Inequalities for derivatives with the Fourier transformстатья
Статья опубликована в высокорейтинговом журнале
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Дата последнего поиска статьи во внешних источниках: 4 июня 2021 г.
Аннотация:In this paper we study sharp constants in inequalities of the following form$$\|x^{(k)}\cd\|_{L_q(\mathbb R)}\le K\|Fx\cd\|_{L_p(\mathbb R)}^\alpha\|x^{(n)}\cd\|_{L_r(\mathbb R)}^\beta,$$where $Fx\cd$ is the Fourier transform of $x\cd$. The sharp value of $K$ in the general case (that is, for all $n\in\mathbb N$ and $0\le k<n$) was known only for $q=r=2$ and $p\ge2$. We obtain the sharp constant in the general case for $q=\infty$, $r=2$, and $1\le p\le\infty$. We also generalized this two cases on multidimensional situation. The sharp constants is obtained for fractional degrees of the Laplace operator $(-\Delta)^{k/2}$ and derivatives $D^\alpha$ of order $\alpha=(\alpha_1,\ldots,\alpha_d)\in\mathbb R^d_+$.