Homogeneous function in bounded mean oscillation BMO($\mathbb R^n$) space

Question

Let me recall some notations: The mean oscillation of a locally integrable function $u$ (i.e. a function belonging to $ L^1_{\textrm{loc}}(\mathbb{R}^n))$ over a cube Q in $\mathbb R^n$ (which has sides parallel to axis) is defined as the following integral: $\frac{1}{|Q|}\int_{Q}|u(y)-u_Q|\,\mathrm{d}y$,

where $|Q|$ is the volume of $Q$, i.e. its Lebesgue measure, $u_Q$ is the average value of $u$ on the cube $Q$, i.e. $u_Q=\frac{1}{|Q|}\int_{Q} u(y)\,\mathrm{d}y$.

A BMO function is any function u belonging to $L^1_{\textrm{loc}}(\mathbb{R}^n)$ whose mean oscillation has a finite supremume over the set of all cubes $Q$ contained in $\mathbb R^n$.

I could find many examples for functions in BMO. But I could not find a function $u:\mathbb R^n\to\mathbb R$ which is not constant, so that $u$ in BMO and $u(tx)=u(x)$ for almost everywhere $t\in[0;1]$, and for every $x\in \mathbb R^n$.

I also want to find such function $u$ in BMO so that $u(tx)=u(t)$ for almost every $t\neq0$, and every $x$

So my question is that: does exits such function $u$, and could you give me any example.

Answer 1

There are many such functions.

Consider: $$f(x) = [x_i \geqslant 0 \text{ for all $i = 1, \ldots, n$}].$$

Or (thanks robjohn) for all $i = 1, \ldots n$ the functions $$g_i(x) = \frac{x_i}{|x_i|}.$$

For another example for your new function. Say you are working in $\mathbf R^2$. Then define $f(x, y) = 1$ everywhere except on the line $x = y$. There we let $f = 0$.

So, $f(tx, ty)$ is $1$ as long as $tx \neq ty$ or $x \neq y$. It is $0$ if $tx = ty$. That is if $x = y$. So, your function is not constant.

Of course, this is only a measure $0$ set where the function is different, however, you can swipe the line from $x = y$ to $x = -y$ and get both in positive measure.

To make this explicit consider the set in $\mathbf R^2$, $A := \{(x, y) \mid |x| \leq |y|\}$, that is this region:

Now consider the function $f(x, y) := [(x, y) \in A]$. This notation for the indicator function is the Iverson Bracket.

Answer 2

Let $S_{n-1}=\{x\in\mathbb{R}^n:|x|=1\}$ be the unit sphere in $\mathbb{R}^n$ and $\phi\colon S_{n-1}\to\mathbb{R}$ a bounded even function, that is $$ \sup_{\omega\in S_{n-1}}|\phi(\omega)|=\|\phi\|_\infty<\infty,\quad\phi(-\omega)=\phi(\omega)\quad\forall\omega\in S_{n-1}. $$ Define $$ u(x)=\begin{cases} \phi(x/|x|) & x\ne0,\\ 0 & x=0. \end{cases} $$ Then $$ u\in L^\infty(\mathbb{R}^n)\implies u\in BMO(\mathbb{R}^n)\text{ and }u(t\,x)=u(x)\quad\forall t\ne0,\forall x\in\mathbb{R}^n. $$