  Let $T$ be an operator and suppose that there exists a positive constant $C$
such that $$\left(\int_I|Tf(x)|^q\, dx\right)^{1/q}\leq C\left(\int_I|f(x)|^q\,
dx\right)^{1/q}$$ for every $q$ which is near enough to $1$ and for every
interval $I$ in $\mathbb{R}$ and $f\in L^{\infty}(\mathbb{R})$. Then we show
that $T$ maps $uL^{\infty}$ to ${\rm{BMO}}_u$.
