Let $D$ be a Jordan domain. We assume that $D$ is a Hölder domain. Namely, there exists a Hölder continuous map $f$ such that $D=f(\mathbb{D})$, where $\mathbb{D}$ is an open unit disk.

It is known that quasidisks are Hölder domains. Hence, the boundaries of Hölder domains can be very wild. For example, the Koch snowflake is a Hölder domain.

My question

How strange is $D$?

I would like to know an example of Jordan domain which is Hölder and does not possesses both volume doubling property and the uniform poincare inequality. These two notions are fundamental in geometric analysis.

Let $D$ be a Jordan domain. We assume that $D$ is a Hölder domain. Namely, there exists a Hölder continuous Riemann map $f$ such that $D=f(\mathbb{D})$, where $\mathbb{D}$ is an open unit disk.

It is known that quasidisks are Hölder domains. Hence, the boundaries of Hölder domains can be very wild. For example, the Koch snowflake is a Hölder domain.

My question

How strange is $D$?

I would like to know an example of Jordan domain which is Hölder and does not possesses both volume doubling property and the uniform poincare inequality. These two notions are fundamental in geometric analysis.

Let $D$ be a Jordan domain. We assume that $D$ is a Hölder domain. Namely, there exists a Hölder older continuous map $f$ such that $D=f(\mathbb{D})$, where $\mathbb{D}$ is an open unit disk.

It is known that quasidisks are Hölder domains. Hence, the boundaries of Hölder domains can be very wild. For example, the Koch snowflake is a Hölder domain.

My question

How strange is $D$?

I would like to know an example of Jordan domain which is Hölder and does not possesses both volume doubling property and the uniform poincare inequality. These two notions are fundamental in geometric analysis.

Let $D$ be a Jordan domain. We assume that $D$ is a Hölder domain. Namely, there exists a Hölder continuous map $f$ such that $D=f(\mathbb{D})$, where $\mathbb{D}$ is an open unit disk.

It is known that quasidisks are Hölder domains. Hence, the boundaries of Hölder domains can be very wild. For example, the Koch snowflake is a Hölder domain.

My question

How strange is $D$?

I would like to know an example of Jordan domain which is Hölder and does not possesses both volume doubling property and the uniform poincare inequality. These two notions are fundamental in geometric analysis.

Let $D$ be a Jordan domain. We assume that $D$ is a Holder domain. Namely, there exists a Holder continuous map $f$ such that $D=f(\mathbb{D})$, where $\mathbb{D}$ is an open unit disk.

It is known that quasidisks are Holder domains. Hence, the boundaries of Holder domains can be very wild. For example, the Koch snowflake is a Holder domain.

My question

How strange is $D$?

I would like to know an example of Jordan domain which is Holder and does not possesses both volume doubling property and the uniform poincare inequality. These two notions are fundamental in geometric analysis.

Let $D$ be a Jordan domain. We assume that $D$ is a Hölder domain. Namely, there exists a Hölder older continuous map $f$ such that $D=f(\mathbb{D})$, where $\mathbb{D}$ is an open unit disk.

It is known that quasidisks are Hölder domains. Hence, the boundaries of Hölder domains can be very wild. For example, the Koch snowflake is a Hölder domain.

My question

How strange is $D$?

I would like to know an example of Jordan domain which is Hölder and does not possesses both volume doubling property and the uniform poincare inequality. These two notions are fundamental in geometric analysis.