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Riku
Riku
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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.

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

How can we prove that the Hausdorff dimension of the essential graph of $u$ 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.

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.

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Riku
Riku
  • 839
  • 6
  • 17

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

How can we prove that the Hausdorff dimension of the graphessential graph of $u$ 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.

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

How can we prove that the Hausdorff dimension of the graph of $u$ 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.

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

How can we prove that the Hausdorff dimension of the essential graph of $u$ 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.

Source Link
Riku
Riku
  • 839
  • 6
  • 17

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.

How can we prove that the Hausdorff dimension of the graph of $u$ 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.