Abstract
In this work we mainly prove the following interior gradient estimates in weighted Lorentz spaces
g
−
1
[
M
1
(
μ
)
]
∈
L
w
,
l
o
c
q
,
r
(
Ω
)
⟹
|
D
u
|
∈
L
w
,
l
o
c
q
,
r
(
Ω
)
,
where
g
(
t
)
=
t
a
(
t
)
for
t
≥
0
and
M
1
(
μ
)
(
x
)
is the first-order fractional maximal function
M
1
(
μ
)
(
x
)
:=
sup
r
>
0
r
|
μ
|
(
B
r
(
x
)
)
|
B
r
(
x
)
|
,
for a class of non-homogeneous divergence quasilinear elliptic equations with measure data in the subquadratic case
−
div
[
a
(
(
A
D
u
⋅
D
u
)
1
2
)
A
D
u
]
=
μ
in
Ω
,
whose model cases are the classical elliptic
p
-Laplacian equation with measure data
−
div
(
|
D
u
|
p
−
2
D
u
)
=
μ
for
1
<
p
<
2
and the elliptic
p
-Laplacian equation with the logarithmic term and measure data
−
div
(
|
D
u
|
p
−
2
D
u
)
=
μ
for
1
<
p
<
2
It deserves to be specially noted that the subquadratic case is a little different from the superquadratic case since as an example, the modulus of ellipticity in the above-mentioned special cases tends to infinity when
|
D
u
|
→
0
for
1
<
p
<
2
.