Is $e^{|x|^2}$ BMO?

Question

In $\mathbb{R}^{n}$, is there a pratical way to check whether a function is BMO?, for example, are the functions $$x\rightarrow e^{|x|^2},\quad x\rightarrow (|x|^{4}+1)e^{-|x|^2}$$ BMO?

Thanks.

Answer 1

You can manually check that the first function is not in BMO and that the second is. The second is in BMO since, BMO contains $L^\infty(\mathbb{R}^n)$. For the first function try taking the average over a ball $B_N$ or radius $1$ and center at $|x| = N$. The average will be of the order of magnitude of $e^{(N - 1)^2}$, but over the section of the ball given by $B_N \cap B(0,N)^c$ the oscillation is larger that $e^{N^2} - e^{(N - 1)^2}$. Using that $B_N \cap B(0,N)^c$ is larger than half the ball $B_N$ you can conclude.

Alternatively, there is a very useful characterization of BMO functions given by the John-Niremberg Theorem. It asserts that BMO functions as locally exp-integrable functions.