nu_{h}$. Being in general a signed measure, it is natural to ask when it is a (positive) measure; in Section 4, we will characterize all the positive energy measures. It is well-known that the derivative is not continuous; nevertheless, in Section 5, we show that a certain restriction of the derivative is continuous. In Section 6, we will express, on each cell, the average value of derivatives by a weighted average of values on vertices of the cell. How the energy distributes is mysterious; in Section 7, we will provide some graphs and hypothesis about this question.

In this section we discuss the self-similarity identities for energy measures and the energy Laplacian. We will see that the Radon-Nikodym derivatives $R_{i}=\frac{d\nu_{i}}{d\nu}$ play a crucial role in these identities. Individually, none of the $\nu_{i}$ is self-similar; nevertheless, together they form a self-similar family, in the sense of Mauldin-Williams [7]. We begin with considering the symmetric function $h_{0}$. We have the following equalities,

Next we establish the relation of $\nu_{i}$’s from cells to subcells.

We now give the “self-similarity” of the energy Laplacian $\Delta_{\nu}$.

To compute $\Delta_{\nu}u$ on the cell $F_{w}SG$, zoom in by looking at $u\circ F_{w}$ as a function on $SG$, compute its Laplacian $\Delta_{\nu}(u\circ F_{w})$ and multiply by $\frac{1}{Q_{w}}$,

and then zoom out via $F_{w}^{-1}$

By the definition of energy measures, we see that every energy measure is continuous; by the self similarity of SG, we see that it makes sense to consider the measure from cells to subcells. It is natural to study how the measure varies for a sequence of cells converging to a point.

We have proven the decay rates of the measures of a sequence of cells converging to a point which is not equivalent to a skew-symmetric cell. Surprisingly, in the case which the cell is equivalent to skew-symmetric at that point, we have the first condition for the Radon-Nikodym derivatives to be $0$.