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Abstract:
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In this work we investigate the various norms of known integral operators such
as Bergman projection and certain integral operators induced with Green function.
We have observed acting of the Bergman projection on the weighted Lebesgue spaces
with image in spaces of analytic functions, Bloch and Besov spaces. We obtained
the estimate for the norm of the Bergman projection onto the Besov spaces, when
1 < p < ∞, and this estimate is asimptoticaly sharp for p → +∞. We have found
norm for the special Hilbert case p = 2. We have also considered some weak type
inequalities for the norm of the Bergman projection onto Besov space B1 .We ob-
served the adjoint operator for the Bergman projection fot the mentioned case B1.
Precisely, in this case we obtained two-sided bounds of norm for the adjoint oper-
ator. It was represented a method for determinig the norm of the Bergman type
projection defined in weighted Lebesgue spaces and with image in Bloch space in
the unit ball B in Cn.
Specially, we have considered the norm of the integral operator induced with Green
function in a context of the weak solutions for the Poisson’s equation with ho-
mogenuous condition in the unit ball in higher dimensions. The norm for the men-
tioned operator was determined for the Hilbert case and when operator is defined on
L∞−spaces (and also L1−case, which is dual).Final results are related to estimating
the norm of gradient for the weak solutions of Poisson’s equation. |