Order estimates of derivatives of the multidimensional periodic Dirichlet $\alpha$-kernel in a mixed norm

È. M. Galeev

Sbornik. Mathematics, Tome 45 (1983) no. 1, pp. 31-43

Résumé

In this paper the author establishes a sharp order estimate, in the mixed norm of $L_p(\mathbf T^n)$ for $1$ and in $L_\infty(\mathbf T^n)$ ($\mathbf T^n =[-\pi,\pi]^n$ is the $n$-dimensional torus), of the derivatives of order $\beta \in \mathbf R^n$ of the multidimensional Dirichlet $\alpha$-kernel $D_{\alpha,\mu}$ and the function $F_{\alpha,\mu}$, $\alpha>0$, $\mu>0$, which are sums of exponentials $e^{i(k,t)}$ lying respectively inside and outside a “graduated hyperbolic cross”, i.e., the set $\{k\in\square_s\mid(\alpha,s)\leqslant \mu\}$, where $\square_s=\{k\in\mathbf Z^n\mid2^{s_{j-1}} \leqslant|k_j|2^{s_j},\, j=1,\ldots,n\}$, $s>0$. Bibliography: 11 titles.