  This paper continues the development of regularity results for quasilinear
measure data problems \begin{align*} \begin{cases} -\mathrm{div}(A(x,\nabla u))
&= \mu \quad \text{in} \ \ \Omega, \\ \quad \quad \qquad u &=0 \quad \text{on}
\ \ \partial \Omega, \end{cases} \end{align*} in Lorentz and Lorentz-Morrey
spaces, where $\Omega \subset \mathbb{R}^n$ ($n \ge 2$), $\mu$ is a finite
Radon measure on $\Omega$, and $A$ is a monotone Carath\'eodory vector valued
operator acting between $W^{1,p}_0(\Omega)$ and its dual $W^{-1,p'}(\Omega)$.
It emphasizes that this paper studies the `very singular' case $1<p \le
\frac{3n-2}{2n-1}$ and the problem is considered under the weak assumption,
where the $p$-capacity uniform thickness condition is imposed on the complement
of domain $\Omega$. There are two main results obtained in our study pertaining
to the global gradient estimates of solutions in Lorentz and Lorentz-Morrey
spaces involving the use of maximal and fractional maximal operators. The idea
for writing this working paper comes directly from the recent results by others
in the same research topic, where global estimates for gradient of solutions
for the `very singular' case still remains a challenge, specifically related to
Lorentz and Lorentz-Morrey spaces.
