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Zusammenfassung:
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The study of integral means of the composition of functions defined
on the unit disk D in the complex plane dates back to the 1920s, with one of the
earliest results in this area being Littlewood’s subordination principle.
When investigating the norm of composition operators on certain spaces of
holomorphic functions, a natural need arises to study the relationship between
the integral means of the composition f ◦ φ and those of the function f itself.
Littlewood’s principle is one of the main tools used to establish this connection.
However, it is not the only one. In this dissertation, additional methods for
studying the relationship between these integral means are presented. By applying
these methods, two-sided estimates for the norm of the composition operator Cφ on
spaces of mixed norm Hp,q,α are obtained in the form K1 ≤ ∥ Cφ ∥Hp,q,α→Hp,q,α ≤ K2,
where the constants K1 and K2 depend on the parameters p, q, α and |φ(0)|.
Furthermore, the monotonicity of the integral mean of a holomorphic function
f on the unit disk D, denoted by Mp,q,α[f ](ρ, R, s) , is investigated, where 0 <
p, q, α < ∞, 0 ≤ ρ < R ≤ 1 and 0 ≤ s ≤ 1. One consequence of this result is
the monotonicity of the norm ∥f ∥p,q,α in mixed norm spaces with respect to the
parameters p, q, α.
One of the operators that can be represented as an integral of weighted
composition operators Tt is the Hilbert matrix operator H acting on the weighted
Bergman spaces Ap
γ . Moreover, it is known that the operator H is bounded if and
only if 1 < γ + 2 < p, and in this case, the following lower bound for the norm
of the operator holds: ∥H∥Ap
γ →Ap
γ ≥ π/ sin (γ+2)π
p . When γ > 0 and p ≥ 2(γ + 2),
it is known that the norm is equal to this constant. In studying the norm of the
operator H, after applying Minkowski’s theorem, the application of Minkowski’s
inequality reduces the problem to estimating the norm of the operator Tt. As a
result of this analysis, in the case where γ < 0 a new upper bound for the norm of
the operator H is obtained, while in the case where γ > 0, the interval on which
the norm equals the constant π/ sin (γ+2)π
p is extended.
Finally, the dissertation presents a refinement of Littlewood’s subordination
principle under an additional injectivity assumption, together with applications
of the new inequality to the Rogosinski theorem and to norm estimates for
compositions of functions on weighted Bergman spaces. |