  A function space, $L^{\theta,\infty)}(\Omega)$, $0 \leq \theta <\infty$, is
defined. It is proved that $L^{\theta,\infty)}(\Omega)$ is a Banach space which
is a generalization of exponential class. An alternative definition of
$L^{\theta,\infty)}(\Omega)$ space is given. As an application, we obtain weak
monotonicity property for very weak solutions of $\mathcal{A}$-harmonic
equation with variable coefficients under some suitable conditions related to
$L^{\theta,\infty)}(\Omega)$, which provides a generalization of a known result
due to Moscariello. A weighted space $L^{\theta,\infty)}_w(\Omega)$) is also
defined, and the boundedness for the Hardy-Littlewood maximal operator $M_w$
and a Calder\'{o}n-Zygmund operator $T$ with respect to
$L^{\theta,\infty)}_w(\Omega)$ are obtained.
