# Hausdorff dimension of the graph of a BV function (in 1 dimensional setting)

Let $$u: \Omega\subset \mathbb{R} \to \mathbb{R}$$ be a function of bounded variation.

Question 1.

How can we prove that the Hausdorff dimension of the essential graph of $$u$$ equal to $$1$$?

Question 2.

Is Question 1 equivalent to asking the following?

How can we prove that there exists a representative $$\tilde u$$ of $$u$$ such that the Hausdorff dimension of $$\tilde u$$ is equal to 1.

Note. I've asked a more general question Hausdorff dimension of the graph of a BV function, which has received a very nice (partial) answer, however, I'd like to see a simpler proof in the 1-dimensional case.

• You can cover the graph by roughly 1/r balls of radius r. – Anthony Quas Apr 10 '19 at 18:03
• @Riku Probably it makes sense to specify what you mean by graph, because a function can be zero a.e. and still its graph may have dimension strictly greater than 1, see my answer here mathoverflow.net/questions/327331 – Skeeve Apr 10 '19 at 18:11
• @AnthonyQuas What do you mean? – Riku Apr 11 '19 at 11:16
• @Skeeve You're right. What would the appropriate definition of graph be? – Riku Apr 11 '19 at 11:22
• @Riku there is a notion of essential range. Maybe it would be appropriate to define similarly essential graph, i.e. to say that essential graph of $u$ is the essential range of the function $x\mapsto(x,u(x))$. – Skeeve Apr 11 '19 at 13:51

I will assume $$\Omega$$ is an interval $$[a,b]$$, say. I assume this was intended as part of your question.
Let $$\text{Var}f=M$$. Let $$r>0$$. The function has at most countably many discontinuities, which are necessarily of jump type. The magnitude of the discontinuities sums to at most $$M$$. In particular, there are at most $$M/r$$ points at which the function jumps by $$r$$ or more. Let these be $$z_1,\ldots, z_k$$. Let $$z_0=a$$ and $$z_1=b$$. Then on each interval $$(z_i,z_{i+1})$$, there exist $$z_i:=x^i_0 such that $$r<\text{Var}_{[x^i_j,x^i_{j+1}]}(f)<2r$$ for each $$i,j$$, except that no lower bound is imposed for $$j=n_i-1$$ (that is the variation is split into chunks of size roughly $$r$$). Now the part of the graph lying over $$[x_j^i,x_{j+1}^i]$$ can be covered by approximately $$\lceil (x^i_{j+1}-x^i_j)/r\rceil$$ balls of radius $$r$$, so that approximately $$N_i=(z_{i+1}-z_i)/r+n_i+1$$ balls are needed to cover the section of the graph lying over $$[z_i,z_{i+1}]$$. Notice that $$n_i\le \text{Var}_{(z_i,z_{i+1})}f/r+1$$, so that summing the $$N_i$$'s we obtain a cover with approximate $$(b-a)/r+\text{Var}_{[a,b]}f/r+k$$ balls of size $$r$$. Since $$k\le M/r$$, we see the number of balls is bounded by approximately $$(b-a)/r+2M/r$$. Hence the Hausdorff dimension is at most 1.
• Thank you. 1) How do you prove that the function has at most countably many discontinuities, which are necessarily of jump type? 2) Why are there at most M/r points at which the function jumps by r or more? 3) You conclude that $\dim graph(f) \le 1$. How do you show that it is also $\ge 1$? – Riku Apr 11 '19 at 21:31