Bernstein-type inequalities for rational functions in weighted Bergman spaces and applications

Rachid Zarouf (1)

1 LATP - Laboratoire d'Analyse, Topologie, Probabilités

{\small Generally speaking, given two Banach spaces $X$ and $Y$ of holomorphic functions on the unit disc $\mathbb{D},$ we are searching for the {}``best possible'' constants $\mathcal{C}_{n,\, r}(X,\, Y)$ and $\mathcal{B}_{n,\, r}(X,\, Y)$ such that \[ \left\Vert f^{'}\right\Vert _{X}\leq\mathcal{C}_{n,\, r}(X,\, Y)\left\Vert f\right\Vert _{Y}\;\mbox{and}\;\left\Vert f\right\Vert _{X}\leq\mathcal{B}_{n,\, r}(X,\, Y)\left\Vert f\right\Vert _{Y},\] for all rational functions $f$ in $\mathbb{D}$ having at most $n$ poles all outside of $\frac{1}{r}\mathbb{D}$, $r\in[0,\,1)$. For the special case $X=Y,$ we set $\mathcal{C}_{n,\, r}(X):=\mathcal{C}_{n,\, r}(X,\, X)$.}{\small \par} \vspace{0.02cm} {\small Throughout this paper the letter $c$ denotes a positive constant that may change from one step to the next. For two positive functions $a$ and $b$, we say that $a$ is dominated by $b$, denoted by $a=O(b),$ if there is a constant $c>0$ such that $a\leq cb;$ and we say that $a$ and $b$ are equivalent, denoted by $a\asymp b$, if both $a=O(b)$ and $b=O(a)$ hold.}{\small \par} \vspace{0.02cm} {\small We show that $\mathcal{C}_{n,\, r}\left(L_{a}^{2},\, H^{2}\right)\asymp\mathcal{B}_{n,\, r}\left(B_{2,\,2}^{\frac{1}{2}},\, H^{2}\right)\asymp\sqrt{\frac{n}{1-r}}\;$, for all $n\geq1$ and $r\in[0,\,1)$, where $H^{2}$ is the classical Hardy space, $L_{a}^{2}$ is the classical Bergman space and $B_{2,\,2}^{\frac{1}{2}}$ is a Besov space also known as the Dirichlet space. Moreover, there exists a limit : $\mbox{lim}{}_{n\rightarrow\infty}\frac{\mathcal{B}_{n,\, r}\left(B_{2,\,2}^{\frac{1}{2}},\, H^{2}\right)}{\sqrt{n}}=\mbox{lim}{}_{n\rightarrow\infty}\frac{\mathcal{C}_{n,\, r}\left(L_{a}^{2},\, H^{2}\right)}{\sqrt{n}}=\sqrt{\frac{1+r}{1-r}}.$ }{\small \par} \vspace{0.02cm} {\small Then, we apply our estimate of $\mathcal{B}_{n,\, r}\left(B_{2,\,2}^{\frac{1}{2}},\, H^{2}\right)$ to an effective Nevanlinna-Pick interpolation problem in the Dirichlet space. }{\small \par} {\small Finally, if $X=Y$ and if $X$ is a radial-weighted Bergman space then we show that $\mathcal{C}_{n,\, r}\left(X\right)\asymp\frac{n}{1-r}\;$ for all $n\geq1$ and $r\in[0,\,1)$.}{\small \par}

Domaines: Analyse fonctionnelle [math.FA]

Rachid Zarouf. Bernstein-type inequalities for rational functions in weighted Bergman spaces and applications. 2010. ⟨hal-00463664v2⟩