Weighted norm inequalities for derivatives and their applications
Let \(-\infty<a<b<\infty\), \(1<p<\infty\), and let \(\varrho\) be a positive and continuous weight function on \([a,b]\). Let \(L_p(\varrho)\) be the weighted Lebesgue space of complex valued functions \(f\) with the finite norm \(\|f\|_{L_p(\varrho)}=(\int_a^b|f(x)|^p\varrho(x)dx)^{1/p}\) and for \(n\in \mathbb N\) let \(W_p^n(\varrho)\) be the weighted space of all complex valued functions \(f\) on \([a,b]\) which have continuous derivatives \(D^kf\) up to order \(n-2\), absolutely continuous derivative \(D^{n-1}f\), and the derivatives \(D^kf\), \(k\leq1\), vanish in \(a\) and \(D^nf\in L_p(\varrho)\). The space \(W_p^n(\varrho)\) endowed with the norm \(\|f\|_{W_p^n(\varrho)}=\|D^nf\|_{L_p(\varrho)}\) is a Banach space.NEWLINENEWLINENEWLINEGiven a positive integer \(m\), a set of weights \(\varrho_j\), \(j=1,\dots,m\), and a set of functions \(G_j\in C^n\), \(j=1,\dots,m\), satisfying a series of technical conditions, the authors prove the inequality \(\|\prod_{j=1}^m G_j\circ f_j\|_{W_p^n(\tau_n\gamma)}^p\leq\prod_{j=1}^m G_j(\|f_j\|_{W_p^n(\tau_n\varrho_j)}^p)\), where \(\tau_n(x)=(b-x)^{n-1}/(n-1)!\) and \(\gamma\) is a weight function constructed from the weights \(\varrho_j\) and the functions \(G_j\). Another theorem establishes the inequality \(\|\prod_{j=1}^m \ast f_j\|_{W_p^n(\omega)}\leq\prod_{j=1}^m\|f_j\|_{W_p^{n_j}(\varrho_j)}\), where \(\prod_{j=1}^m \ast f_j\) denotes the Laplace convolution product defined as \([\prod_{j=1}^1\ast f_j](x)=f_1(x)\), \([\prod_{j=1}^2\ast f_j](x)=[f_1\ast f_2](x)=\int_0 f_1(t)f_2(x-t)dt\), \([\prod_{j=1}^m\ast f_j]=[\prod_{j=1}^{m-1}\ast f_j]\ast f_m\), and \(\omega\) is constructed from the weights \(\varrho_j\).NEWLINENEWLINEExamples show how the results can be used for estimating the Riemann-Liouville fractional derivatives and solutions to the Volterra integral equations of the first and second kind. The paper follows the papers [\textit{Dinh Thanh Duc} and \textit{Nguyen Du Vi Nhan}, Kodai Math. J. 34, No. 2, 191--201 (2011; Zbl 1229.26028); \textit{Nguyen Du Vi Nhan}, \textit{Dinh Thanh Duc} and \textit{Vu Kim Tuan}, Comput. Math. Appl. 61, No. 4, 832--839 (2011; Zbl 1217.44001); \textit{Akira Yamada}, Math. Inequal. Appl. 14, No. 3, 523--528 (2011; Zbl 1222.26024)].
scientific article; zbMATH DE number 6203618