Let
$0<\unicode[STIX]{x1D6FC}<n,1\leq p<q<\infty$
with
$1/p-1/q=\unicode[STIX]{x1D6FC}/n$
,
$\unicode[STIX]{x1D714}\in A_{p,q}$
,
$\unicode[STIX]{x1D708}\in A_{\infty }$
and let
$f$
be a locally integrable function. In this paper, it is proved that
$f$
is in bounded mean oscillation
$\mathit{BMO}$
space if and only if
$$\begin{eqnarray}\sup _{B}\frac{|B|^{\unicode[STIX]{x1D6FC}/n}}{\unicode[STIX]{x1D714}^{p}(B)^{1/p}}\bigg(\int _{B}|f(x)-f_{\unicode[STIX]{x1D708},B}|^{q}\unicode[STIX]{x1D714}(x)^{q}\,dx\bigg)^{1/q}<\infty ,\end{eqnarray}$$
where
$\unicode[STIX]{x1D714}^{p}(B)=\int _{B}\unicode[STIX]{x1D714}(x)^{p}\,dx$
and
$f_{\unicode[STIX]{x1D708},B}=(1/\unicode[STIX]{x1D708}(B))\int _{B}f(y)\unicode[STIX]{x1D708}(y)\,dy$
. We also show that
$f$
belongs to Lipschitz space
$Lip_{\unicode[STIX]{x1D6FC}}$
if and only if
$$\begin{eqnarray}\sup _{B}\frac{1}{\unicode[STIX]{x1D714}^{p}(B)^{1/p}}\bigg(\int _{B}|f(x)-f_{\unicode[STIX]{x1D708},B}|^{q}\unicode[STIX]{x1D714}(x)^{q}\,dx\bigg)^{1/q}<\infty .\end{eqnarray}$$
As applications, we characterize these spaces by the boundedness of commutators of some operators on weighted Lebesgue spaces.