About integral of Weber--Shafheitlin

I. S. Belov

Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 10 (2003), pp. 481-489.

Résumé

Let $L_{\lambda }^p$ be the function space at half-line with the norm $\Vert f{\Vert }_{p,\lambda }^p={\int }_0^{\mathrm{\infty }}|f(x){|}^p{x}^{-\lambda }dx$. In the work the operators $A_{\mu }$ of multiplicative convolution with Bessel function $A_{\mu }f(x)={\int }_0^{\mathrm{\infty }}{J}_{\mu }(xt)f(t)t^{-\lambda }dt$ are considered and their following propeties are proved. The operators $A_{\mu }$, $\mu \ge 0$, are bounded on $L^2(\lambda )$, $-1\le \lambda \le 1$. $A_{\mu }$, $\mu >0$, are bounded on $L_{\lambda }^p$, $1\le p\le \mathrm{\infty }$, but $A_0$ is unbounded on $L_1^p$, $1\le p\le \mathrm{\infty }$. The operators $A_{\mu }$ are unbounded on $L_{\lambda }^p$ $p\ne 2$, $1\le \lambda 1$. With some relations between values $(\mu ,\nu ,\lambda ,p)$ the products $A_{\nu }{A}_{\mu }$ are bounded on $L_{\lambda }^p$.