In this paper, we give some results concerning Bernstein--Nikol'skii
inequality for weighted Lebesgue spaces. The main result is as follows: Let $1 < u,p < \infty$, $0<q+ 1/p <v + 1/u <1,$
$v-q\geq 0$, $\kappa >0$, $f \in L^u_v(\R)$ and $\supp\widehat{f} \subset [-\kappa, \kappa]$. Then $D^mf \in L^p_q(\R)$, $\supp\widehat{D^m f}=\supp\widehat{f}$ 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 $\|f\|_{L^p_q} = \big(\int_{\R} |f(x)|^p |x|^{pq} dx\big)^{1/p}
< \infty.$ Moreover, $ \lim_{m\to \infty}\|D^mf\|_{L^p_{q}}^{1/m}=
\sup \big\{ |x|: \, x \in \textnormal{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.
Linear bound for the dyadic paraproduct on weighted Lebesgue space L2(w)
