Abstract
AbstractGiven a regular measure $$\eta \in M([0,1))$$
η
∈
M
(
[
0
,
1
)
)
and an analytic function $$g\in {\mathcal H}(\mathbb {D})$$
g
∈
H
(
D
)
, we define $$H(\eta ,g)(z)=\int _0^1g(tz)d\eta (t)$$
H
(
η
,
g
)
(
z
)
=
∫
0
1
g
(
t
z
)
d
η
(
t
)
and study its boundedness from $$X\times Y$$
X
×
Y
into Z where $$X\subset M([0,1))$$
X
⊂
M
(
[
0
,
1
)
)
and $$Y,Z\subset {\mathcal H}(\mathbb {D})$$
Y
,
Z
⊂
H
(
D
)
are the Hardy spaces. We shall analyze the case $$X=L^p([0,1))$$
X
=
L
p
(
[
0
,
1
)
)
and characterize the functions $$g\in {\mathcal H}(\mathbb {D})$$
g
∈
H
(
D
)
such that $$H_g$$
H
g
maps $$L^p([0,1))$$
L
p
(
[
0
,
1
)
)
into $$H^p(\mathbb {D})$$
H
p
(
D
)
where $$H_g(\eta )=H(\eta , g)$$
H
g
(
η
)
=
H
(
η
,
g
)
.
Publisher
Springer Science and Business Media LLC
Subject
Applied Mathematics,General Mathematics,Statistics and Probability