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Estimates of the first partial derivatives of (α,β)-harmonic functions on the unit disc
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Journal Article
Estimates of the first partial derivatives of (α,β)-harmonic functions on the unit disc
2025
Overview
Suppose
α
,
β
∈
R
∖
Z
−
such that
α
+
β
>
−
1
and
1
≤
p
≤
∞
. Let
u
=
P
α
,
β
[
f
]
be an
(
α
,
β
)
-harmonic function on
D
, the unit disc of
C
, with the boundary
f
being absolutely continuous and
f
˙
∈
L
p
(
0
,
2
π
)
, where
f
˙
(
e
i
θ
)
:
=
d
d
θ
f
(
e
i
θ
)
. In this paper, we investigate the membership of the partial derivatives
∂
z
u
and
∂
z
‾
u
in the space
H
G
p
(
D
)
, the generalized Hardy space. We prove, if
α
+
β
>
0
, then both
∂
z
u
and
∂
z
‾
u
are in
H
G
p
(
D
)
. For
α
+
β
<
0
, we show if
∂
z
u
or
∂
z
‾
u
∈
H
G
1
(
D
)
then
u
=
0
or
u
is a polyharmonic function.
Publisher
Springer International Publishing,Springer Nature B.V,SpringerOpen
