Nikolskii constants in $L^{p}(\mathbb{R},|x|^{2\alpha+1}\,dx)$ spaces

Abstract

Recently Arestov, Babenko, Deikalova, and Horv\'ath have established a series
of interesting results correspondent to the sharp Nikolskii constant
$\mathcal{L}_\textup{even}(\alpha,p)$ in the weighted inequality
\[
\sup_{x\in [0,\infty)}|f(x)|\le
\mathcal{L}_\textup{even}(\alpha,p)\sigma^{(2\alpha+2)/p}
\biggl(2\int_{0}^{\infty}|f(x)|^{p}x^{2\alpha+1}\,dx\biggr)^{1/p}
\]
for the subspace $\mathcal{E}^{\sigma}\cap
L^{p}(\mathbb{R}_{+},x^{2\alpha+1}\,dx)$ of even entire functions $f$ of
exponential type at most $\sigma>0$, where $1\le p<\infty$ and $\alpha\ge -1/2$.

We prove that, for the same $\alpha$ and $p$
\[
\mathcal{L}_\textup{even}(\alpha,p)=\mathcal{L}(\alpha,p),
\]
where $\mathcal{L}(\alpha,p)$ is the sharp constant in the Nikolskii inequality
\[
\sup_{x\in \mathbb{R}}|f(x)|\le \mathcal{L}(\alpha,p)\sigma^{(2\alpha+2)/p}
\biggl(\int_{\mathbb{R}}|f(x)|^{p}|x|^{2\alpha+1}\,dx\biggr)^{1/p}
\]
for any (not necessary even) functions $f\in
\mathcal{E}_{p,\alpha}^{\sigma}:=\mathcal{E}^{\sigma}\cap
L^{p}(\mathbb{R},|x|^{2\alpha+1}\,dx)$.

Also we give bounds of the normalized Nikolskii constant
\[
\mathcal{L}^{*}(\alpha,p):=
(2^{2\alpha+2}\Gamma(\alpha+1)\Gamma(\alpha+2))^{1/p}\mathcal{L}(\alpha,p),
\]
which are as follows:
\[
\mathcal{L}^{*}(\alpha,p)\le \lceil p/2\rceil^{\frac{2\alpha+2}{p}},\quad p\in
(0,\infty),
\]
and for fixed $p\in [1,\infty)$
\[
\mathcal{L}^{*}(\alpha,p)\ge (p/2)^{\frac{2\alpha+2}{p}\,(1+o(1))},\quad
\alpha\to \infty.
\]
The upper estimate is sharp if and only if $p=2$. In this case,
$\mathcal{L}^{*}(\alpha,2)=1$ for each $\alpha\ge -1/2$.

Our approach relies on the one-dimensional Dunkl harmonic analysis. To prove
the identity $\mathcal{L}_\textup{even}(\alpha,p)=\mathcal{L}(\alpha,p)$ we use
the even positive Dunkl-type generalized translation operator $T^{t}$ such that
is bounded on $L^{p}(\mathbb{R},|t|^{2\alpha+1}\,dt)$ with constant one and
invariant on the subspace $\mathcal{E}_{p,\alpha}^{\sigma}$.

The proof of the upper estimate of the constant $\mathcal{L}^{*}(\alpha,p)$ is
based on estimation of norms of the reproducing kernel for the subspace
$\mathcal{E}_{p,\alpha}^{1}$ and the multiplicative inequality for the
Nikolskii constant. To obtain the lower estimate we consider the normalized
Bessel function $j_{\nu}\in \mathcal{E}_{p,\alpha}^{1}$ of order $\nu\sim
(2\alpha+2)/p$.