|
Abstract:
|
In this thesis we study sharp estimates of gradients and
operator norm estimates in harmonic function theory.
First, we obtain Schwarz-type inequalities for holomorphic mappings from the
unit ball
B
n
to the unit ball
B
m
,
and then analoguous inequalities for holomorphic
functions on the disk
D
without zeros and pluriharmonic functions from the unit
ball
B
n
to
(
−
1
,
1)
.
These extend results from [
32
] and [
18
]. Also, we give a new proof
of the fact that positive harmonic function in the upper-half plane is a contraction
with resprect to hyperbolic metrics on both
H
and
R
+
([
47
]). Besides that, in the
second chapter, we construct the examples to show that the analoguous does not
hold for the higher-dimensional upper-half spaces. All mentioned results are from
the authors’ paper [55].
In the third chapter we intend to calculate the exact seminorm of the weighted
Berezin transform considered as an operator from
L
∞
(
B
n
)
to the ”smooth” Bloch
space ([57]).
The fourth chapter contains results concerning Bergman projection. We solve
the problem posed by Kalaj and Marković in [
28
] on determining the exact seminorm
of the Bergman projections from
L
∞
(
B
n
)
to the
B
(
B
n
)
.
The crucial obstacle is
the fact that
B
(
B
n
)
is equipped with
M−
invariant gradient seminorm. Also, we
provide the sharp gradient estimates of the Bergman projection of an
L
p
function
in the unit ball
B
n
,
as well as its consequences on Cauchy projection and certain
gradient estimates for the functions from the Hardy and Bergman spaces.We obtain
the exact values of the Bloch’s seminorms and norms for the Cauchy projection on
L
∞
(
S
n
)
.
These results are based on the papers [56] and [58].
The last chapter contains the proof of the one part of Hollenbeck-Verbitsky
conjecture from [
26
]. Exactly, we find the exact norms of
(
|
P
+
|
s
+
|
P
−
|
s
)
1
s
for
0
< s
≤
2
on
L
p
(
T
)
,
where
P
+
is the Riesz projection and
P
−
=
I
−
P
+
.
Also we
give the appropriate dual estimates and prove that they are sharp. The paper [
45
]
is motivated by the results from [25] and [33]. |