A Sufficient Condition For An Operator To Map $u{L}^{\mathrm{\infty }}$ to ${\mathrm{B}\mathrm{M}\mathrm{O}}_u$

Let $T$ be an operator and suppose that there exists a positive constant $C$ such that $${({\int }_I\mathrm{\mid }Tf(x){\mathrm{\mid }}^{q}dx)}^{1\mathrm{/}q}\le C{({\int }_I\mathrm{\mid }f(x){\mathrm{\mid }}^{q}dx)}^{1\mathrm{/}q}$$ for every $q$ which is near enough to $1$ and for every interval $I$ in $\mathbb{R}$ and $f\in L^{\mathrm{\infty }}(\mathbb{R})$. Then we show that $T$ maps $u{L}^{\mathrm{\infty }}$ to ${\mathrm{B}\mathrm{M}\mathrm{O}}_u$.