Speaker
Description
For $r\in(0,1)$ let $f_r(z) = f(rz)$. A well-known Weissler inequality states that for the Hardy spaces $H^p$ and $H^q$, $p\leq q$,
$\|f_r\|_{H^q}\leq \|f\|_{H^p}\qquad \Longleftrightarrow \qquad r\leq \sqrt{\frac pq}.$
We consider Bergman spaces $A^p(w)$ with the weight $w(r)\geq0$, consisting of functions $f$ analytic in the unit
disk $\mathbb{D}$ and such that
$\|f\|_{A^p(w)} :=\left( \frac1{2\pi}\int\limits_{\mathbb{D}} |f(z)|^p w(|z|) \, dz\right)^{1/p} <\infty.$
Obtaining a counterpart of the Weissler inequality for Bergman spaces is a problem that is currently being actively researched. For classical Bergman weights $w_{\alpha}(r) = 2(\alpha-1)(1-r^2)^{\alpha-2}$, $\alpha>1$, a significant progress was achieved by K.Seip, F. Bayart, O.F. Brevig, J. Ortega-Cerdà, K.-M. Perfekt and, recently, A. Kulikov, and P. Melentijević.
In our work, we considered a counterpart of the Weissler inequality for Bergman spaces $A^{2n}(w)$ and $A^{2}(w)$ with general radial weights. We found the conditions on the weight $w$ in terms of its moments ensuring that the counterpart is true for $n\in\mathbb{N}$ and $0 < r$ $\leq \frac{1}{\sqrt{n}} $.
We also examined the case of noninteger exponents for the particular function $f(z) = e^z$. Then the corresponding inequality can be considered as a certain analog of the Bernoulli inequality. An example of a monotonic weight was constructed for which these inequalities are no longer true.
The talk is based on a joint work with A.D. Baranov, I.R. Kayumov and R.Sh. Khasyanov.
