In this paper we discuss two different topics concerning
A
\mathcal {A}
- harmonic functions. These are weak solutions of the partial differential equation
div
(
A
(
x
,
∇
u
)
)
=
0
,
\begin{equation*}\text {div}(\mathcal {A}(x,\nabla u))=0,\end{equation*}
where
α
(
x
)
|
ξ
|
p
−
1
≤
⟨
A
(
x
,
ξ
)
,
ξ
⟩
≤
β
(
x
)
|
ξ
|
p
−
1
\alpha (x)|\xi |^{p-1}\le \langle \mathcal {A}(x,\xi ),\xi \rangle \le \beta (x) |\xi |^{p-1}
for some fixed
p
∈
(
1
,
∞
)
p\in (1,\infty )
, the function
β
\beta
is bounded and
α
(
x
)
>
0
\alpha (x)>0
for a.e.
x
x
. First, we present a new approach to the regularity of
A
\mathcal {A}
-harmonic functions for
p
>
n
−
1
p>n-1
. Secondly, we establish results on the existence of nontangential limits for
A
\mathcal {A}
-harmonic functions in the Sobolev space
W
1
,
q
(
B
)
W^{1,q}(\mathbb {B})
, for some
q
>
1
q>1
, where
B
\mathbb {B}
is the unit ball in
R
n
\mathbb {R}^n
. Here
q
q
is allowed to be different from
p
p
.