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DOI: 10.46698/h8083-6917-3687-w

A Bernstein-Nikol'skii Inequality for Weighted Lebesgue Spaces

Bang, H. H. ,  Huy, V. N.
Vladikavkaz Mathematical Journal 2020. Vol. 22. Issue 3.
Abstract:
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.
Keywords: weighted Lebesgue spaces, Bernstein inequality, Nikol'skii inequality
Language: English Download the full text  
For citation: Bang, H. H. and Huy, V. N. A Bernstein-Nikol'skii Inequality for Weighted Lebesgue Spaces, Vladikavkaz Math. J., 2020, vol. 22, no. 3, pp. 18-29. DOI 10.46698/h8083-6917-3687-w
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