A Bernstein–Nikol'skii inequality for weighted Lebesgue spaces

In this paper, we give some results concerning Bernstein–Nikol'skii inequality for weighted Lebesgue spaces. The main result is as follows: Let $1 < u,p < \infty$, $0<q+ 1/p <v + 1/u <1,$ $v-q\geq 0$, $\kappa >0$, $f \in L^u_v(\mathbb{R})$ and $\mathrm{supp}\,\widehat{f} \subset [-\kappa, \kappa]$. Then $D^mf \in L^p_q(\mathbb{R})$, $\mathrm{supp}\,\widehat{D^m f}=\mathrm{supp}\,\widehat{f}$ and there exists a constant $C$ independent of $f$, $m$, $\kappa$ such that $\|D^mf\|_{L^p_{q}} \leq C m^{-\varrho} \kappa^{m+\varrho} \|f\|_{ L^u_v}, $ for all $m = 1,2,\dots $, where $\varrho=v + \frac{1}{u} -\frac{1}{p} - q>0,$ and the weighted Lebesgue space $L^p_q$ consists of all measurable functions such that $\|f\|_{L^p_q} = \big(\int_{\mathbb{R}} |f(x)|^p |x|^{pq} dx\big)^{1/p} < \infty.$ Moreover, $ \lim_{m\to \infty}\|D^mf\|_{L^p_{q}}^{1/m}= \sup \big\{ |x|: x \in \mathrm{supp}\,\widehat{f}\big \}.$ The advantage of our result is that $m^{-\varrho}$ appears on the right hand side of the inequality ($\varrho >0$), which has never appeared in related articles by other authors. The corresponding result for the $n$-dimensional case is also obtained.