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

D. V. Gorbachev ; N. N. Dobrovolsky

Čebyševskij sbornik, Tome 19 (2018) no. 2, pp. 67-79.

Résumé

Recently Arestov, Babenko, Deikalova, and Horváth have established a series of interesting results correspondent to the sharp Nikolskii constant $\mathcal{L}_{\mathrm{even}}(\alpha,p)$ in the weighted inequality $$ \sup_{x\in [0,\infty)}|f(x)|\le \mathcal{L}_{\mathrm{even}}(\alpha,p)\sigma^{(2\alpha+2)/p} \biggl(2\int_{0}^{\infty}|f(x)|^{p}x^{2\alpha+1}\,dx\biggr)^{1/p} $$ 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 $p$ $$ \mathcal{L}_{\mathrm{even}}(\alpha,p)=\mathcal{L}(\alpha,p), $$ 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 $\mathcal{L}_{\mathrm{even}}(\alpha,p)=\mathcal{L}(\alpha,p)$ 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 $\nu\sim (2\alpha+2)/p$.

Détail

Keywords: weighted Nikolskii inequality, sharp constant, entire function of exponential type, Dunkl transform, generalized translation operator, reproducing kernel, Bessel function.

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