Convergence exponent of singular integral in generalized Hilbert--Kamke problem

A. Zrein

Fundamentalʹnaâ i prikladnaâ matematika, Tome 1 (1995) no. 4, pp. 939-951

Résumé

In this article we find exact value of the convergence exponent of singular integral in the problem of simultaneous representation of increasing set of natural numbers $N_1,\ldots,N_r$ by sum of terms $[x^{n_1+\theta}],[x^{n_2+\theta}],\ldots,[x^{n_r+\theta}]$ ($n_1$ — natural numbers, $0\leq\theta\leq1$). We consider integral: $$ \theta_0=\int\limits_{\mathbb R^r}|I(\alpha_1,\ldots,\alpha_r)|^k\,d\alpha_1\ldots d\alpha_r, $$ where $k$ is an unrestricted index and $$ I(\alpha_1,\ldots,\alpha _r)=\int\limits_{0}^{1}\exp\biggl\{2\pi i\sum_{j=1}^{r}\alpha_jx^{n_j+\theta}\biggr\}\,dx. $$ It is proved that $\theta_0$ converges when $k>k_0$ and diverges when $k\leq k_0$ where $$ k_0=\max \left\{n_1+\cdots+n_r+r\theta,\frac{r(r+1)}{2}+1\right\}. $$