On contractive inequalities in the Dirichlet range

For $0<\alpha, p<\infty$ we introduce the classes $A_\alpha^p$ consisting of analytic functions in the unit disk $\mathbb{D}$ for which
$$\|f\|_{\alpha,p}^p=|f(0)|^p+\int_0^1\left(\frac{d}{dt}M_p^p(r,f)\right)(1-r^2)^{\alpha-1}<\infty,$$
where $M^p(r,f)$ denotes the radial $L^p$-means. For $\alpha>1$ the class $A_\alpha^p$ coincides with the standard Bergman space and for $\alpha=1$ we recover the Hardy space $H^1$.
We investigate the following problem: for which parameters $\alpha, \beta, p, q$ satisfying
\begin{equation} \label{eq:1} 0<\alpha<\beta<\infty,\,\, 0<p<q<\infty,\,\, \textrm{and}\,\, \frac{\alpha}{p}=\frac{\beta}{q} \end{equation}
the inequality
\begin{equation}\label{eq:2} \|f\|_{\beta,q}\leq \|f\|_{\alpha,p} \end{equation}
holds for every function $f$ in $A_\alpha^p$?
The assumptions \ref{eq:1} imposed on the parameters are motivated, in particular, by the conformal invariance of classes $A_\alpha^p$ with index $\alpha/p$: if $f\in A_\alpha^p$ and $w\in\mathbb{D}$, then the function
$$T_{w,\alpha/p}f(z)=f\left(\frac{w-z}{1-\bar{w}z}\right)\frac{(1-|w|^2)^{\alpha/p}}{(1-\bar{w}z)^{2\alpha/p}}$$
also belongs to $A_\alpha^p$ and $\|f\|_{\alpha,p}=\|T_{w,\alpha/p}f\|_{\alpha,p}$.
Kulikov [2] established inequality \ref{eq:2} for all $0<p<q<\infty$ and $\alpha\geq1$.Later Llinares in [3] did the same for $\beta=1,\,p=2$ and all $\alpha\in(0,1)$. Our main result is that the inequality \ref{eq:2} holds for all parameters $\alpha, \beta, p, q$ that satisfy \ref{eq:1}. The proof is based on the result from [2], an analytic continuation trick, and the conformal invariance of the classes $A_\alpha^p.$
We also explore the relation between the classes Apα and the classical Besov spaces.
The talk is based on a joint work with Ole Brevig, Aleksei Kulikov and Kristian Seip.

Список литературы

1. Bonk M., Extremalprobleme bei Bloch-Funktionen, Ph.D. thesis, TU Braunschweig, 1988
2. Kayumov I. R., Wirths K.-J., “On the sum of squares of the coefficients of Bloch functions”, Monat. Math., 190 (2019), 123–135
3. Ruscheweyh S., “Two remarks on bounded analytic functions”, Bulg. Math. Publ., 11 (1985), 200–202

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