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

Abstract

Let $T$ be an operator and suppose that there exists a positive constant $C$ such that $${\left({\int }_I|Tf(x){|}^{q}dx\right)}^{1/q}\le C{\left({\int }_I|f(x){|}^{q}dx\right)}^{1/q}$$ 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 $u{L}^{\infty }$ to ${\mathrm{BMO}}_u$.